idx int64 1 56k | question stringlengths 15 155 | answer stringlengths 2 29.2k ⌀ | question_cut stringlengths 15 100 | answer_cut stringlengths 2 200 ⌀ | conversation stringlengths 47 29.3k | conversation_cut stringlengths 47 301 |
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36,201 | How to model the probability of a truth claim given an arrangement of eyewitness accounts supporting specific instances of that claim? | In my opinion, the most important thing to consider is if there are ways in which eyewitness accounts may not capture the full possibilities of the event (or lack of it), i.e., if they are statistically biased in any way, and if so, how.
Of course, "eyewitness" can be replaced with any non-human objective measurement o... | How to model the probability of a truth claim given an arrangement of eyewitness accounts supporting | In my opinion, the most important thing to consider is if there are ways in which eyewitness accounts may not capture the full possibilities of the event (or lack of it), i.e., if they are statistical | How to model the probability of a truth claim given an arrangement of eyewitness accounts supporting specific instances of that claim?
In my opinion, the most important thing to consider is if there are ways in which eyewitness accounts may not capture the full possibilities of the event (or lack of it), i.e., if they ... | How to model the probability of a truth claim given an arrangement of eyewitness accounts supporting
In my opinion, the most important thing to consider is if there are ways in which eyewitness accounts may not capture the full possibilities of the event (or lack of it), i.e., if they are statistical |
36,202 | PCA as a Cure for the Curse of Dimensionality | In a way, PCA does not use the outcome you are trying to model/predict, i.e. it is an unsupervised technique. From that perspective, its parameters are not parameters that get trained in your supervised model. Of course, using PCA for dimensionality reduction is not in any way guaranteed to preserve "the signal" for th... | PCA as a Cure for the Curse of Dimensionality | In a way, PCA does not use the outcome you are trying to model/predict, i.e. it is an unsupervised technique. From that perspective, its parameters are not parameters that get trained in your supervis | PCA as a Cure for the Curse of Dimensionality
In a way, PCA does not use the outcome you are trying to model/predict, i.e. it is an unsupervised technique. From that perspective, its parameters are not parameters that get trained in your supervised model. Of course, using PCA for dimensionality reduction is not in any ... | PCA as a Cure for the Curse of Dimensionality
In a way, PCA does not use the outcome you are trying to model/predict, i.e. it is an unsupervised technique. From that perspective, its parameters are not parameters that get trained in your supervis |
36,203 | PCA as a Cure for the Curse of Dimensionality | But then I have computed 30x30 elements for my eigenvector matrix and 3 parameters for my model, I have fitted 900+3 parameters to the data.
The possible solutions for the parameters relating to the features are strongly limited. You are effectively only fitting 3 parameters. Because the potential solutions $\hat{Y} =... | PCA as a Cure for the Curse of Dimensionality | But then I have computed 30x30 elements for my eigenvector matrix and 3 parameters for my model, I have fitted 900+3 parameters to the data.
The possible solutions for the parameters relating to the | PCA as a Cure for the Curse of Dimensionality
But then I have computed 30x30 elements for my eigenvector matrix and 3 parameters for my model, I have fitted 900+3 parameters to the data.
The possible solutions for the parameters relating to the features are strongly limited. You are effectively only fitting 3 paramete... | PCA as a Cure for the Curse of Dimensionality
But then I have computed 30x30 elements for my eigenvector matrix and 3 parameters for my model, I have fitted 900+3 parameters to the data.
The possible solutions for the parameters relating to the |
36,204 | Sampling Normal variables with linear constraints and given variances - Fraser (1951) | Suppose $(Y_1, \cdots, Y_{n-1}) \sim N_{n-1}\left(\boldsymbol{0}_{n-1}, \boldsymbol{T}\right)$. For simplicity, let $v_i=1$ for all $i$ and let $b_i^{\ast}$ denote the $i$th column of $\boldsymbol{B}$ excluding observation $b_{ni}$. That reference states that the elements of $\boldsymbol{T} = (\tau_{rs})$ are found b... | Sampling Normal variables with linear constraints and given variances - Fraser (1951) | Suppose $(Y_1, \cdots, Y_{n-1}) \sim N_{n-1}\left(\boldsymbol{0}_{n-1}, \boldsymbol{T}\right)$. For simplicity, let $v_i=1$ for all $i$ and let $b_i^{\ast}$ denote the $i$th column of $\boldsymbol{B} | Sampling Normal variables with linear constraints and given variances - Fraser (1951)
Suppose $(Y_1, \cdots, Y_{n-1}) \sim N_{n-1}\left(\boldsymbol{0}_{n-1}, \boldsymbol{T}\right)$. For simplicity, let $v_i=1$ for all $i$ and let $b_i^{\ast}$ denote the $i$th column of $\boldsymbol{B}$ excluding observation $b_{ni}$. ... | Sampling Normal variables with linear constraints and given variances - Fraser (1951)
Suppose $(Y_1, \cdots, Y_{n-1}) \sim N_{n-1}\left(\boldsymbol{0}_{n-1}, \boldsymbol{T}\right)$. For simplicity, let $v_i=1$ for all $i$ and let $b_i^{\ast}$ denote the $i$th column of $\boldsymbol{B} |
36,205 | Is there a real example in which a correlation finally leads to the discovery of a non-trivial causal relationship? | Lung cancer was not even recognised medically until the 18th century, and as recently as 1900 only about 140 cases were known in the published medical literature. ... Tobacco was apparently not even suspected as a cause of lung tumours until the final decade of the 19th century. ... Scholars started noting the parallel... | Is there a real example in which a correlation finally leads to the discovery of a non-trivial causa | Lung cancer was not even recognised medically until the 18th century, and as recently as 1900 only about 140 cases were known in the published medical literature. ... Tobacco was apparently not even s | Is there a real example in which a correlation finally leads to the discovery of a non-trivial causal relationship?
Lung cancer was not even recognised medically until the 18th century, and as recently as 1900 only about 140 cases were known in the published medical literature. ... Tobacco was apparently not even suspe... | Is there a real example in which a correlation finally leads to the discovery of a non-trivial causa
Lung cancer was not even recognised medically until the 18th century, and as recently as 1900 only about 140 cases were known in the published medical literature. ... Tobacco was apparently not even s |
36,206 | Prove that $t_{n-1, \alpha/2}$ is strictly decreasing in $n$ | This looks like a good opportunity to discuss important relationships among the Student $t$ distributions. The analysis needed to demonstrate them is elementary, requiring only the basic concepts of differential Calculus, and with the right strategy can be reduced to a simple algebraic calculation.
There is a classic... | Prove that $t_{n-1, \alpha/2}$ is strictly decreasing in $n$ | This looks like a good opportunity to discuss important relationships among the Student $t$ distributions. The analysis needed to demonstrate them is elementary, requiring only the basic concepts of | Prove that $t_{n-1, \alpha/2}$ is strictly decreasing in $n$
This looks like a good opportunity to discuss important relationships among the Student $t$ distributions. The analysis needed to demonstrate them is elementary, requiring only the basic concepts of differential Calculus, and with the right strategy can be r... | Prove that $t_{n-1, \alpha/2}$ is strictly decreasing in $n$
This looks like a good opportunity to discuss important relationships among the Student $t$ distributions. The analysis needed to demonstrate them is elementary, requiring only the basic concepts of |
36,207 | A challenging question of ANN | As long as we are talking only about additive neurons (i.e. all inputs to the neuron are summed together before being passed to the activation function), "unipolar" and "bipolar" can be used interchangeably. We can always transform a "unipolar" output to a "bipolar" one by multiplying by 2 and subtracting one:
$$
o_{bi... | A challenging question of ANN | As long as we are talking only about additive neurons (i.e. all inputs to the neuron are summed together before being passed to the activation function), "unipolar" and "bipolar" can be used interchan | A challenging question of ANN
As long as we are talking only about additive neurons (i.e. all inputs to the neuron are summed together before being passed to the activation function), "unipolar" and "bipolar" can be used interchangeably. We can always transform a "unipolar" output to a "bipolar" one by multiplying by 2... | A challenging question of ANN
As long as we are talking only about additive neurons (i.e. all inputs to the neuron are summed together before being passed to the activation function), "unipolar" and "bipolar" can be used interchan |
36,208 | A challenging question of ANN | If neuron had three outputs, say [-1,0,1] then it could draw three areas with linear boundaries as shown here for the first layer and solution would be (E).
The second layer simply picks the south and north region as one category, and west and east regions as another.
A neuron with two outputs, whether it's [0,1] or [... | A challenging question of ANN | If neuron had three outputs, say [-1,0,1] then it could draw three areas with linear boundaries as shown here for the first layer and solution would be (E).
The second layer simply picks the south an | A challenging question of ANN
If neuron had three outputs, say [-1,0,1] then it could draw three areas with linear boundaries as shown here for the first layer and solution would be (E).
The second layer simply picks the south and north region as one category, and west and east regions as another.
A neuron with two ou... | A challenging question of ANN
If neuron had three outputs, say [-1,0,1] then it could draw three areas with linear boundaries as shown here for the first layer and solution would be (E).
The second layer simply picks the south an |
36,209 | Tail probability bounds on $P(|Z| > t)$ tend to be useless for small $t>0$. Why is that? | The first thing to note here is that the upper bounds on the probabilities are going above one in some cases, and obviously you can truncate these to make them one in that Chebychev bound and the Mills bound go above one is really just a matter of convention --- i.e., the expression does not bother to differentiate bet... | Tail probability bounds on $P(|Z| > t)$ tend to be useless for small $t>0$. Why is that? | The first thing to note here is that the upper bounds on the probabilities are going above one in some cases, and obviously you can truncate these to make them one in that Chebychev bound and the Mill | Tail probability bounds on $P(|Z| > t)$ tend to be useless for small $t>0$. Why is that?
The first thing to note here is that the upper bounds on the probabilities are going above one in some cases, and obviously you can truncate these to make them one in that Chebychev bound and the Mills bound go above one is really ... | Tail probability bounds on $P(|Z| > t)$ tend to be useless for small $t>0$. Why is that?
The first thing to note here is that the upper bounds on the probabilities are going above one in some cases, and obviously you can truncate these to make them one in that Chebychev bound and the Mill |
36,210 | Tail probability bounds on $P(|Z| > t)$ tend to be useless for small $t>0$. Why is that? | The distribution of $\bar X$ near the true mean will depend on the distribution of $X$ near the true mean, so you won't be able to bound it just by making assumptions about moments or other expectations. Also, there's more theoretical interest in tail bounds, so more work has gone into refining them.
Results for $\bar... | Tail probability bounds on $P(|Z| > t)$ tend to be useless for small $t>0$. Why is that? | The distribution of $\bar X$ near the true mean will depend on the distribution of $X$ near the true mean, so you won't be able to bound it just by making assumptions about moments or other expectatio | Tail probability bounds on $P(|Z| > t)$ tend to be useless for small $t>0$. Why is that?
The distribution of $\bar X$ near the true mean will depend on the distribution of $X$ near the true mean, so you won't be able to bound it just by making assumptions about moments or other expectations. Also, there's more theoret... | Tail probability bounds on $P(|Z| > t)$ tend to be useless for small $t>0$. Why is that?
The distribution of $\bar X$ near the true mean will depend on the distribution of $X$ near the true mean, so you won't be able to bound it just by making assumptions about moments or other expectatio |
36,211 | GLM interpretation | Since you have only 1 other gender, it would make a bit more sense to include this with males and have "female" and "not female", although it won't change the interpretation much.
Assuming that "Y" for glasses is coded as 1, and "N" is coded as 0, then there is clear evidence from these data that males are less likely ... | GLM interpretation | Since you have only 1 other gender, it would make a bit more sense to include this with males and have "female" and "not female", although it won't change the interpretation much.
Assuming that "Y" fo | GLM interpretation
Since you have only 1 other gender, it would make a bit more sense to include this with males and have "female" and "not female", although it won't change the interpretation much.
Assuming that "Y" for glasses is coded as 1, and "N" is coded as 0, then there is clear evidence from these data that mal... | GLM interpretation
Since you have only 1 other gender, it would make a bit more sense to include this with males and have "female" and "not female", although it won't change the interpretation much.
Assuming that "Y" fo |
36,212 | PCA and variable contributions to first n dimensions | We have a dedicated thread for that very specific purpose: Using principal component analysis (PCA) for feature selection.
Just a few points regarding the interpretation of those visual displays, and some reflexions on the question at hand:
This graphical output is a visual aid to see which variable contribute the mos... | PCA and variable contributions to first n dimensions | We have a dedicated thread for that very specific purpose: Using principal component analysis (PCA) for feature selection.
Just a few points regarding the interpretation of those visual displays, and | PCA and variable contributions to first n dimensions
We have a dedicated thread for that very specific purpose: Using principal component analysis (PCA) for feature selection.
Just a few points regarding the interpretation of those visual displays, and some reflexions on the question at hand:
This graphical output is ... | PCA and variable contributions to first n dimensions
We have a dedicated thread for that very specific purpose: Using principal component analysis (PCA) for feature selection.
Just a few points regarding the interpretation of those visual displays, and |
36,213 | Generating a min{p, 0.5} coin from a p-coin - Bernoulli factory type problem | Note: This answer is wrong/incomplete and will need an update according to the comments
Y̶o̶u̶ ̶c̶a̶n̶ ̶a̶p̶p̶r̶o̶a̶c̶h̶ ̶t̶h̶e̶ ̶f̶u̶n̶c̶t̶i̶o̶n̶ ̶b̶u̶t̶ ̶n̶o̶t̶ ̶g̶e̶t̶ ̶e̶q̶u̶a̶l̶
Suppose we are given a coin with arbitrary (unknown) head probability $p$, I am wondering if there is an easy-to-implement algorithm for... | Generating a min{p, 0.5} coin from a p-coin - Bernoulli factory type problem | Note: This answer is wrong/incomplete and will need an update according to the comments
Y̶o̶u̶ ̶c̶a̶n̶ ̶a̶p̶p̶r̶o̶a̶c̶h̶ ̶t̶h̶e̶ ̶f̶u̶n̶c̶t̶i̶o̶n̶ ̶b̶u̶t̶ ̶n̶o̶t̶ ̶g̶e̶t̶ ̶e̶q̶u̶a̶l̶
Suppose we are g | Generating a min{p, 0.5} coin from a p-coin - Bernoulli factory type problem
Note: This answer is wrong/incomplete and will need an update according to the comments
Y̶o̶u̶ ̶c̶a̶n̶ ̶a̶p̶p̶r̶o̶a̶c̶h̶ ̶t̶h̶e̶ ̶f̶u̶n̶c̶t̶i̶o̶n̶ ̶b̶u̶t̶ ̶n̶o̶t̶ ̶g̶e̶t̶ ̶e̶q̶u̶a̶l̶
Suppose we are given a coin with arbitrary (unknown) head p... | Generating a min{p, 0.5} coin from a p-coin - Bernoulli factory type problem
Note: This answer is wrong/incomplete and will need an update according to the comments
Y̶o̶u̶ ̶c̶a̶n̶ ̶a̶p̶p̶r̶o̶a̶c̶h̶ ̶t̶h̶e̶ ̶f̶u̶n̶c̶t̶i̶o̶n̶ ̶b̶u̶t̶ ̶n̶o̶t̶ ̶g̶e̶t̶ ̶e̶q̶u̶a̶l̶
Suppose we are g |
36,214 | Generating a min{p, 0.5} coin from a p-coin - Bernoulli factory type problem | Since the function min(λ, c) meets the Keane–O'Brien theorem, this implies that there are polynomials that converge from above and below to the function. This is discussed, for example, in Thomas and Blanchet (2012) and in Łatuszyński et al. (2009/2011). However, neither paper shows how to automate the task of finding ... | Generating a min{p, 0.5} coin from a p-coin - Bernoulli factory type problem | Since the function min(λ, c) meets the Keane–O'Brien theorem, this implies that there are polynomials that converge from above and below to the function. This is discussed, for example, in Thomas and | Generating a min{p, 0.5} coin from a p-coin - Bernoulli factory type problem
Since the function min(λ, c) meets the Keane–O'Brien theorem, this implies that there are polynomials that converge from above and below to the function. This is discussed, for example, in Thomas and Blanchet (2012) and in Łatuszyński et al. (... | Generating a min{p, 0.5} coin from a p-coin - Bernoulli factory type problem
Since the function min(λ, c) meets the Keane–O'Brien theorem, this implies that there are polynomials that converge from above and below to the function. This is discussed, for example, in Thomas and |
36,215 | Generating a min{p, 0.5} coin from a p-coin - Bernoulli factory type problem | I don't know how to generate that function, but there is a simple way to generate increasingly good approximations of it by consuming an increasingly large of inputs. Namely, you observe $N$ inputs, where $N$ is an odd integer. Then:
If the majority of those inputs is $0$ you "decide" that $p<1/2$. So you take an addt... | Generating a min{p, 0.5} coin from a p-coin - Bernoulli factory type problem | I don't know how to generate that function, but there is a simple way to generate increasingly good approximations of it by consuming an increasingly large of inputs. Namely, you observe $N$ inputs, w | Generating a min{p, 0.5} coin from a p-coin - Bernoulli factory type problem
I don't know how to generate that function, but there is a simple way to generate increasingly good approximations of it by consuming an increasingly large of inputs. Namely, you observe $N$ inputs, where $N$ is an odd integer. Then:
If the m... | Generating a min{p, 0.5} coin from a p-coin - Bernoulli factory type problem
I don't know how to generate that function, but there is a simple way to generate increasingly good approximations of it by consuming an increasingly large of inputs. Namely, you observe $N$ inputs, w |
36,216 | How to conduct a multilevel model/regression for panel data in Python? | Linear regression will not be suitable for a multilevel model.
A mixed effects model is a good way to fit most multilevel models.
In python you can use mixedlm in statsmodels. For example:
In [1]: import statsmodels.api as sm
In [2]: import statsmodels.formula.api as smf
In [3]: data = sm.datasets.get_rdataset("dieto... | How to conduct a multilevel model/regression for panel data in Python? | Linear regression will not be suitable for a multilevel model.
A mixed effects model is a good way to fit most multilevel models.
In python you can use mixedlm in statsmodels. For example:
In [1]: imp | How to conduct a multilevel model/regression for panel data in Python?
Linear regression will not be suitable for a multilevel model.
A mixed effects model is a good way to fit most multilevel models.
In python you can use mixedlm in statsmodels. For example:
In [1]: import statsmodels.api as sm
In [2]: import statsmo... | How to conduct a multilevel model/regression for panel data in Python?
Linear regression will not be suitable for a multilevel model.
A mixed effects model is a good way to fit most multilevel models.
In python you can use mixedlm in statsmodels. For example:
In [1]: imp |
36,217 | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neither or both? | Most U.S. health surveys (NHIS and its kiddo MEPS, NHANES, NSDUH) are stratified cluster surveys. The common representation of the public use data sets is a two-stage design with ~50 strata at the first stage of sampling (at which clusters are sampled), usually with two clusters per stratum, and people sampled at the s... | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neith | Most U.S. health surveys (NHIS and its kiddo MEPS, NHANES, NSDUH) are stratified cluster surveys. The common representation of the public use data sets is a two-stage design with ~50 strata at the fir | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neither or both?
Most U.S. health surveys (NHIS and its kiddo MEPS, NHANES, NSDUH) are stratified cluster surveys. The common representation of the public use data sets is a two-stage design with ~50 strata at the first stage... | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neith
Most U.S. health surveys (NHIS and its kiddo MEPS, NHANES, NSDUH) are stratified cluster surveys. The common representation of the public use data sets is a two-stage design with ~50 strata at the fir |
36,218 | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neither or both? | Stratified sampling is most efficient (in terms of variance of the estimate) when you have homogeneity WITHIN strata and heterogeneity BETWEEN strata. Think US states if your variable of interest were some social issue. Texans are very similar to each other but wildly different from New Yorkers (who are again similar t... | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neith | Stratified sampling is most efficient (in terms of variance of the estimate) when you have homogeneity WITHIN strata and heterogeneity BETWEEN strata. Think US states if your variable of interest were | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neither or both?
Stratified sampling is most efficient (in terms of variance of the estimate) when you have homogeneity WITHIN strata and heterogeneity BETWEEN strata. Think US states if your variable of interest were some so... | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neith
Stratified sampling is most efficient (in terms of variance of the estimate) when you have homogeneity WITHIN strata and heterogeneity BETWEEN strata. Think US states if your variable of interest were |
36,219 | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neither or both? | Here's how the terms are usually used in survey research.
Stratified sampling is when you take the entire sample frame and preemptively divide it into a number of "buckets" based on some criteria you already know. So if you are sampling people in the US and you already know their race you might divide the sample into w... | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neith | Here's how the terms are usually used in survey research.
Stratified sampling is when you take the entire sample frame and preemptively divide it into a number of "buckets" based on some criteria you | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neither or both?
Here's how the terms are usually used in survey research.
Stratified sampling is when you take the entire sample frame and preemptively divide it into a number of "buckets" based on some criteria you already ... | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neith
Here's how the terms are usually used in survey research.
Stratified sampling is when you take the entire sample frame and preemptively divide it into a number of "buckets" based on some criteria you |
36,220 | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neither or both? | As I understand it, Cluster sampling is best when the population is homogeneous, and the differences between the means of the clusters is small, and the variance within a cluster is large. The aim is to use the cluster as a proxy for the population as a whole. The benefit is practical. For example, it is easier to pick... | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neith | As I understand it, Cluster sampling is best when the population is homogeneous, and the differences between the means of the clusters is small, and the variance within a cluster is large. The aim is | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neither or both?
As I understand it, Cluster sampling is best when the population is homogeneous, and the differences between the means of the clusters is small, and the variance within a cluster is large. The aim is to use t... | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neith
As I understand it, Cluster sampling is best when the population is homogeneous, and the differences between the means of the clusters is small, and the variance within a cluster is large. The aim is |
36,221 | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neither or both? | Other answers have been giving good and clear examples. I'd like to try a different wording for this.
Consider you are going to sample a city's population to know its average income.
Some of things that will "stratify" your population:
Income level (high, medium, low)
Type of job (skilled labor, unskilled labor, etc.... | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neith | Other answers have been giving good and clear examples. I'd like to try a different wording for this.
Consider you are going to sample a city's population to know its average income.
Some of things t | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neither or both?
Other answers have been giving good and clear examples. I'd like to try a different wording for this.
Consider you are going to sample a city's population to know its average income.
Some of things that will... | Are the differences between sampling clusters and sampling strata, conceptual, methodological, neith
Other answers have been giving good and clear examples. I'd like to try a different wording for this.
Consider you are going to sample a city's population to know its average income.
Some of things t |
36,222 | What does it mean to have a "gaussian prior?" | Prior is a belief you have on some quantity, typically on a set of parameters, without having any look at the data. If data is involved, the belief you have is updated and is called as posterior.
In ridge regression, a gaussian prior on regression coefficients means that the coefficients are assumed to be distributed a... | What does it mean to have a "gaussian prior?" | Prior is a belief you have on some quantity, typically on a set of parameters, without having any look at the data. If data is involved, the belief you have is updated and is called as posterior.
In r | What does it mean to have a "gaussian prior?"
Prior is a belief you have on some quantity, typically on a set of parameters, without having any look at the data. If data is involved, the belief you have is updated and is called as posterior.
In ridge regression, a gaussian prior on regression coefficients means that th... | What does it mean to have a "gaussian prior?"
Prior is a belief you have on some quantity, typically on a set of parameters, without having any look at the data. If data is involved, the belief you have is updated and is called as posterior.
In r |
36,223 | Multivariate Wasserstein metric for $n$-dimensions | Wasserstein in 1D is a special case of optimal transport. Both the R wasserstein1d and Python scipy.stats.wasserstein_distance are intended solely for the 1D special case. The algorithm behind both functions rank discrete data according to their c.d.f.'s so that the distances and amounts to move are multiplied together... | Multivariate Wasserstein metric for $n$-dimensions | Wasserstein in 1D is a special case of optimal transport. Both the R wasserstein1d and Python scipy.stats.wasserstein_distance are intended solely for the 1D special case. The algorithm behind both fu | Multivariate Wasserstein metric for $n$-dimensions
Wasserstein in 1D is a special case of optimal transport. Both the R wasserstein1d and Python scipy.stats.wasserstein_distance are intended solely for the 1D special case. The algorithm behind both functions rank discrete data according to their c.d.f.'s so that the di... | Multivariate Wasserstein metric for $n$-dimensions
Wasserstein in 1D is a special case of optimal transport. Both the R wasserstein1d and Python scipy.stats.wasserstein_distance are intended solely for the 1D special case. The algorithm behind both fu |
36,224 | What is limiting about a linear model? | I will cite an educational reference to indicate the possible drawbacks. To quote for the case of a Simple Linear Regression Model:
Objective: model the expected value of a continuous variable, Y, as a linear function of the continuous predictor, X, E(Yi) = β0 + β1xi
Model structure: ${Y_i = β_0 + β_1x_i + \epsilon_i}... | What is limiting about a linear model? | I will cite an educational reference to indicate the possible drawbacks. To quote for the case of a Simple Linear Regression Model:
Objective: model the expected value of a continuous variable, Y, as | What is limiting about a linear model?
I will cite an educational reference to indicate the possible drawbacks. To quote for the case of a Simple Linear Regression Model:
Objective: model the expected value of a continuous variable, Y, as a linear function of the continuous predictor, X, E(Yi) = β0 + β1xi
Model struct... | What is limiting about a linear model?
I will cite an educational reference to indicate the possible drawbacks. To quote for the case of a Simple Linear Regression Model:
Objective: model the expected value of a continuous variable, Y, as |
36,225 | What is limiting about a linear model? | $$Y=\beta_0+\beta_1x_1+\beta_2 x_2^2+\beta_3 e^{5x_3}+\cdots+\epsilon$$
are linear models.
Visualy, I would expect such flexibility would let me model any sort of shape between the response and the predictors if I plot my data. I haven't yet learnt more advanced models, but what would be a drawback/incapability of just... | What is limiting about a linear model? | $$Y=\beta_0+\beta_1x_1+\beta_2 x_2^2+\beta_3 e^{5x_3}+\cdots+\epsilon$$
are linear models.
Visualy, I would expect such flexibility would let me model any sort of shape between the response and the pr | What is limiting about a linear model?
$$Y=\beta_0+\beta_1x_1+\beta_2 x_2^2+\beta_3 e^{5x_3}+\cdots+\epsilon$$
are linear models.
Visualy, I would expect such flexibility would let me model any sort of shape between the response and the predictors if I plot my data. I haven't yet learnt more advanced models, but what w... | What is limiting about a linear model?
$$Y=\beta_0+\beta_1x_1+\beta_2 x_2^2+\beta_3 e^{5x_3}+\cdots+\epsilon$$
are linear models.
Visualy, I would expect such flexibility would let me model any sort of shape between the response and the pr |
36,226 | What is limiting about a linear model? | Just an example: Step functions cannot be represented by linear regressions: A factory at the sea side has a wall to protect it from the waves. Waves smaller then 5 meters stay behind the wall and do no harm. Waves above 5 meters lead to water coming into the cooler, short-circuit it and there is a loss worth of 10 Mil... | What is limiting about a linear model? | Just an example: Step functions cannot be represented by linear regressions: A factory at the sea side has a wall to protect it from the waves. Waves smaller then 5 meters stay behind the wall and do | What is limiting about a linear model?
Just an example: Step functions cannot be represented by linear regressions: A factory at the sea side has a wall to protect it from the waves. Waves smaller then 5 meters stay behind the wall and do no harm. Waves above 5 meters lead to water coming into the cooler, short-circuit... | What is limiting about a linear model?
Just an example: Step functions cannot be represented by linear regressions: A factory at the sea side has a wall to protect it from the waves. Waves smaller then 5 meters stay behind the wall and do |
36,227 | What is limiting about a linear model? | There's little limiting about a linear model per se. In fact, there is Cybenko's universal approximation theorem for neural networks! This has the output of a single layer network as a linear function of some constructed predictors. The problem lies in finding the right set of predictors, generalization out of sample a... | What is limiting about a linear model? | There's little limiting about a linear model per se. In fact, there is Cybenko's universal approximation theorem for neural networks! This has the output of a single layer network as a linear function | What is limiting about a linear model?
There's little limiting about a linear model per se. In fact, there is Cybenko's universal approximation theorem for neural networks! This has the output of a single layer network as a linear function of some constructed predictors. The problem lies in finding the right set of pre... | What is limiting about a linear model?
There's little limiting about a linear model per se. In fact, there is Cybenko's universal approximation theorem for neural networks! This has the output of a single layer network as a linear function |
36,228 | Kernels in SVM primal form | First, some terminology clarification, which is important for further understanding:
In your second formula, applying $\phi(\mathbf{x}^{(i)})$ is not using the kernel trick! Kernel trick is computing $K(\mathbf{x}^{(i)}, \mathbf{x}^{(j)})$ without computing $\phi(\mathbf{x}^{(i)})$ or $\phi(\mathbf{x}^{(j)})$, and even... | Kernels in SVM primal form | First, some terminology clarification, which is important for further understanding:
In your second formula, applying $\phi(\mathbf{x}^{(i)})$ is not using the kernel trick! Kernel trick is computing | Kernels in SVM primal form
First, some terminology clarification, which is important for further understanding:
In your second formula, applying $\phi(\mathbf{x}^{(i)})$ is not using the kernel trick! Kernel trick is computing $K(\mathbf{x}^{(i)}, \mathbf{x}^{(j)})$ without computing $\phi(\mathbf{x}^{(i)})$ or $\phi(\... | Kernels in SVM primal form
First, some terminology clarification, which is important for further understanding:
In your second formula, applying $\phi(\mathbf{x}^{(i)})$ is not using the kernel trick! Kernel trick is computing |
36,229 | What is the difference between intervention and conditional distribution? | It is true that $C$ is not a function of $E;$ moreover, as the causal diagram $C\to E$ clearly says, we are thinking of $C$ as a cause of $E.$ However, in a causal diagram, you can use the $\newcommand{\doop}{\operatorname{do}} \doop$ operator on any variable you like; doing so deletes all arrows going into that node. ... | What is the difference between intervention and conditional distribution? | It is true that $C$ is not a function of $E;$ moreover, as the causal diagram $C\to E$ clearly says, we are thinking of $C$ as a cause of $E.$ However, in a causal diagram, you can use the $\newcomman | What is the difference between intervention and conditional distribution?
It is true that $C$ is not a function of $E;$ moreover, as the causal diagram $C\to E$ clearly says, we are thinking of $C$ as a cause of $E.$ However, in a causal diagram, you can use the $\newcommand{\doop}{\operatorname{do}} \doop$ operator on... | What is the difference between intervention and conditional distribution?
It is true that $C$ is not a function of $E;$ moreover, as the causal diagram $C\to E$ clearly says, we are thinking of $C$ as a cause of $E.$ However, in a causal diagram, you can use the $\newcomman |
36,230 | Explanation of Joint Probability and Independence | Informally, independence mean that knowing the value of one random variable gives you no extra information about the other
But if $0 \lt X+Y \lt 1$, then knowing $X=\frac34$ tells you $Y < \frac14$. Meanwhile knowing $X=\frac13$ tells you $Y$ can take values up to $\frac23$. So the value of $X$ is affecting the dist... | Explanation of Joint Probability and Independence | Informally, independence mean that knowing the value of one random variable gives you no extra information about the other
But if $0 \lt X+Y \lt 1$, then knowing $X=\frac34$ tells you $Y < \frac14$. | Explanation of Joint Probability and Independence
Informally, independence mean that knowing the value of one random variable gives you no extra information about the other
But if $0 \lt X+Y \lt 1$, then knowing $X=\frac34$ tells you $Y < \frac14$. Meanwhile knowing $X=\frac13$ tells you $Y$ can take values up to $\f... | Explanation of Joint Probability and Independence
Informally, independence mean that knowing the value of one random variable gives you no extra information about the other
But if $0 \lt X+Y \lt 1$, then knowing $X=\frac34$ tells you $Y < \frac14$. |
36,231 | Explanation of Joint Probability and Independence | The necessary (not sufficient) condition for independence is that $f(x,y)$ should be factored into something like $g(x)h(y)$. For that to happen, $I(x,y)$ should be factored like $I_A(x)I_B(y)$, but the author says that there is no way to do it, basically because of the line $0<x+y<1$.
Assume $I(x,y)=I_A(x)I_B(y)$, whe... | Explanation of Joint Probability and Independence | The necessary (not sufficient) condition for independence is that $f(x,y)$ should be factored into something like $g(x)h(y)$. For that to happen, $I(x,y)$ should be factored like $I_A(x)I_B(y)$, but t | Explanation of Joint Probability and Independence
The necessary (not sufficient) condition for independence is that $f(x,y)$ should be factored into something like $g(x)h(y)$. For that to happen, $I(x,y)$ should be factored like $I_A(x)I_B(y)$, but the author says that there is no way to do it, basically because of the... | Explanation of Joint Probability and Independence
The necessary (not sufficient) condition for independence is that $f(x,y)$ should be factored into something like $g(x)h(y)$. For that to happen, $I(x,y)$ should be factored like $I_A(x)I_B(y)$, but t |
36,232 | Explanation of Joint Probability and Independence | A sufficient test for detecting non-independence of random variables is the eyeball test (described briefly in this answer of mine on stats.SE and in more detail in an answer on math.SE) which says that if the support of the joint density is not a rectangle with sides parallel to the coordinate axes, then the random va... | Explanation of Joint Probability and Independence | A sufficient test for detecting non-independence of random variables is the eyeball test (described briefly in this answer of mine on stats.SE and in more detail in an answer on math.SE) which says th | Explanation of Joint Probability and Independence
A sufficient test for detecting non-independence of random variables is the eyeball test (described briefly in this answer of mine on stats.SE and in more detail in an answer on math.SE) which says that if the support of the joint density is not a rectangle with sides p... | Explanation of Joint Probability and Independence
A sufficient test for detecting non-independence of random variables is the eyeball test (described briefly in this answer of mine on stats.SE and in more detail in an answer on math.SE) which says th |
36,233 | Explanation of Joint Probability and Independence | I don't know if this makes it any easier to understand, but consider a function that's equal to $x^2y^3$ when $x<y$ and $0$ otherwise. Now consider the function $x^2y^3 (|y-x|+y-x)/(2y-2x)$. If you plot out the second function, you'll find that it's the same as the first function. It's in more complicated form, but it'... | Explanation of Joint Probability and Independence | I don't know if this makes it any easier to understand, but consider a function that's equal to $x^2y^3$ when $x<y$ and $0$ otherwise. Now consider the function $x^2y^3 (|y-x|+y-x)/(2y-2x)$. If you pl | Explanation of Joint Probability and Independence
I don't know if this makes it any easier to understand, but consider a function that's equal to $x^2y^3$ when $x<y$ and $0$ otherwise. Now consider the function $x^2y^3 (|y-x|+y-x)/(2y-2x)$. If you plot out the second function, you'll find that it's the same as the firs... | Explanation of Joint Probability and Independence
I don't know if this makes it any easier to understand, but consider a function that's equal to $x^2y^3$ when $x<y$ and $0$ otherwise. Now consider the function $x^2y^3 (|y-x|+y-x)/(2y-2x)$. If you pl |
36,234 | The difference between DID and fixed effect model | Let's begin with an understanding of the standard fixed effects estimator before extending our intuition to make sense of how difference-in-differences (DD) estimation may offer any improvements.
Assume you have repeated observations of individuals across time. For example, let’s say we want to estimate the following m... | The difference between DID and fixed effect model | Let's begin with an understanding of the standard fixed effects estimator before extending our intuition to make sense of how difference-in-differences (DD) estimation may offer any improvements.
Assu | The difference between DID and fixed effect model
Let's begin with an understanding of the standard fixed effects estimator before extending our intuition to make sense of how difference-in-differences (DD) estimation may offer any improvements.
Assume you have repeated observations of individuals across time. For exam... | The difference between DID and fixed effect model
Let's begin with an understanding of the standard fixed effects estimator before extending our intuition to make sense of how difference-in-differences (DD) estimation may offer any improvements.
Assu |
36,235 | the approximation power of Gaussian mixture models? | Goodfellow et al. 2016, p. 65 states:
A Gaussian mixture model is a universal approximator of densities, in
the sense that any smooth density can be approximated with any specific
nonzero amount of error by a Gaussian mixture model with enough
components.
M. Carreira ascribes this property to kernel density esti... | the approximation power of Gaussian mixture models? | Goodfellow et al. 2016, p. 65 states:
A Gaussian mixture model is a universal approximator of densities, in
the sense that any smooth density can be approximated with any specific
nonzero amount o | the approximation power of Gaussian mixture models?
Goodfellow et al. 2016, p. 65 states:
A Gaussian mixture model is a universal approximator of densities, in
the sense that any smooth density can be approximated with any specific
nonzero amount of error by a Gaussian mixture model with enough
components.
M. Ca... | the approximation power of Gaussian mixture models?
Goodfellow et al. 2016, p. 65 states:
A Gaussian mixture model is a universal approximator of densities, in
the sense that any smooth density can be approximated with any specific
nonzero amount o |
36,236 | the approximation power of Gaussian mixture models? | Dalal and Hall's "Approximating Priors by Mixtures of Natural Conjugate Priors" discusses estimating arbitrary densities with mixtures of normals (or, for that matter, mixtures of other conjugate prior densities). This paper and papers which cite it discuss the details of how to approximate densities arbitrarily well w... | the approximation power of Gaussian mixture models? | Dalal and Hall's "Approximating Priors by Mixtures of Natural Conjugate Priors" discusses estimating arbitrary densities with mixtures of normals (or, for that matter, mixtures of other conjugate prio | the approximation power of Gaussian mixture models?
Dalal and Hall's "Approximating Priors by Mixtures of Natural Conjugate Priors" discusses estimating arbitrary densities with mixtures of normals (or, for that matter, mixtures of other conjugate prior densities). This paper and papers which cite it discuss the detail... | the approximation power of Gaussian mixture models?
Dalal and Hall's "Approximating Priors by Mixtures of Natural Conjugate Priors" discusses estimating arbitrary densities with mixtures of normals (or, for that matter, mixtures of other conjugate prio |
36,237 | p-values change after mean centering with interaction terms. How to test for significance? | But I do not understand what it means by "correct test for significance". Can someone explain what he's referring to?
If I were you I would post a comment to that answer by @EdM, otherwise, unless they actually see this question and answers themself, we can only make an informed guess. Having said that, what I think i... | p-values change after mean centering with interaction terms. How to test for significance? | But I do not understand what it means by "correct test for significance". Can someone explain what he's referring to?
If I were you I would post a comment to that answer by @EdM, otherwise, unless th | p-values change after mean centering with interaction terms. How to test for significance?
But I do not understand what it means by "correct test for significance". Can someone explain what he's referring to?
If I were you I would post a comment to that answer by @EdM, otherwise, unless they actually see this question... | p-values change after mean centering with interaction terms. How to test for significance?
But I do not understand what it means by "correct test for significance". Can someone explain what he's referring to?
If I were you I would post a comment to that answer by @EdM, otherwise, unless th |
36,238 | p-values change after mean centering with interaction terms. How to test for significance? | The reported p-values for the coefficient for z will differ between the uncentered and x-centered models. That might seem troubling at first, but that's OK. The correct test for significance of a predictor involved in an interaction must involve both its individual coefficient and its interaction coefficient, and the r... | p-values change after mean centering with interaction terms. How to test for significance? | The reported p-values for the coefficient for z will differ between the uncentered and x-centered models. That might seem troubling at first, but that's OK. The correct test for significance of a pred | p-values change after mean centering with interaction terms. How to test for significance?
The reported p-values for the coefficient for z will differ between the uncentered and x-centered models. That might seem troubling at first, but that's OK. The correct test for significance of a predictor involved in an interact... | p-values change after mean centering with interaction terms. How to test for significance?
The reported p-values for the coefficient for z will differ between the uncentered and x-centered models. That might seem troubling at first, but that's OK. The correct test for significance of a pred |
36,239 | Is the PCA estimator used in regression root-n-consistent? | It will not be consistent if $Y$ is explained by any of the the discarded components.
Consider the case with $p=2$ and $ d = 1$ where the first principal component is $(1,0)$ and the second is $(0,1)$ where the first explains $\%99$ percent of the variancewith $(\beta_1, \beta_2) = (0, 100)$. There is no way $\beta_2$... | Is the PCA estimator used in regression root-n-consistent? | It will not be consistent if $Y$ is explained by any of the the discarded components.
Consider the case with $p=2$ and $ d = 1$ where the first principal component is $(1,0)$ and the second is $(0,1) | Is the PCA estimator used in regression root-n-consistent?
It will not be consistent if $Y$ is explained by any of the the discarded components.
Consider the case with $p=2$ and $ d = 1$ where the first principal component is $(1,0)$ and the second is $(0,1)$ where the first explains $\%99$ percent of the variancewith... | Is the PCA estimator used in regression root-n-consistent?
It will not be consistent if $Y$ is explained by any of the the discarded components.
Consider the case with $p=2$ and $ d = 1$ where the first principal component is $(1,0)$ and the second is $(0,1) |
36,240 | What is the relationship between the Harrel's C and the AUC? | In the case of a binary outcome and a continuous predictor, the AUC of the ROC or c-index is simply a function of how well the ordered values of the continuous predictor correlate to the corresponding event status.
In a Cox model or other time to event method, those persons with a higher predictor value (hazard ratio) ... | What is the relationship between the Harrel's C and the AUC? | In the case of a binary outcome and a continuous predictor, the AUC of the ROC or c-index is simply a function of how well the ordered values of the continuous predictor correlate to the corresponding | What is the relationship between the Harrel's C and the AUC?
In the case of a binary outcome and a continuous predictor, the AUC of the ROC or c-index is simply a function of how well the ordered values of the continuous predictor correlate to the corresponding event status.
In a Cox model or other time to event method... | What is the relationship between the Harrel's C and the AUC?
In the case of a binary outcome and a continuous predictor, the AUC of the ROC or c-index is simply a function of how well the ordered values of the continuous predictor correlate to the corresponding |
36,241 | What is the relationship between the Harrel's C and the AUC? | AUROC is the same as concordance probability (Harrel's C) for the binary outcome. If the outcome is not binary, or it is censored, then in general whatever measure you compute for this outcome would not be called AUC/ROC. | What is the relationship between the Harrel's C and the AUC? | AUROC is the same as concordance probability (Harrel's C) for the binary outcome. If the outcome is not binary, or it is censored, then in general whatever measure you compute for this outcome would n | What is the relationship between the Harrel's C and the AUC?
AUROC is the same as concordance probability (Harrel's C) for the binary outcome. If the outcome is not binary, or it is censored, then in general whatever measure you compute for this outcome would not be called AUC/ROC. | What is the relationship between the Harrel's C and the AUC?
AUROC is the same as concordance probability (Harrel's C) for the binary outcome. If the outcome is not binary, or it is censored, then in general whatever measure you compute for this outcome would n |
36,242 | Uniform distribution and ordered statistics | As detailed in (the simulation Bible) Devroye's Non-uniform Variate Generation (1985), Chapter V, Section 2, the vector $(S_1,\ldots,S_{n})$ is jointly uniform:
Theorem 2.1 $\quad$ If $U_{(1)}\le\cdots\le U_{(n)}$ are the Uniform $\mathcal U(0,1)$ order statistics of an $n$-sample, and the$$S_i=U_{(i)}-U_{(i-1)}\qquad... | Uniform distribution and ordered statistics | As detailed in (the simulation Bible) Devroye's Non-uniform Variate Generation (1985), Chapter V, Section 2, the vector $(S_1,\ldots,S_{n})$ is jointly uniform:
Theorem 2.1 $\quad$ If $U_{(1)}\le\cdo | Uniform distribution and ordered statistics
As detailed in (the simulation Bible) Devroye's Non-uniform Variate Generation (1985), Chapter V, Section 2, the vector $(S_1,\ldots,S_{n})$ is jointly uniform:
Theorem 2.1 $\quad$ If $U_{(1)}\le\cdots\le U_{(n)}$ are the Uniform $\mathcal U(0,1)$ order statistics of an $n$-... | Uniform distribution and ordered statistics
As detailed in (the simulation Bible) Devroye's Non-uniform Variate Generation (1985), Chapter V, Section 2, the vector $(S_1,\ldots,S_{n})$ is jointly uniform:
Theorem 2.1 $\quad$ If $U_{(1)}\le\cdo |
36,243 | How does one compare two nested quasibinomial GLMs? | Yes, a deviance test is still valid. Some more details: Since the general theory is not specific for binomial models, I will start out with some general theory, but use binomial examples (and R.)
GLM's is based on the exponential dispersion model
$$
f(y_i;\theta_i,\phi)= \exp\left\{ w_i [y_i \theta_i -\gamma(\theta... | How does one compare two nested quasibinomial GLMs? | Yes, a deviance test is still valid. Some more details: Since the general theory is not specific for binomial models, I will start out with some general theory, but use binomial examples (and R.)
GLM' | How does one compare two nested quasibinomial GLMs?
Yes, a deviance test is still valid. Some more details: Since the general theory is not specific for binomial models, I will start out with some general theory, but use binomial examples (and R.)
GLM's is based on the exponential dispersion model
$$
f(y_i;\theta_i... | How does one compare two nested quasibinomial GLMs?
Yes, a deviance test is still valid. Some more details: Since the general theory is not specific for binomial models, I will start out with some general theory, but use binomial examples (and R.)
GLM' |
36,244 | Causes of bimodal distributions when bootstrapping a meta-analysis model | Thanks for providing the data and code. I re-fitted the model you are working with and the second variance component (for which cor_mat is specified) drifts off to a really large value, which is strange. However, profiling this variance component (with profile(rmamv_model, sigma2=2)) indicates no problems, so I don't t... | Causes of bimodal distributions when bootstrapping a meta-analysis model | Thanks for providing the data and code. I re-fitted the model you are working with and the second variance component (for which cor_mat is specified) drifts off to a really large value, which is stran | Causes of bimodal distributions when bootstrapping a meta-analysis model
Thanks for providing the data and code. I re-fitted the model you are working with and the second variance component (for which cor_mat is specified) drifts off to a really large value, which is strange. However, profiling this variance component ... | Causes of bimodal distributions when bootstrapping a meta-analysis model
Thanks for providing the data and code. I re-fitted the model you are working with and the second variance component (for which cor_mat is specified) drifts off to a really large value, which is stran |
36,245 | Causes of bimodal distributions when bootstrapping a meta-analysis model | Without having access to a reproducible example is extremely difficult to give a definite answer to this bootstrapping behaviour. Assuming that there are indeed no outliers, I suspect that we observe a mild case of Stein's phenomenon especially as a mixed-effect methodology suggests we some clustering in our data.
Havi... | Causes of bimodal distributions when bootstrapping a meta-analysis model | Without having access to a reproducible example is extremely difficult to give a definite answer to this bootstrapping behaviour. Assuming that there are indeed no outliers, I suspect that we observe | Causes of bimodal distributions when bootstrapping a meta-analysis model
Without having access to a reproducible example is extremely difficult to give a definite answer to this bootstrapping behaviour. Assuming that there are indeed no outliers, I suspect that we observe a mild case of Stein's phenomenon especially as... | Causes of bimodal distributions when bootstrapping a meta-analysis model
Without having access to a reproducible example is extremely difficult to give a definite answer to this bootstrapping behaviour. Assuming that there are indeed no outliers, I suspect that we observe |
36,246 | Errors and residuals in linear regression | I think there is a lot of confusion in this question (caused of course by authors that describe what they think linear regression means very bad/imprecise/up to wrong).
First of all we are given some data $(x_i, y_i)_{i=1,...,N}, x_i \in \mathbb{R}^d, y_i \in \mathbb{R}$ and we want to "make sense of it in form of a li... | Errors and residuals in linear regression | I think there is a lot of confusion in this question (caused of course by authors that describe what they think linear regression means very bad/imprecise/up to wrong).
First of all we are given some | Errors and residuals in linear regression
I think there is a lot of confusion in this question (caused of course by authors that describe what they think linear regression means very bad/imprecise/up to wrong).
First of all we are given some data $(x_i, y_i)_{i=1,...,N}, x_i \in \mathbb{R}^d, y_i \in \mathbb{R}$ and we... | Errors and residuals in linear regression
I think there is a lot of confusion in this question (caused of course by authors that describe what they think linear regression means very bad/imprecise/up to wrong).
First of all we are given some |
36,247 | Finding UMVUE of $\theta e^{-\theta}$ where $X_i\sim\text{Pois}(\theta)$ | The Poisson distribution is a one-parameter exponential family distribution, with natural sufficient statistic given by the sample total $T(\mathbf{x}) = \sum_{i=1}^n x_i$. The canonical form is:
$$p(\mathbf{x}|\theta) = \exp \Big( \ln (\theta) T(\mathbf{x}) - n\theta \Big) \cdot h(\mathbf{x}) \quad \quad \quad h(\mat... | Finding UMVUE of $\theta e^{-\theta}$ where $X_i\sim\text{Pois}(\theta)$ | The Poisson distribution is a one-parameter exponential family distribution, with natural sufficient statistic given by the sample total $T(\mathbf{x}) = \sum_{i=1}^n x_i$. The canonical form is:
$$p | Finding UMVUE of $\theta e^{-\theta}$ where $X_i\sim\text{Pois}(\theta)$
The Poisson distribution is a one-parameter exponential family distribution, with natural sufficient statistic given by the sample total $T(\mathbf{x}) = \sum_{i=1}^n x_i$. The canonical form is:
$$p(\mathbf{x}|\theta) = \exp \Big( \ln (\theta) T... | Finding UMVUE of $\theta e^{-\theta}$ where $X_i\sim\text{Pois}(\theta)$
The Poisson distribution is a one-parameter exponential family distribution, with natural sufficient statistic given by the sample total $T(\mathbf{x}) = \sum_{i=1}^n x_i$. The canonical form is:
$$p |
36,248 | Finding UMVUE of $\theta e^{-\theta}$ where $X_i\sim\text{Pois}(\theta)$ | Here is a simulation in R that I did using a the average of $n = 20$ observations, where $\lambda = 5.$ The parameter par is $P(X = 1) = \lambda e^{-\lambda}.$
The estimate par.fcn, which tries to estimate $P(X = 1)$ merely as a function of the average, is biased. My version of your UMVUE for $P(X=1)$ using a function ... | Finding UMVUE of $\theta e^{-\theta}$ where $X_i\sim\text{Pois}(\theta)$ | Here is a simulation in R that I did using a the average of $n = 20$ observations, where $\lambda = 5.$ The parameter par is $P(X = 1) = \lambda e^{-\lambda}.$
The estimate par.fcn, which tries to est | Finding UMVUE of $\theta e^{-\theta}$ where $X_i\sim\text{Pois}(\theta)$
Here is a simulation in R that I did using a the average of $n = 20$ observations, where $\lambda = 5.$ The parameter par is $P(X = 1) = \lambda e^{-\lambda}.$
The estimate par.fcn, which tries to estimate $P(X = 1)$ merely as a function of the av... | Finding UMVUE of $\theta e^{-\theta}$ where $X_i\sim\text{Pois}(\theta)$
Here is a simulation in R that I did using a the average of $n = 20$ observations, where $\lambda = 5.$ The parameter par is $P(X = 1) = \lambda e^{-\lambda}.$
The estimate par.fcn, which tries to est |
36,249 | Simple explanation of dynamic linear models | I also have to speak regularly to people who do not have a technical background, and here is how I would approach it:
First, unless your audience knows about the normal distribution, I would not even mention DLM, I would just talk about state space models. I would still give them a DLM set of equations as an example (l... | Simple explanation of dynamic linear models | I also have to speak regularly to people who do not have a technical background, and here is how I would approach it:
First, unless your audience knows about the normal distribution, I would not even | Simple explanation of dynamic linear models
I also have to speak regularly to people who do not have a technical background, and here is how I would approach it:
First, unless your audience knows about the normal distribution, I would not even mention DLM, I would just talk about state space models. I would still give ... | Simple explanation of dynamic linear models
I also have to speak regularly to people who do not have a technical background, and here is how I would approach it:
First, unless your audience knows about the normal distribution, I would not even |
36,250 | Simple explanation of dynamic linear models | I recommend that you go through a few examples. The most common question is "what does the state variable represent?" The answer to that depends on the model, but most DLMs can be thought of as a regression with a time-varying coefficient. In this context, those time-varying coefficients are your states usually.
If yo... | Simple explanation of dynamic linear models | I recommend that you go through a few examples. The most common question is "what does the state variable represent?" The answer to that depends on the model, but most DLMs can be thought of as a regr | Simple explanation of dynamic linear models
I recommend that you go through a few examples. The most common question is "what does the state variable represent?" The answer to that depends on the model, but most DLMs can be thought of as a regression with a time-varying coefficient. In this context, those time-varying ... | Simple explanation of dynamic linear models
I recommend that you go through a few examples. The most common question is "what does the state variable represent?" The answer to that depends on the model, but most DLMs can be thought of as a regr |
36,251 | Simple explanation of dynamic linear models | Best I can do is a tad lengthy but perhaps easier to follow (1st try):
So for a (static) linear regression, the usual format is y = mx +b, like the equation of a line (b is a constant, m is the slope, x are your predictors, y is your response).
The magic of regression is that it amps up this equation to the matrix leve... | Simple explanation of dynamic linear models | Best I can do is a tad lengthy but perhaps easier to follow (1st try):
So for a (static) linear regression, the usual format is y = mx +b, like the equation of a line (b is a constant, m is the slope, | Simple explanation of dynamic linear models
Best I can do is a tad lengthy but perhaps easier to follow (1st try):
So for a (static) linear regression, the usual format is y = mx +b, like the equation of a line (b is a constant, m is the slope, x are your predictors, y is your response).
The magic of regression is that... | Simple explanation of dynamic linear models
Best I can do is a tad lengthy but perhaps easier to follow (1st try):
So for a (static) linear regression, the usual format is y = mx +b, like the equation of a line (b is a constant, m is the slope, |
36,252 | Proper way of estimating the covariance error ellipse in 2D | I believe I've found the reason for the discrepancy between these two methods. Both seem to be correct, they just estimate different statistical concepts.
The first method describes an error ellipse, characterized by some number of standard deviations.
The second method describes a confidence ellipse, characterized by ... | Proper way of estimating the covariance error ellipse in 2D | I believe I've found the reason for the discrepancy between these two methods. Both seem to be correct, they just estimate different statistical concepts.
The first method describes an error ellipse, | Proper way of estimating the covariance error ellipse in 2D
I believe I've found the reason for the discrepancy between these two methods. Both seem to be correct, they just estimate different statistical concepts.
The first method describes an error ellipse, characterized by some number of standard deviations.
The sec... | Proper way of estimating the covariance error ellipse in 2D
I believe I've found the reason for the discrepancy between these two methods. Both seem to be correct, they just estimate different statistical concepts.
The first method describes an error ellipse, |
36,253 | Can confidence interval of positive values be negative? | The first question has a simple answer: yes.
I interpret your question to mean, "can a strictly positive sample (where all data points are positive) have a 68% confidence interval for the normal distribution with a negative lower bound?"
Proof by construction: Let $X = [1, 1, 7]^T$. The mean is $3$, the sample standard... | Can confidence interval of positive values be negative? | The first question has a simple answer: yes.
I interpret your question to mean, "can a strictly positive sample (where all data points are positive) have a 68% confidence interval for the normal distr | Can confidence interval of positive values be negative?
The first question has a simple answer: yes.
I interpret your question to mean, "can a strictly positive sample (where all data points are positive) have a 68% confidence interval for the normal distribution with a negative lower bound?"
Proof by construction: Let... | Can confidence interval of positive values be negative?
The first question has a simple answer: yes.
I interpret your question to mean, "can a strictly positive sample (where all data points are positive) have a 68% confidence interval for the normal distr |
36,254 | Why Kullback-Leibler in Stochastic Neighbor Embedding | Dimensionality reduction techniques are often motivated by finding new representations of the data to discover hidden variables or to discover structure. The aim of SNE is to take a different approach (compared to PCA for example) by preserving local structures, which is done by taking advantage of KL-divergence's asym... | Why Kullback-Leibler in Stochastic Neighbor Embedding | Dimensionality reduction techniques are often motivated by finding new representations of the data to discover hidden variables or to discover structure. The aim of SNE is to take a different approach | Why Kullback-Leibler in Stochastic Neighbor Embedding
Dimensionality reduction techniques are often motivated by finding new representations of the data to discover hidden variables or to discover structure. The aim of SNE is to take a different approach (compared to PCA for example) by preserving local structures, whi... | Why Kullback-Leibler in Stochastic Neighbor Embedding
Dimensionality reduction techniques are often motivated by finding new representations of the data to discover hidden variables or to discover structure. The aim of SNE is to take a different approach |
36,255 | Why Kullback-Leibler in Stochastic Neighbor Embedding | Stochastic Neighbor Embedding under f-divergences
https://arxiv.org/pdf/1811.01247.pdf
This paper tries five different f- divergence functions : KL, RKL, JS, CH (Chi-Square), HL (Hellinger).
Also, the paper goes over which divergence emphasize what in terms of precision and recall. | Why Kullback-Leibler in Stochastic Neighbor Embedding | Stochastic Neighbor Embedding under f-divergences
https://arxiv.org/pdf/1811.01247.pdf
This paper tries five different f- divergence functions : KL, RKL, JS, CH (Chi-Square), HL (Hellinger).
Also, the | Why Kullback-Leibler in Stochastic Neighbor Embedding
Stochastic Neighbor Embedding under f-divergences
https://arxiv.org/pdf/1811.01247.pdf
This paper tries five different f- divergence functions : KL, RKL, JS, CH (Chi-Square), HL (Hellinger).
Also, the paper goes over which divergence emphasize what in terms of preci... | Why Kullback-Leibler in Stochastic Neighbor Embedding
Stochastic Neighbor Embedding under f-divergences
https://arxiv.org/pdf/1811.01247.pdf
This paper tries five different f- divergence functions : KL, RKL, JS, CH (Chi-Square), HL (Hellinger).
Also, the |
36,256 | Feature Importance of a feature in lightgbm is high but reduces evaluation score | If you look in the lightgbm docs for feature_importance function, you will see that it has a parameter importance_type. The two valid values for this parameters are split(default one) and gain. It is not necessarily important that both split and gain produce same feature importances. There is a new library for feature ... | Feature Importance of a feature in lightgbm is high but reduces evaluation score | If you look in the lightgbm docs for feature_importance function, you will see that it has a parameter importance_type. The two valid values for this parameters are split(default one) and gain. It is | Feature Importance of a feature in lightgbm is high but reduces evaluation score
If you look in the lightgbm docs for feature_importance function, you will see that it has a parameter importance_type. The two valid values for this parameters are split(default one) and gain. It is not necessarily important that both spl... | Feature Importance of a feature in lightgbm is high but reduces evaluation score
If you look in the lightgbm docs for feature_importance function, you will see that it has a parameter importance_type. The two valid values for this parameters are split(default one) and gain. It is |
36,257 | Feature Importance of a feature in lightgbm is high but reduces evaluation score | You should use verbose_eval and early_stopping_rounds to track the actual performance of the model upon training. For example, verbose_eval = 10 will print out the performance of the model at every 10 iterations. It is both possible that the feature harms your model or the model is overfitted. In overfitting situations... | Feature Importance of a feature in lightgbm is high but reduces evaluation score | You should use verbose_eval and early_stopping_rounds to track the actual performance of the model upon training. For example, verbose_eval = 10 will print out the performance of the model at every 10 | Feature Importance of a feature in lightgbm is high but reduces evaluation score
You should use verbose_eval and early_stopping_rounds to track the actual performance of the model upon training. For example, verbose_eval = 10 will print out the performance of the model at every 10 iterations. It is both possible that t... | Feature Importance of a feature in lightgbm is high but reduces evaluation score
You should use verbose_eval and early_stopping_rounds to track the actual performance of the model upon training. For example, verbose_eval = 10 will print out the performance of the model at every 10 |
36,258 | Can the 'bin size' in a histogram be thought of as a regularity constraint? | Yes, this is a reasonable way to think about it (assuming the histogram is normalized to obtain a proper pdf). Bin width constrains the smoothness of the density estimate (speaking loosely, since histograms are discontinuous functions). It controls the extent to which finer structure can be modeled, and also the extent... | Can the 'bin size' in a histogram be thought of as a regularity constraint? | Yes, this is a reasonable way to think about it (assuming the histogram is normalized to obtain a proper pdf). Bin width constrains the smoothness of the density estimate (speaking loosely, since hist | Can the 'bin size' in a histogram be thought of as a regularity constraint?
Yes, this is a reasonable way to think about it (assuming the histogram is normalized to obtain a proper pdf). Bin width constrains the smoothness of the density estimate (speaking loosely, since histograms are discontinuous functions). It cont... | Can the 'bin size' in a histogram be thought of as a regularity constraint?
Yes, this is a reasonable way to think about it (assuming the histogram is normalized to obtain a proper pdf). Bin width constrains the smoothness of the density estimate (speaking loosely, since hist |
36,259 | Can the 'bin size' in a histogram be thought of as a regularity constraint? | Kernel density estimators are oftentimes rationalised as a "continuous" version of a histogram. Many books on nonparametric kernel estimation also discuss histograms. See, e.g., chapter 2 in
Racine, Jeffrey S. "Nonparametric econometrics: A primer." Foundations and Trends® in Econometrics 3.1 (2008): 1-88. | Can the 'bin size' in a histogram be thought of as a regularity constraint? | Kernel density estimators are oftentimes rationalised as a "continuous" version of a histogram. Many books on nonparametric kernel estimation also discuss histograms. See, e.g., chapter 2 in
Racine, J | Can the 'bin size' in a histogram be thought of as a regularity constraint?
Kernel density estimators are oftentimes rationalised as a "continuous" version of a histogram. Many books on nonparametric kernel estimation also discuss histograms. See, e.g., chapter 2 in
Racine, Jeffrey S. "Nonparametric econometrics: A pri... | Can the 'bin size' in a histogram be thought of as a regularity constraint?
Kernel density estimators are oftentimes rationalised as a "continuous" version of a histogram. Many books on nonparametric kernel estimation also discuss histograms. See, e.g., chapter 2 in
Racine, J |
36,260 | Can the 'bin size' in a histogram be thought of as a regularity constraint? | It is reasonable, because what you're doing by putting samples in bins is approximating the data. In my experience depending on your goal and data available, those bins can vary drastically and have a big impact on how the data is handled further. For some cases you might not need a lot bins or maybe you lack data, so ... | Can the 'bin size' in a histogram be thought of as a regularity constraint? | It is reasonable, because what you're doing by putting samples in bins is approximating the data. In my experience depending on your goal and data available, those bins can vary drastically and have a | Can the 'bin size' in a histogram be thought of as a regularity constraint?
It is reasonable, because what you're doing by putting samples in bins is approximating the data. In my experience depending on your goal and data available, those bins can vary drastically and have a big impact on how the data is handled furth... | Can the 'bin size' in a histogram be thought of as a regularity constraint?
It is reasonable, because what you're doing by putting samples in bins is approximating the data. In my experience depending on your goal and data available, those bins can vary drastically and have a |
36,261 | How to find nearest neighbors using cosine similarity for all items from a large embeddings matrix? | Actually, we can use cosine similarity in knn via sklearn.
The source code is here.
This works for me:
model = NearestNeighbors(n_neighbors=n_neighbor,
metric='cosine',
algorithm='brute',
n_jobs=-1)
model.fit(user_item_matrix_sparse) | How to find nearest neighbors using cosine similarity for all items from a large embeddings matrix? | Actually, we can use cosine similarity in knn via sklearn.
The source code is here.
This works for me:
model = NearestNeighbors(n_neighbors=n_neighbor,
metric='cosine',
| How to find nearest neighbors using cosine similarity for all items from a large embeddings matrix?
Actually, we can use cosine similarity in knn via sklearn.
The source code is here.
This works for me:
model = NearestNeighbors(n_neighbors=n_neighbor,
metric='cosine',
a... | How to find nearest neighbors using cosine similarity for all items from a large embeddings matrix?
Actually, we can use cosine similarity in knn via sklearn.
The source code is here.
This works for me:
model = NearestNeighbors(n_neighbors=n_neighbor,
metric='cosine',
|
36,262 | How to find nearest neighbors using cosine similarity for all items from a large embeddings matrix? | one weak point is sorting, and second is creating new collection
so you could just make another collection (list) and then keep topK items until some point this way instead of sorting everytnig and allocate new portion of memory you could do it more robust
use this https://stackoverflow.com/questions/27672494/how-to-us... | How to find nearest neighbors using cosine similarity for all items from a large embeddings matrix? | one weak point is sorting, and second is creating new collection
so you could just make another collection (list) and then keep topK items until some point this way instead of sorting everytnig and al | How to find nearest neighbors using cosine similarity for all items from a large embeddings matrix?
one weak point is sorting, and second is creating new collection
so you could just make another collection (list) and then keep topK items until some point this way instead of sorting everytnig and allocate new portion o... | How to find nearest neighbors using cosine similarity for all items from a large embeddings matrix?
one weak point is sorting, and second is creating new collection
so you could just make another collection (list) and then keep topK items until some point this way instead of sorting everytnig and al |
36,263 | How to find nearest neighbors using cosine similarity for all items from a large embeddings matrix? | A Nearest Neighbours model is fairly fast to build, because the algorithm uses the triangle inequality. Sadly, Scikit-Learn's ball tree does not support cosine distances, so you will end up with a KDTree, which is less efficient for high-dimensional data.
from sklearn.neighbors import NearestNeighbors
embeddings = get... | How to find nearest neighbors using cosine similarity for all items from a large embeddings matrix? | A Nearest Neighbours model is fairly fast to build, because the algorithm uses the triangle inequality. Sadly, Scikit-Learn's ball tree does not support cosine distances, so you will end up with a KDT | How to find nearest neighbors using cosine similarity for all items from a large embeddings matrix?
A Nearest Neighbours model is fairly fast to build, because the algorithm uses the triangle inequality. Sadly, Scikit-Learn's ball tree does not support cosine distances, so you will end up with a KDTree, which is less e... | How to find nearest neighbors using cosine similarity for all items from a large embeddings matrix?
A Nearest Neighbours model is fairly fast to build, because the algorithm uses the triangle inequality. Sadly, Scikit-Learn's ball tree does not support cosine distances, so you will end up with a KDT |
36,264 | How to determine what type of layers do I need for my Deep learning model? | I have some good news and some bad news for you. The good news is that there are many problems for which we know which architecture works best, because of previous research. The bad news is that since today we don't have a good theory of generalization for Deep Networks, we lack theoretical guidance about how to select... | How to determine what type of layers do I need for my Deep learning model? | I have some good news and some bad news for you. The good news is that there are many problems for which we know which architecture works best, because of previous research. The bad news is that since | How to determine what type of layers do I need for my Deep learning model?
I have some good news and some bad news for you. The good news is that there are many problems for which we know which architecture works best, because of previous research. The bad news is that since today we don't have a good theory of general... | How to determine what type of layers do I need for my Deep learning model?
I have some good news and some bad news for you. The good news is that there are many problems for which we know which architecture works best, because of previous research. The bad news is that since |
36,265 | How to determine what type of layers do I need for my Deep learning model? | If you want to use Deep Learning, you must know what is it good at in the current state of art, and what problem is still challenging. Essentially, the classes of problems are (no my list is definitely not exhaustive):
a. image recognition and classification
b. natural language processing: translation and
c. ... | How to determine what type of layers do I need for my Deep learning model? | If you want to use Deep Learning, you must know what is it good at in the current state of art, and what problem is still challenging. Essentially, the classes of problems are (no my list is definit | How to determine what type of layers do I need for my Deep learning model?
If you want to use Deep Learning, you must know what is it good at in the current state of art, and what problem is still challenging. Essentially, the classes of problems are (no my list is definitely not exhaustive):
a. image recognition a... | How to determine what type of layers do I need for my Deep learning model?
If you want to use Deep Learning, you must know what is it good at in the current state of art, and what problem is still challenging. Essentially, the classes of problems are (no my list is definit |
36,266 | Difference between simulated annealing and multiple greedy | The method you describe is called random restart hill climbing (or sometimes shotgun hill climbing), and it is a different algorithm from simulated annealing.
Yes, generally as the number of iterations $k$ increases both methods will eventually give a location $w_i$ which reaches a global optimum $w^*$. This is for th... | Difference between simulated annealing and multiple greedy | The method you describe is called random restart hill climbing (or sometimes shotgun hill climbing), and it is a different algorithm from simulated annealing.
Yes, generally as the number of iteration | Difference between simulated annealing and multiple greedy
The method you describe is called random restart hill climbing (or sometimes shotgun hill climbing), and it is a different algorithm from simulated annealing.
Yes, generally as the number of iterations $k$ increases both methods will eventually give a location... | Difference between simulated annealing and multiple greedy
The method you describe is called random restart hill climbing (or sometimes shotgun hill climbing), and it is a different algorithm from simulated annealing.
Yes, generally as the number of iteration |
36,267 | Can I treat the mean of a set of z-scores as a z-score? | Maybe someone else can explain the math behind it, but consider this quick demonstration: I generate five vectors, each 100 numbers long. Each of these vectors is on a different scale, so I standardize them (i.e., create z-scored variables). That is, the mean is zero and the standard deviation is 1 for each of these fi... | Can I treat the mean of a set of z-scores as a z-score? | Maybe someone else can explain the math behind it, but consider this quick demonstration: I generate five vectors, each 100 numbers long. Each of these vectors is on a different scale, so I standardiz | Can I treat the mean of a set of z-scores as a z-score?
Maybe someone else can explain the math behind it, but consider this quick demonstration: I generate five vectors, each 100 numbers long. Each of these vectors is on a different scale, so I standardize them (i.e., create z-scored variables). That is, the mean is z... | Can I treat the mean of a set of z-scores as a z-score?
Maybe someone else can explain the math behind it, but consider this quick demonstration: I generate five vectors, each 100 numbers long. Each of these vectors is on a different scale, so I standardiz |
36,268 | Can I treat the mean of a set of z-scores as a z-score? | Nope. The central limit theorem should provide some insight. Or you can appeal to the variance of a sum. If $X_1, X_2, \ldots, X_p$ comprise your $p$ independent z-scores to average together, (mean 0, variance 1), then the mean has variance:
$$\mbox{var} (\bar{X}) = \frac{1}{p^2} \sum_{i=1}^p \mbox{var}(X_i) = 1/p$$
Th... | Can I treat the mean of a set of z-scores as a z-score? | Nope. The central limit theorem should provide some insight. Or you can appeal to the variance of a sum. If $X_1, X_2, \ldots, X_p$ comprise your $p$ independent z-scores to average together, (mean 0, | Can I treat the mean of a set of z-scores as a z-score?
Nope. The central limit theorem should provide some insight. Or you can appeal to the variance of a sum. If $X_1, X_2, \ldots, X_p$ comprise your $p$ independent z-scores to average together, (mean 0, variance 1), then the mean has variance:
$$\mbox{var} (\bar{X})... | Can I treat the mean of a set of z-scores as a z-score?
Nope. The central limit theorem should provide some insight. Or you can appeal to the variance of a sum. If $X_1, X_2, \ldots, X_p$ comprise your $p$ independent z-scores to average together, (mean 0, |
36,269 | Are these estimators of $P(X<Y)$ asymptotically normally distributed? | The statistic $T$ is an example of a general $U$-statistic, introduced
by Hoeffding in his 1948 paper A class of statistics with asymptotically
normal distribution. Moreover, it is among the most famous of that class,
namely the Mann-Whitney U test. It has lower mean
squared error than $S$, for it equals $E\left(S\mid ... | Are these estimators of $P(X<Y)$ asymptotically normally distributed? | The statistic $T$ is an example of a general $U$-statistic, introduced
by Hoeffding in his 1948 paper A class of statistics with asymptotically
normal distribution. Moreover, it is among the most famo | Are these estimators of $P(X<Y)$ asymptotically normally distributed?
The statistic $T$ is an example of a general $U$-statistic, introduced
by Hoeffding in his 1948 paper A class of statistics with asymptotically
normal distribution. Moreover, it is among the most famous of that class,
namely the Mann-Whitney U test. ... | Are these estimators of $P(X<Y)$ asymptotically normally distributed?
The statistic $T$ is an example of a general $U$-statistic, introduced
by Hoeffding in his 1948 paper A class of statistics with asymptotically
normal distribution. Moreover, it is among the most famo |
36,270 | Are these estimators of $P(X<Y)$ asymptotically normally distributed? | Here is a fairly straightforward and intuitive argument for both estimators being asymptotically normal.
Call $$T(X|y_1)=\frac{1}{n} \sum_{i=1}^{n} I(x_i>y_1)$$
As with $S$, central limit theorem implies this is asymptotically normal.
Furthermore, take $$T = \frac{1}{n}\sum_{i=1}^{n} T(X|y_i)$$
Since each $T(X|y_i)$ is... | Are these estimators of $P(X<Y)$ asymptotically normally distributed? | Here is a fairly straightforward and intuitive argument for both estimators being asymptotically normal.
Call $$T(X|y_1)=\frac{1}{n} \sum_{i=1}^{n} I(x_i>y_1)$$
As with $S$, central limit theorem impl | Are these estimators of $P(X<Y)$ asymptotically normally distributed?
Here is a fairly straightforward and intuitive argument for both estimators being asymptotically normal.
Call $$T(X|y_1)=\frac{1}{n} \sum_{i=1}^{n} I(x_i>y_1)$$
As with $S$, central limit theorem implies this is asymptotically normal.
Furthermore, ta... | Are these estimators of $P(X<Y)$ asymptotically normally distributed?
Here is a fairly straightforward and intuitive argument for both estimators being asymptotically normal.
Call $$T(X|y_1)=\frac{1}{n} \sum_{i=1}^{n} I(x_i>y_1)$$
As with $S$, central limit theorem impl |
36,271 | Why do some statistics symbols have a "squared", e.g. Variance $\sigma^2$, "R squared" $R^2$ or heritability $H^2$ | While we might analyze the many different formulae that we find in statistics, and see that second moments have a special place...
...maybe a more special place in statistics than in physics (Which also occasionally uses square terms for simplicity as well, for instance 'radius of gyration' $r_g^2$. And also a term lik... | Why do some statistics symbols have a "squared", e.g. Variance $\sigma^2$, "R squared" $R^2$ or heri | While we might analyze the many different formulae that we find in statistics, and see that second moments have a special place...
...maybe a more special place in statistics than in physics (Which al | Why do some statistics symbols have a "squared", e.g. Variance $\sigma^2$, "R squared" $R^2$ or heritability $H^2$
While we might analyze the many different formulae that we find in statistics, and see that second moments have a special place...
...maybe a more special place in statistics than in physics (Which also oc... | Why do some statistics symbols have a "squared", e.g. Variance $\sigma^2$, "R squared" $R^2$ or heri
While we might analyze the many different formulae that we find in statistics, and see that second moments have a special place...
...maybe a more special place in statistics than in physics (Which al |
36,272 | Why do some statistics symbols have a "squared", e.g. Variance $\sigma^2$, "R squared" $R^2$ or heritability $H^2$ | Narrow sense heritability is denoted $h^2$ because people (not sure who but see Felsenstein, 2016, Ch. IX, problem 7) first introduced the symbol $h$ for the correlation between the additive genetic effect $x$ and the phenotype $z=x+e$,
\begin{align}
h=\mbox{corr}(x,z)
&=\frac{\mbox{Cov}(x,z)}{\sqrt{\mbox{Var}(x)\mbo... | Why do some statistics symbols have a "squared", e.g. Variance $\sigma^2$, "R squared" $R^2$ or heri | Narrow sense heritability is denoted $h^2$ because people (not sure who but see Felsenstein, 2016, Ch. IX, problem 7) first introduced the symbol $h$ for the correlation between the additive genetic e | Why do some statistics symbols have a "squared", e.g. Variance $\sigma^2$, "R squared" $R^2$ or heritability $H^2$
Narrow sense heritability is denoted $h^2$ because people (not sure who but see Felsenstein, 2016, Ch. IX, problem 7) first introduced the symbol $h$ for the correlation between the additive genetic effect... | Why do some statistics symbols have a "squared", e.g. Variance $\sigma^2$, "R squared" $R^2$ or heri
Narrow sense heritability is denoted $h^2$ because people (not sure who but see Felsenstein, 2016, Ch. IX, problem 7) first introduced the symbol $h$ for the correlation between the additive genetic e |
36,273 | Applying duality and KKT conditions to LASSO problem | 1) You're going the wrong direction by invoking duality directly. To get from
$\text{arg min}_{\beta: \|\beta\|_1 \leq t} \|y - X\beta\|_2^2$
to
$\text{arg min}_{\beta} \|y - X\beta\|_2^2 + \lambda\|\beta\|_1$
you just need to invoke Lagrange multipliers. (See, e.g. Section 5.1 of [1])
LMs are often discussed in t... | Applying duality and KKT conditions to LASSO problem | 1) You're going the wrong direction by invoking duality directly. To get from
$\text{arg min}_{\beta: \|\beta\|_1 \leq t} \|y - X\beta\|_2^2$
to
$\text{arg min}_{\beta} \|y - X\beta\|_2^2 + \lambda | Applying duality and KKT conditions to LASSO problem
1) You're going the wrong direction by invoking duality directly. To get from
$\text{arg min}_{\beta: \|\beta\|_1 \leq t} \|y - X\beta\|_2^2$
to
$\text{arg min}_{\beta} \|y - X\beta\|_2^2 + \lambda\|\beta\|_1$
you just need to invoke Lagrange multipliers. (See, e... | Applying duality and KKT conditions to LASSO problem
1) You're going the wrong direction by invoking duality directly. To get from
$\text{arg min}_{\beta: \|\beta\|_1 \leq t} \|y - X\beta\|_2^2$
to
$\text{arg min}_{\beta} \|y - X\beta\|_2^2 + \lambda |
36,274 | Disagreement between p-values and confidence intervals | Caveat: This answer assumes that the question is about interpreting bootstrapped p-values and CIs. A comparison between a traditional p-value (not bootstrapped) and a bootstrapped CI would be a different issue.
With a traditional (not bootstrapped) t-test, the 95%CI and the p-value's position relative to the .05 cutoff... | Disagreement between p-values and confidence intervals | Caveat: This answer assumes that the question is about interpreting bootstrapped p-values and CIs. A comparison between a traditional p-value (not bootstrapped) and a bootstrapped CI would be a differ | Disagreement between p-values and confidence intervals
Caveat: This answer assumes that the question is about interpreting bootstrapped p-values and CIs. A comparison between a traditional p-value (not bootstrapped) and a bootstrapped CI would be a different issue.
With a traditional (not bootstrapped) t-test, the 95%C... | Disagreement between p-values and confidence intervals
Caveat: This answer assumes that the question is about interpreting bootstrapped p-values and CIs. A comparison between a traditional p-value (not bootstrapped) and a bootstrapped CI would be a differ |
36,275 | Disagreement between p-values and confidence intervals | If the p-value of the null hypothesis is smaller than 0.05, then zero should not be contained in the confidence interval at 0.05 of the parameter that you are assuming to be zero in the null hypothesis. This is the same thing. So there is a bug or you don't test the same hypothesis.
EDIT, as the other answers and the... | Disagreement between p-values and confidence intervals | If the p-value of the null hypothesis is smaller than 0.05, then zero should not be contained in the confidence interval at 0.05 of the parameter that you are assuming to be zero in the null hypothesi | Disagreement between p-values and confidence intervals
If the p-value of the null hypothesis is smaller than 0.05, then zero should not be contained in the confidence interval at 0.05 of the parameter that you are assuming to be zero in the null hypothesis. This is the same thing. So there is a bug or you don't test th... | Disagreement between p-values and confidence intervals
If the p-value of the null hypothesis is smaller than 0.05, then zero should not be contained in the confidence interval at 0.05 of the parameter that you are assuming to be zero in the null hypothesi |
36,276 | If $\mathbb{E}|X_n|=O(a_n)$, how large is $Y_n = X_n\ln\left(\frac{1}{X_n}\right)$? | I believe much might be revealed by contemplating series of random variables like the following:
$$X_n = \left\{\eqalign{\frac{1}{n} & \text{ with probability } 1 - \frac{1}{f(n)^2}e^{-g(n)}\\
f(n)e^{g(n)} & \text{ with probability } \frac{1}{f(n)^2}e^{-g(n)}.}\right.$$
Later we will identify suitable functions $f$ and... | If $\mathbb{E}|X_n|=O(a_n)$, how large is $Y_n = X_n\ln\left(\frac{1}{X_n}\right)$? | I believe much might be revealed by contemplating series of random variables like the following:
$$X_n = \left\{\eqalign{\frac{1}{n} & \text{ with probability } 1 - \frac{1}{f(n)^2}e^{-g(n)}\\
f(n)e^{ | If $\mathbb{E}|X_n|=O(a_n)$, how large is $Y_n = X_n\ln\left(\frac{1}{X_n}\right)$?
I believe much might be revealed by contemplating series of random variables like the following:
$$X_n = \left\{\eqalign{\frac{1}{n} & \text{ with probability } 1 - \frac{1}{f(n)^2}e^{-g(n)}\\
f(n)e^{g(n)} & \text{ with probability } \f... | If $\mathbb{E}|X_n|=O(a_n)$, how large is $Y_n = X_n\ln\left(\frac{1}{X_n}\right)$?
I believe much might be revealed by contemplating series of random variables like the following:
$$X_n = \left\{\eqalign{\frac{1}{n} & \text{ with probability } 1 - \frac{1}{f(n)^2}e^{-g(n)}\\
f(n)e^{ |
36,277 | If $\mathbb{E}|X_n|=O(a_n)$, how large is $Y_n = X_n\ln\left(\frac{1}{X_n}\right)$? | Since $X_n$ are positive random variables, we do not need the absolute value. We have
$$\{\mathbb{E}X_n\}=O(a_n) \implies \lim_{n\to \infty}\frac{\mathbb{E}X_n}{a_n} < K \in \mathbb R_{++}$$
Then, since also
$$a_n \to 0 \implies \mathbb{E}X_n \to 0 \implies X_n \to 0,\;\;\; n\to \infty$$
since they are positive r.v's... | If $\mathbb{E}|X_n|=O(a_n)$, how large is $Y_n = X_n\ln\left(\frac{1}{X_n}\right)$? | Since $X_n$ are positive random variables, we do not need the absolute value. We have
$$\{\mathbb{E}X_n\}=O(a_n) \implies \lim_{n\to \infty}\frac{\mathbb{E}X_n}{a_n} < K \in \mathbb R_{++}$$
Then, sin | If $\mathbb{E}|X_n|=O(a_n)$, how large is $Y_n = X_n\ln\left(\frac{1}{X_n}\right)$?
Since $X_n$ are positive random variables, we do not need the absolute value. We have
$$\{\mathbb{E}X_n\}=O(a_n) \implies \lim_{n\to \infty}\frac{\mathbb{E}X_n}{a_n} < K \in \mathbb R_{++}$$
Then, since also
$$a_n \to 0 \implies \math... | If $\mathbb{E}|X_n|=O(a_n)$, how large is $Y_n = X_n\ln\left(\frac{1}{X_n}\right)$?
Since $X_n$ are positive random variables, we do not need the absolute value. We have
$$\{\mathbb{E}X_n\}=O(a_n) \implies \lim_{n\to \infty}\frac{\mathbb{E}X_n}{a_n} < K \in \mathbb R_{++}$$
Then, sin |
36,278 | Is testing model assumptions considered p-hacking/fishing? | It's not quite the same thing in the sense that the practice of testing whether assumptions were violated was originally intended to make sure an appropriate analysis was done, but as it turns out, it does have some of the same consequences (see e.g. this question). But it is in a milder form than the more extreme vari... | Is testing model assumptions considered p-hacking/fishing? | It's not quite the same thing in the sense that the practice of testing whether assumptions were violated was originally intended to make sure an appropriate analysis was done, but as it turns out, it | Is testing model assumptions considered p-hacking/fishing?
It's not quite the same thing in the sense that the practice of testing whether assumptions were violated was originally intended to make sure an appropriate analysis was done, but as it turns out, it does have some of the same consequences (see e.g. this quest... | Is testing model assumptions considered p-hacking/fishing?
It's not quite the same thing in the sense that the practice of testing whether assumptions were violated was originally intended to make sure an appropriate analysis was done, but as it turns out, it |
36,279 | Is testing model assumptions considered p-hacking/fishing? | I do not believe that checking the assumptions of any model qualifies as p-hacking /fishing. In the first article, the author is talking about analysts who are repeatedly performing analyses on a data set and only reporting the best result. In other words, they are purposely portraying a biased picture of what is happe... | Is testing model assumptions considered p-hacking/fishing? | I do not believe that checking the assumptions of any model qualifies as p-hacking /fishing. In the first article, the author is talking about analysts who are repeatedly performing analyses on a data | Is testing model assumptions considered p-hacking/fishing?
I do not believe that checking the assumptions of any model qualifies as p-hacking /fishing. In the first article, the author is talking about analysts who are repeatedly performing analyses on a data set and only reporting the best result. In other words, they... | Is testing model assumptions considered p-hacking/fishing?
I do not believe that checking the assumptions of any model qualifies as p-hacking /fishing. In the first article, the author is talking about analysts who are repeatedly performing analyses on a data |
36,280 | How to report a linear mixed effects model for those who are unfamiliar and skeptical? | I partly take side with the reviewer on this one. You are interested in the effect of your parameter of interest — given the rest of the model. It is hard to interpret the results and to check the the validity of the model if you only report a single parameter of interest.
I would provide:
the formula of your model
be... | How to report a linear mixed effects model for those who are unfamiliar and skeptical? | I partly take side with the reviewer on this one. You are interested in the effect of your parameter of interest — given the rest of the model. It is hard to interpret the results and to check the the | How to report a linear mixed effects model for those who are unfamiliar and skeptical?
I partly take side with the reviewer on this one. You are interested in the effect of your parameter of interest — given the rest of the model. It is hard to interpret the results and to check the the validity of the model if you onl... | How to report a linear mixed effects model for those who are unfamiliar and skeptical?
I partly take side with the reviewer on this one. You are interested in the effect of your parameter of interest — given the rest of the model. It is hard to interpret the results and to check the the |
36,281 | Expressing the LASSO regression constraint via the penalty parameter | The answer to your question follows from consideration of Lagrangian duality. This is worked in the post which I consider to be a duplicate in my comment to OP's post. In what follows, I work out what I find to be a more insightful derivation.
When we're solving a lasso, really, we're trying trying to jointly minimize ... | Expressing the LASSO regression constraint via the penalty parameter | The answer to your question follows from consideration of Lagrangian duality. This is worked in the post which I consider to be a duplicate in my comment to OP's post. In what follows, I work out what | Expressing the LASSO regression constraint via the penalty parameter
The answer to your question follows from consideration of Lagrangian duality. This is worked in the post which I consider to be a duplicate in my comment to OP's post. In what follows, I work out what I find to be a more insightful derivation.
When we... | Expressing the LASSO regression constraint via the penalty parameter
The answer to your question follows from consideration of Lagrangian duality. This is worked in the post which I consider to be a duplicate in my comment to OP's post. In what follows, I work out what |
36,282 | What is .hat in regression output | Those would be the diagonal elements of the hat-matrix which describe the leverage each point has on its fitted values.
If one fits $\vec{Y} = \mathbf{X} \vec{\beta} + \vec{\epsilon}$ then $\mathbf{H} = \mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T$.
In this example:
$$\begin{pmatrix}Y_1\\
\vdots\\
Y_{32}\end{pma... | What is .hat in regression output | Those would be the diagonal elements of the hat-matrix which describe the leverage each point has on its fitted values.
If one fits $\vec{Y} = \mathbf{X} \vec{\beta} + \vec{\epsilon}$ then $\mathbf{H} | What is .hat in regression output
Those would be the diagonal elements of the hat-matrix which describe the leverage each point has on its fitted values.
If one fits $\vec{Y} = \mathbf{X} \vec{\beta} + \vec{\epsilon}$ then $\mathbf{H} = \mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T$.
In this example:
$$\begin{pma... | What is .hat in regression output
Those would be the diagonal elements of the hat-matrix which describe the leverage each point has on its fitted values.
If one fits $\vec{Y} = \mathbf{X} \vec{\beta} + \vec{\epsilon}$ then $\mathbf{H} |
36,283 | Compressed Sensing relationship to L1 Regularization | There is essentially no difference. It's just statistician's terminology vs electrical engineer's terminology.
Compressed sensing (more precisely, basis pursuit denoising [1]) is this problem:
$\text{arg min}_x \frac{1}{2}\|Ax - b\| + \lambda \|x\|_1$
while the Lasso[2] is this problem
$\text{arg min}_{\beta} \frac{1}... | Compressed Sensing relationship to L1 Regularization | There is essentially no difference. It's just statistician's terminology vs electrical engineer's terminology.
Compressed sensing (more precisely, basis pursuit denoising [1]) is this problem:
$\text | Compressed Sensing relationship to L1 Regularization
There is essentially no difference. It's just statistician's terminology vs electrical engineer's terminology.
Compressed sensing (more precisely, basis pursuit denoising [1]) is this problem:
$\text{arg min}_x \frac{1}{2}\|Ax - b\| + \lambda \|x\|_1$
while the Lass... | Compressed Sensing relationship to L1 Regularization
There is essentially no difference. It's just statistician's terminology vs electrical engineer's terminology.
Compressed sensing (more precisely, basis pursuit denoising [1]) is this problem:
$\text |
36,284 | CLT with random variables that are not integrable | Among the many ways to solve this one, constructing the sequence by perturbing a standard Normal variable seems like the simplest and most elegant.
At the end I comment on the connection with the Central Limit Theorem.
Characteristic Functions
Allow me a digression before I present a solution. The inspiration for the... | CLT with random variables that are not integrable | Among the many ways to solve this one, constructing the sequence by perturbing a standard Normal variable seems like the simplest and most elegant.
At the end I comment on the connection with the Cent | CLT with random variables that are not integrable
Among the many ways to solve this one, constructing the sequence by perturbing a standard Normal variable seems like the simplest and most elegant.
At the end I comment on the connection with the Central Limit Theorem.
Characteristic Functions
Allow me a digression bef... | CLT with random variables that are not integrable
Among the many ways to solve this one, constructing the sequence by perturbing a standard Normal variable seems like the simplest and most elegant.
At the end I comment on the connection with the Cent |
36,285 | Importance sampling: unbiased estimator of the normalizing constant | Why the author takes $\mathfrak{Z}=∫φ(x)dx$?
Since $p$ is a density, its integral is equal to $1$. If $\mathfrak{Z}$ is the normalising constant of $\varphi$, it has to satisfy
$$\int p(x)\text{d}x=\int \frac{\varphi(x)}{\mathfrak{Z}}\text{d}x=1$$
2- I'm not able to prove mathematically that why $\mathbb{E}[\hat{\ma... | Importance sampling: unbiased estimator of the normalizing constant | Why the author takes $\mathfrak{Z}=∫φ(x)dx$?
Since $p$ is a density, its integral is equal to $1$. If $\mathfrak{Z}$ is the normalising constant of $\varphi$, it has to satisfy
$$\int p(x)\text{d}x= | Importance sampling: unbiased estimator of the normalizing constant
Why the author takes $\mathfrak{Z}=∫φ(x)dx$?
Since $p$ is a density, its integral is equal to $1$. If $\mathfrak{Z}$ is the normalising constant of $\varphi$, it has to satisfy
$$\int p(x)\text{d}x=\int \frac{\varphi(x)}{\mathfrak{Z}}\text{d}x=1$$
2... | Importance sampling: unbiased estimator of the normalizing constant
Why the author takes $\mathfrak{Z}=∫φ(x)dx$?
Since $p$ is a density, its integral is equal to $1$. If $\mathfrak{Z}$ is the normalising constant of $\varphi$, it has to satisfy
$$\int p(x)\text{d}x= |
36,286 | What is the mean and variance of the median of a set of i.i.d normal random variables? | The median is the central order statistic when the number of observations is odd. If $n$ is even then the median is either an order statistic, or the mean of 2 order statistics (or something else) depending on which definition of median you use.
So the exact distribution of the median can be worked out based on the di... | What is the mean and variance of the median of a set of i.i.d normal random variables? | The median is the central order statistic when the number of observations is odd. If $n$ is even then the median is either an order statistic, or the mean of 2 order statistics (or something else) de | What is the mean and variance of the median of a set of i.i.d normal random variables?
The median is the central order statistic when the number of observations is odd. If $n$ is even then the median is either an order statistic, or the mean of 2 order statistics (or something else) depending on which definition of me... | What is the mean and variance of the median of a set of i.i.d normal random variables?
The median is the central order statistic when the number of observations is odd. If $n$ is even then the median is either an order statistic, or the mean of 2 order statistics (or something else) de |
36,287 | Meaning of proper prior | A prior distribution that integrates to 1 is a proper prior, by contrast with an improper prior which doesn't.
For example, consider estimation of the mean, $\mu$ in a normal distribution. the following two prior distributions:
$\qquad f(\mu) = N(\mu_0,\tau^2)\,,\: -\infty<\mu<\infty$
$\qquad f(\mu) \propto c\,,\qquad\... | Meaning of proper prior | A prior distribution that integrates to 1 is a proper prior, by contrast with an improper prior which doesn't.
For example, consider estimation of the mean, $\mu$ in a normal distribution. the followi | Meaning of proper prior
A prior distribution that integrates to 1 is a proper prior, by contrast with an improper prior which doesn't.
For example, consider estimation of the mean, $\mu$ in a normal distribution. the following two prior distributions:
$\qquad f(\mu) = N(\mu_0,\tau^2)\,,\: -\infty<\mu<\infty$
$\qquad f(... | Meaning of proper prior
A prior distribution that integrates to 1 is a proper prior, by contrast with an improper prior which doesn't.
For example, consider estimation of the mean, $\mu$ in a normal distribution. the followi |
36,288 | How does Python Scikit Learn handle linear separation problem in logistic regression? | Yes, sklearn.linear_model.LogisticRegression uses penalized logistic regression which "solves" the problem of perfect separation. If you set C to something too large, you might still end up with bad results, though. | How does Python Scikit Learn handle linear separation problem in logistic regression? | Yes, sklearn.linear_model.LogisticRegression uses penalized logistic regression which "solves" the problem of perfect separation. If you set C to something too large, you might still end up with bad r | How does Python Scikit Learn handle linear separation problem in logistic regression?
Yes, sklearn.linear_model.LogisticRegression uses penalized logistic regression which "solves" the problem of perfect separation. If you set C to something too large, you might still end up with bad results, though. | How does Python Scikit Learn handle linear separation problem in logistic regression?
Yes, sklearn.linear_model.LogisticRegression uses penalized logistic regression which "solves" the problem of perfect separation. If you set C to something too large, you might still end up with bad r |
36,289 | What distinction is there between statistical inference and causal inference? | Causal inference is the process of ascribing causal relationships to associations between variables. Statistical inference is the process of using statistical methods to characterize the association between variables. Causality is at the root of scientific explanation which is considered to be causal explanation. Ho... | What distinction is there between statistical inference and causal inference? | Causal inference is the process of ascribing causal relationships to associations between variables. Statistical inference is the process of using statistical methods to characterize the association | What distinction is there between statistical inference and causal inference?
Causal inference is the process of ascribing causal relationships to associations between variables. Statistical inference is the process of using statistical methods to characterize the association between variables. Causality is at the ro... | What distinction is there between statistical inference and causal inference?
Causal inference is the process of ascribing causal relationships to associations between variables. Statistical inference is the process of using statistical methods to characterize the association |
36,290 | What distinction is there between statistical inference and causal inference? | "Causal inference" mean reasoning about causation, whereas "statistical inference" means reasoning with statistics (it's more or less synonymous with the word "statistics" itself). So, causal inference is a subset of statistical inference, except that you can do some causal reasoning without statistics per se (e.g., if... | What distinction is there between statistical inference and causal inference? | "Causal inference" mean reasoning about causation, whereas "statistical inference" means reasoning with statistics (it's more or less synonymous with the word "statistics" itself). So, causal inferenc | What distinction is there between statistical inference and causal inference?
"Causal inference" mean reasoning about causation, whereas "statistical inference" means reasoning with statistics (it's more or less synonymous with the word "statistics" itself). So, causal inference is a subset of statistical inference, ex... | What distinction is there between statistical inference and causal inference?
"Causal inference" mean reasoning about causation, whereas "statistical inference" means reasoning with statistics (it's more or less synonymous with the word "statistics" itself). So, causal inferenc |
36,291 | What distinction is there between statistical inference and causal inference? | Causal inference uses techniques like
matching before fitting statistical models. In other words, causal inference puts more emphasis on research design, while statistical inference puts more emphasis on the mathematical/computational part. | What distinction is there between statistical inference and causal inference? | Causal inference uses techniques like
matching before fitting statistical models. In other words, causal inference puts more emphasis on research design, while statistical inference puts more emphasi | What distinction is there between statistical inference and causal inference?
Causal inference uses techniques like
matching before fitting statistical models. In other words, causal inference puts more emphasis on research design, while statistical inference puts more emphasis on the mathematical/computational part. | What distinction is there between statistical inference and causal inference?
Causal inference uses techniques like
matching before fitting statistical models. In other words, causal inference puts more emphasis on research design, while statistical inference puts more emphasi |
36,292 | Characteristic Function of a Compound Poisson Process | I was missing the knowledge of the exponential series:
$$
e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots
$$
I also made a mistake when I separated the uniform in the expectation. Fixing these problems:
$$
\begin{align}
\mathbb{E}(e^{iuY_1})&=\sum_nP(N=n)\mathbb{E}(e^{iuY_1}\mid N=n)\\
&=\sum_nP(N=n)\prod_{... | Characteristic Function of a Compound Poisson Process | I was missing the knowledge of the exponential series:
$$
e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots
$$
I also made a mistake when I separated the uniform in the expectation. Fixing t | Characteristic Function of a Compound Poisson Process
I was missing the knowledge of the exponential series:
$$
e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots
$$
I also made a mistake when I separated the uniform in the expectation. Fixing these problems:
$$
\begin{align}
\mathbb{E}(e^{iuY_1})&=\sum_nP(N=n... | Characteristic Function of a Compound Poisson Process
I was missing the knowledge of the exponential series:
$$
e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots
$$
I also made a mistake when I separated the uniform in the expectation. Fixing t |
36,293 | Characteristic Function of a Compound Poisson Process | Here's another approach that uses a common trick with characteristic functions to avoid having to work out the sums / integrals.
I'll set $\lambda = 1$ without loss of generality, it simplifies notation and can be put back in in obvious ways in what follows. This means all the "$\lambda$"s below are not related to th... | Characteristic Function of a Compound Poisson Process | Here's another approach that uses a common trick with characteristic functions to avoid having to work out the sums / integrals.
I'll set $\lambda = 1$ without loss of generality, it simplifies notat | Characteristic Function of a Compound Poisson Process
Here's another approach that uses a common trick with characteristic functions to avoid having to work out the sums / integrals.
I'll set $\lambda = 1$ without loss of generality, it simplifies notation and can be put back in in obvious ways in what follows. This ... | Characteristic Function of a Compound Poisson Process
Here's another approach that uses a common trick with characteristic functions to avoid having to work out the sums / integrals.
I'll set $\lambda = 1$ without loss of generality, it simplifies notat |
36,294 | Is there a multi-Gaussian version of the Mahalanobis distance ? | After a little research, I found what I was looking for, a paper called "Deriving cluster analytic distance functions from gaussian mixture models" which proposes an extension of the Mahalanobis distance in the context of multi-modal data (a GMM representation) using Fisher Kernel method and other techniques. | Is there a multi-Gaussian version of the Mahalanobis distance ? | After a little research, I found what I was looking for, a paper called "Deriving cluster analytic distance functions from gaussian mixture models" which proposes an extension of the Mahalanobis dista | Is there a multi-Gaussian version of the Mahalanobis distance ?
After a little research, I found what I was looking for, a paper called "Deriving cluster analytic distance functions from gaussian mixture models" which proposes an extension of the Mahalanobis distance in the context of multi-modal data (a GMM representa... | Is there a multi-Gaussian version of the Mahalanobis distance ?
After a little research, I found what I was looking for, a paper called "Deriving cluster analytic distance functions from gaussian mixture models" which proposes an extension of the Mahalanobis dista |
36,295 | Is there a multi-Gaussian version of the Mahalanobis distance ? | I know this is several years ago, but I wanted to point out that it is possible (given some data) to estimate the Kullback-leibler divergence between two GMMs. Depending on your goal, this may prove a very elegant approach.
See:
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEw... | Is there a multi-Gaussian version of the Mahalanobis distance ? | I know this is several years ago, but I wanted to point out that it is possible (given some data) to estimate the Kullback-leibler divergence between two GMMs. Depending on your goal, this may prove | Is there a multi-Gaussian version of the Mahalanobis distance ?
I know this is several years ago, but I wanted to point out that it is possible (given some data) to estimate the Kullback-leibler divergence between two GMMs. Depending on your goal, this may prove a very elegant approach.
See:
https://www.google.com/url... | Is there a multi-Gaussian version of the Mahalanobis distance ?
I know this is several years ago, but I wanted to point out that it is possible (given some data) to estimate the Kullback-leibler divergence between two GMMs. Depending on your goal, this may prove |
36,296 | Gaussian processes with finite sampling area | I think your two questions nail the issue down. It sounds like you can use GPs for some part of the problem but you might need to do more. To explain the issues I see, I will first translate my understanding of your problem into more mathematical language:
The problem
You are interested in some physical quantity $f(x... | Gaussian processes with finite sampling area | I think your two questions nail the issue down. It sounds like you can use GPs for some part of the problem but you might need to do more. To explain the issues I see, I will first translate my unders | Gaussian processes with finite sampling area
I think your two questions nail the issue down. It sounds like you can use GPs for some part of the problem but you might need to do more. To explain the issues I see, I will first translate my understanding of your problem into more mathematical language:
The problem
You ... | Gaussian processes with finite sampling area
I think your two questions nail the issue down. It sounds like you can use GPs for some part of the problem but you might need to do more. To explain the issues I see, I will first translate my unders |
36,297 | Gaussian processes with finite sampling area | There is a topic in geostatistics called Exact Downscaling. The main goal here is to estimate a property at a smaller scale than the observations. Also these observations may or may not be overlapped (does not really matter). Please take a look to this paper:
http://www.ccgalberta.com/ccgresources/report07/2005-101-exa... | Gaussian processes with finite sampling area | There is a topic in geostatistics called Exact Downscaling. The main goal here is to estimate a property at a smaller scale than the observations. Also these observations may or may not be overlapped | Gaussian processes with finite sampling area
There is a topic in geostatistics called Exact Downscaling. The main goal here is to estimate a property at a smaller scale than the observations. Also these observations may or may not be overlapped (does not really matter). Please take a look to this paper:
http://www.ccga... | Gaussian processes with finite sampling area
There is a topic in geostatistics called Exact Downscaling. The main goal here is to estimate a property at a smaller scale than the observations. Also these observations may or may not be overlapped |
36,298 | Is PCA a non-linear transform? | I think the confusion is due to what exactly is meant here to be linear or non-linear.
Using the notation of your quote, operation $w(X)$ maps a data matrix $X$ into a projector $P_k$ on the first $k$ principal axes of $X$. Let us be completely clear about the notation here; for simplicity let us fix $k=1$ and assume t... | Is PCA a non-linear transform? | I think the confusion is due to what exactly is meant here to be linear or non-linear.
Using the notation of your quote, operation $w(X)$ maps a data matrix $X$ into a projector $P_k$ on the first $k$ | Is PCA a non-linear transform?
I think the confusion is due to what exactly is meant here to be linear or non-linear.
Using the notation of your quote, operation $w(X)$ maps a data matrix $X$ into a projector $P_k$ on the first $k$ principal axes of $X$. Let us be completely clear about the notation here; for simplicit... | Is PCA a non-linear transform?
I think the confusion is due to what exactly is meant here to be linear or non-linear.
Using the notation of your quote, operation $w(X)$ maps a data matrix $X$ into a projector $P_k$ on the first $k$ |
36,299 | Neuron saturation occurs only in last layer or all layers? | It seems to me the author didn't mean that it is the only reason for learning slowdown.
Surely sigmoid activation functions in hidden layers are likely to cause vanishing gradients, but for sigmoid in the output layer, it can be avoided by using the cross-entropy loss.
I think the discussions about output layer and s... | Neuron saturation occurs only in last layer or all layers? | It seems to me the author didn't mean that it is the only reason for learning slowdown.
Surely sigmoid activation functions in hidden layers are likely to cause vanishing gradients, but for sigmoid | Neuron saturation occurs only in last layer or all layers?
It seems to me the author didn't mean that it is the only reason for learning slowdown.
Surely sigmoid activation functions in hidden layers are likely to cause vanishing gradients, but for sigmoid in the output layer, it can be avoided by using the cross-ent... | Neuron saturation occurs only in last layer or all layers?
It seems to me the author didn't mean that it is the only reason for learning slowdown.
Surely sigmoid activation functions in hidden layers are likely to cause vanishing gradients, but for sigmoid |
36,300 | Neuron saturation occurs only in last layer or all layers? | From what I understand, using the cross entropy cost function instead of quadratic cost function will only help you for avoiding the vanishing gradients because of the $\sigma'$ term in the output layer. If we look at the backprop equations, we see that the $\sigma'$ terms are multiplied for gradient computations for e... | Neuron saturation occurs only in last layer or all layers? | From what I understand, using the cross entropy cost function instead of quadratic cost function will only help you for avoiding the vanishing gradients because of the $\sigma'$ term in the output lay | Neuron saturation occurs only in last layer or all layers?
From what I understand, using the cross entropy cost function instead of quadratic cost function will only help you for avoiding the vanishing gradients because of the $\sigma'$ term in the output layer. If we look at the backprop equations, we see that the $\s... | Neuron saturation occurs only in last layer or all layers?
From what I understand, using the cross entropy cost function instead of quadratic cost function will only help you for avoiding the vanishing gradients because of the $\sigma'$ term in the output lay |
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