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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37,001 | Comparing a sample distribution to an implied distribution | Look at the ks.test function for one option of comparing data to a given continuous distribution (with predefined parameters). Look at the vcd package for tools to examine discrete distributions. | Comparing a sample distribution to an implied distribution | Look at the ks.test function for one option of comparing data to a given continuous distribution (with predefined parameters). Look at the vcd package for tools to examine discrete distributions. | Comparing a sample distribution to an implied distribution
Look at the ks.test function for one option of comparing data to a given continuous distribution (with predefined parameters). Look at the vcd package for tools to examine discrete distributions. | Comparing a sample distribution to an implied distribution
Look at the ks.test function for one option of comparing data to a given continuous distribution (with predefined parameters). Look at the vcd package for tools to examine discrete distributions. |
37,002 | Log-likelihood of multivariate Poisson distribution | Well, the $\log( y_{\bf t}!)$ terms don't involve ${\boldsymbol \theta}$, so forget about them. Multivariate derivatives are just concatenations of univariate partial derivatives.
By linearity, the elements of the gradient vector are
$$ \frac{ \partial \ell( {\boldsymbol \theta} )}{ \partial \theta_{i}}
= \sum_{ {\b... | Log-likelihood of multivariate Poisson distribution | Well, the $\log( y_{\bf t}!)$ terms don't involve ${\boldsymbol \theta}$, so forget about them. Multivariate derivatives are just concatenations of univariate partial derivatives.
By linearity, the e | Log-likelihood of multivariate Poisson distribution
Well, the $\log( y_{\bf t}!)$ terms don't involve ${\boldsymbol \theta}$, so forget about them. Multivariate derivatives are just concatenations of univariate partial derivatives.
By linearity, the elements of the gradient vector are
$$ \frac{ \partial \ell( {\boldsy... | Log-likelihood of multivariate Poisson distribution
Well, the $\log( y_{\bf t}!)$ terms don't involve ${\boldsymbol \theta}$, so forget about them. Multivariate derivatives are just concatenations of univariate partial derivatives.
By linearity, the e |
37,003 | Poisson with an autoregressive term | I think you are looking for the model in Brandt et al. (2000) there called PEWMA, after the forecast function. R code to fit it is available here.
The paper also has some general discussion of possible conditionally Poisson AR models. Fro more of that, chapter 7 of Cameron and Trivedi (1998) is useful. | Poisson with an autoregressive term | I think you are looking for the model in Brandt et al. (2000) there called PEWMA, after the forecast function. R code to fit it is available here.
The paper also has some general discussion of poss | Poisson with an autoregressive term
I think you are looking for the model in Brandt et al. (2000) there called PEWMA, after the forecast function. R code to fit it is available here.
The paper also has some general discussion of possible conditionally Poisson AR models. Fro more of that, chapter 7 of Cameron and Tr... | Poisson with an autoregressive term
I think you are looking for the model in Brandt et al. (2000) there called PEWMA, after the forecast function. R code to fit it is available here.
The paper also has some general discussion of poss |
37,004 | Poisson with an autoregressive term | Have you considered a Transfer Function between N and E and your other covariates which could encode changes in parameters over time , changes in error variance over time , any necessaery autoregressive structure possibly proxying unspecified seasonal drivers and other omitted structure like level shifts, seasonal puls... | Poisson with an autoregressive term | Have you considered a Transfer Function between N and E and your other covariates which could encode changes in parameters over time , changes in error variance over time , any necessaery autoregressi | Poisson with an autoregressive term
Have you considered a Transfer Function between N and E and your other covariates which could encode changes in parameters over time , changes in error variance over time , any necessaery autoregressive structure possibly proxying unspecified seasonal drivers and other omitted struct... | Poisson with an autoregressive term
Have you considered a Transfer Function between N and E and your other covariates which could encode changes in parameters over time , changes in error variance over time , any necessaery autoregressi |
37,005 | Advice on scientifically sound scale construction | i'm assuming that the purpose of your analysis is to obtain evidence for the validity of your scale/instrument. so, first of all, your instrument was designed based on 4 hypothesized constructs, therefore, you should approach this using confirmatory factor analysis (CFA). exploratory factor analysis (EFA) is appropriat... | Advice on scientifically sound scale construction | i'm assuming that the purpose of your analysis is to obtain evidence for the validity of your scale/instrument. so, first of all, your instrument was designed based on 4 hypothesized constructs, there | Advice on scientifically sound scale construction
i'm assuming that the purpose of your analysis is to obtain evidence for the validity of your scale/instrument. so, first of all, your instrument was designed based on 4 hypothesized constructs, therefore, you should approach this using confirmatory factor analysis (CFA... | Advice on scientifically sound scale construction
i'm assuming that the purpose of your analysis is to obtain evidence for the validity of your scale/instrument. so, first of all, your instrument was designed based on 4 hypothesized constructs, there |
37,006 | Advice on scientifically sound scale construction | A tough situation. Factors 6, 4, and 7 seem fairly robustly measured, but not the others, and I bet internal consistency is going to be low for factors 1, 3, and 5. Is it at all possible to assess reliability through some other method, such as interrater rel.? Or to assess validity through some other method than con... | Advice on scientifically sound scale construction | A tough situation. Factors 6, 4, and 7 seem fairly robustly measured, but not the others, and I bet internal consistency is going to be low for factors 1, 3, and 5. Is it at all possible to assess r | Advice on scientifically sound scale construction
A tough situation. Factors 6, 4, and 7 seem fairly robustly measured, but not the others, and I bet internal consistency is going to be low for factors 1, 3, and 5. Is it at all possible to assess reliability through some other method, such as interrater rel.? Or to ... | Advice on scientifically sound scale construction
A tough situation. Factors 6, 4, and 7 seem fairly robustly measured, but not the others, and I bet internal consistency is going to be low for factors 1, 3, and 5. Is it at all possible to assess r |
37,007 | Math behind multivariate testing for website optimization | This Microsoft page has quite a few resources.
I suggest you read at least this paper from the page: "Controlled Experiments on the Web: Survey and Practical Guide." It'll give you some starting point about the metrics to measure and convey, things to consider for online experiments (including designing elements of web... | Math behind multivariate testing for website optimization | This Microsoft page has quite a few resources.
I suggest you read at least this paper from the page: "Controlled Experiments on the Web: Survey and Practical Guide." It'll give you some starting point | Math behind multivariate testing for website optimization
This Microsoft page has quite a few resources.
I suggest you read at least this paper from the page: "Controlled Experiments on the Web: Survey and Practical Guide." It'll give you some starting point about the metrics to measure and convey, things to consider f... | Math behind multivariate testing for website optimization
This Microsoft page has quite a few resources.
I suggest you read at least this paper from the page: "Controlled Experiments on the Web: Survey and Practical Guide." It'll give you some starting point |
37,008 | Math behind multivariate testing for website optimization | These lectures notes are more about optimizing online advertising than optimizing a website, but the references there (especially lecture 6) might put you in the right direction.
http://www.stanford.edu/class/msande239/
I hope it helps. | Math behind multivariate testing for website optimization | These lectures notes are more about optimizing online advertising than optimizing a website, but the references there (especially lecture 6) might put you in the right direction.
http://www.stanford | Math behind multivariate testing for website optimization
These lectures notes are more about optimizing online advertising than optimizing a website, but the references there (especially lecture 6) might put you in the right direction.
http://www.stanford.edu/class/msande239/
I hope it helps. | Math behind multivariate testing for website optimization
These lectures notes are more about optimizing online advertising than optimizing a website, but the references there (especially lecture 6) might put you in the right direction.
http://www.stanford |
37,009 | If the n(th) moment exists does it mean all smaller moments exist too? | I believe it follows from Hölder's inequality: | If the n(th) moment exists does it mean all smaller moments exist too? | I believe it follows from Hölder's inequality: | If the n(th) moment exists does it mean all smaller moments exist too?
I believe it follows from Hölder's inequality: | If the n(th) moment exists does it mean all smaller moments exist too?
I believe it follows from Hölder's inequality: |
37,010 | If the n(th) moment exists does it mean all smaller moments exist too? | My solution lacks rigor, but here's a rough sketch:
Instead of using the concept of a moment centered around zero, such as $E(X)$ and $E(X^2)$, use the notion of a central moment, defined as $\mu_k=E\left[(X-\mu)^k\right]$.
From your statement, we know the $r$'th moment exists, so we know $\mu_r=E(X-\mu)^r$ exists. Use... | If the n(th) moment exists does it mean all smaller moments exist too? | My solution lacks rigor, but here's a rough sketch:
Instead of using the concept of a moment centered around zero, such as $E(X)$ and $E(X^2)$, use the notion of a central moment, defined as $\mu_k=E\ | If the n(th) moment exists does it mean all smaller moments exist too?
My solution lacks rigor, but here's a rough sketch:
Instead of using the concept of a moment centered around zero, such as $E(X)$ and $E(X^2)$, use the notion of a central moment, defined as $\mu_k=E\left[(X-\mu)^k\right]$.
From your statement, we k... | If the n(th) moment exists does it mean all smaller moments exist too?
My solution lacks rigor, but here's a rough sketch:
Instead of using the concept of a moment centered around zero, such as $E(X)$ and $E(X^2)$, use the notion of a central moment, defined as $\mu_k=E\ |
37,011 | Poisson deviance - and what about zero observed values? | Based on Cardinals' comment:
$\lim\limits_{x \rightarrow 0} \ x \log x = \lim\limits_{x \rightarrow 0} \frac{ \log x}{1 \over x} = \lim\limits_{x \rightarrow 0} \frac{1 \over x}{-1 \over x^2} = \lim\limits_{x \rightarrow 0} -x = 0$ | Poisson deviance - and what about zero observed values? | Based on Cardinals' comment:
$\lim\limits_{x \rightarrow 0} \ x \log x = \lim\limits_{x \rightarrow 0} \frac{ \log x}{1 \over x} = \lim\limits_{x \rightarrow 0} \frac{1 \over x}{-1 \over x^2} = \lim\l | Poisson deviance - and what about zero observed values?
Based on Cardinals' comment:
$\lim\limits_{x \rightarrow 0} \ x \log x = \lim\limits_{x \rightarrow 0} \frac{ \log x}{1 \over x} = \lim\limits_{x \rightarrow 0} \frac{1 \over x}{-1 \over x^2} = \lim\limits_{x \rightarrow 0} -x = 0$ | Poisson deviance - and what about zero observed values?
Based on Cardinals' comment:
$\lim\limits_{x \rightarrow 0} \ x \log x = \lim\limits_{x \rightarrow 0} \frac{ \log x}{1 \over x} = \lim\limits_{x \rightarrow 0} \frac{1 \over x}{-1 \over x^2} = \lim\l |
37,012 | Central limit theorem for sample quantiles | As @whuber said in a comment, the formula you cite for the variance of a sample quantile is correct, as is that you give for the sample median in the normal case. The sample mean and sample median have different distributions, and in fact the median has higher variance in the normal case.
In R, you could estimate th... | Central limit theorem for sample quantiles | As @whuber said in a comment, the formula you cite for the variance of a sample quantile is correct, as is that you give for the sample median in the normal case. The sample mean and sample median ha | Central limit theorem for sample quantiles
As @whuber said in a comment, the formula you cite for the variance of a sample quantile is correct, as is that you give for the sample median in the normal case. The sample mean and sample median have different distributions, and in fact the median has higher variance in the... | Central limit theorem for sample quantiles
As @whuber said in a comment, the formula you cite for the variance of a sample quantile is correct, as is that you give for the sample median in the normal case. The sample mean and sample median ha |
37,013 | Central limit theorem for sample quantiles | For $n=1$ the variance is of course 1. For $n=3$ we can show $1-\sqrt{3}/\pi$. For larger $n$ your estimate $\pi/(2n)$ seems very good from my experiments: | Central limit theorem for sample quantiles | For $n=1$ the variance is of course 1. For $n=3$ we can show $1-\sqrt{3}/\pi$. For larger $n$ your estimate $\pi/(2n)$ seems very good from my experiments: | Central limit theorem for sample quantiles
For $n=1$ the variance is of course 1. For $n=3$ we can show $1-\sqrt{3}/\pi$. For larger $n$ your estimate $\pi/(2n)$ seems very good from my experiments: | Central limit theorem for sample quantiles
For $n=1$ the variance is of course 1. For $n=3$ we can show $1-\sqrt{3}/\pi$. For larger $n$ your estimate $\pi/(2n)$ seems very good from my experiments: |
37,014 | Assuming two Gaussian distributions of equal mean and variance, then how different can we expect the top X members of each group to be? | Let's look at
the top 3 of 100 Gaussians vs. the top 3 of 1000.
Real statisticians will give formulas for this and more;
for the rest of us, here's a little Monte Carlo.
The intent of the code is to give a rough
idea of the distributions of $X_{(N-2)} X_{(N-1)} X_{(N)}$;
running it gives
# top 3 of 100 Gaussians, me... | Assuming two Gaussian distributions of equal mean and variance, then how different can we expect the | Let's look at
the top 3 of 100 Gaussians vs. the top 3 of 1000.
Real statisticians will give formulas for this and more;
for the rest of us, here's a little Monte Carlo.
The intent of the code is to g | Assuming two Gaussian distributions of equal mean and variance, then how different can we expect the top X members of each group to be?
Let's look at
the top 3 of 100 Gaussians vs. the top 3 of 1000.
Real statisticians will give formulas for this and more;
for the rest of us, here's a little Monte Carlo.
The intent of ... | Assuming two Gaussian distributions of equal mean and variance, then how different can we expect the
Let's look at
the top 3 of 100 Gaussians vs. the top 3 of 1000.
Real statisticians will give formulas for this and more;
for the rest of us, here's a little Monte Carlo.
The intent of the code is to g |
37,015 | Assuming two Gaussian distributions of equal mean and variance, then how different can we expect the top X members of each group to be? | The earlier answer doesn't address the question of the gender composition of the top $5$ or top $10$ players. The analytical answer is simple, and it doesn't depend on the underlying distribution (as long as it is the same for men and for women, and continuous, and each person's ability is assumed to be independent of ... | Assuming two Gaussian distributions of equal mean and variance, then how different can we expect the | The earlier answer doesn't address the question of the gender composition of the top $5$ or top $10$ players. The analytical answer is simple, and it doesn't depend on the underlying distribution (as | Assuming two Gaussian distributions of equal mean and variance, then how different can we expect the top X members of each group to be?
The earlier answer doesn't address the question of the gender composition of the top $5$ or top $10$ players. The analytical answer is simple, and it doesn't depend on the underlying d... | Assuming two Gaussian distributions of equal mean and variance, then how different can we expect the
The earlier answer doesn't address the question of the gender composition of the top $5$ or top $10$ players. The analytical answer is simple, and it doesn't depend on the underlying distribution (as |
37,016 | Can slopes in linear regressions be used as the independent or dependent variables in other regression models? | In effect, you are proposing to use linear regression as a mathematical procedure to condense a 10-variate observation into a single variable (the slope). As such it's just another example of similar procedures like (say) using an average of repeated measurements as a regression variable or including principal compone... | Can slopes in linear regressions be used as the independent or dependent variables in other regressi | In effect, you are proposing to use linear regression as a mathematical procedure to condense a 10-variate observation into a single variable (the slope). As such it's just another example of similar | Can slopes in linear regressions be used as the independent or dependent variables in other regression models?
In effect, you are proposing to use linear regression as a mathematical procedure to condense a 10-variate observation into a single variable (the slope). As such it's just another example of similar procedur... | Can slopes in linear regressions be used as the independent or dependent variables in other regressi
In effect, you are proposing to use linear regression as a mathematical procedure to condense a 10-variate observation into a single variable (the slope). As such it's just another example of similar |
37,017 | What is the relation between statistics theory and decision theory? | Statistical decision theory is a subset of statistical theory.
Exploratory statistics is not decision theory but it is statistics.
A theory about how to make (good) decisions is certainly much wider than statistical decision theory. For example, making a good decision in society may have more relation with psychology... | What is the relation between statistics theory and decision theory? | Statistical decision theory is a subset of statistical theory.
Exploratory statistics is not decision theory but it is statistics.
A theory about how to make (good) decisions is certainly much wider | What is the relation between statistics theory and decision theory?
Statistical decision theory is a subset of statistical theory.
Exploratory statistics is not decision theory but it is statistics.
A theory about how to make (good) decisions is certainly much wider than statistical decision theory. For example, maki... | What is the relation between statistics theory and decision theory?
Statistical decision theory is a subset of statistical theory.
Exploratory statistics is not decision theory but it is statistics.
A theory about how to make (good) decisions is certainly much wider |
37,018 | How to compute prediction error from Relevance Vector Machine and Gaussian Process Regression? | Ok, here's the scoop: first off, have a look at my class in Berkeley. The slides for GP regression should be up shortly at http://alex.smola.org/teaching/berkeley2012 where I discuss this in more detail.
For a GP regression you can integrate out the latent GP and since the observation model and the underlying GP are bo... | How to compute prediction error from Relevance Vector Machine and Gaussian Process Regression? | Ok, here's the scoop: first off, have a look at my class in Berkeley. The slides for GP regression should be up shortly at http://alex.smola.org/teaching/berkeley2012 where I discuss this in more deta | How to compute prediction error from Relevance Vector Machine and Gaussian Process Regression?
Ok, here's the scoop: first off, have a look at my class in Berkeley. The slides for GP regression should be up shortly at http://alex.smola.org/teaching/berkeley2012 where I discuss this in more detail.
For a GP regression ... | How to compute prediction error from Relevance Vector Machine and Gaussian Process Regression?
Ok, here's the scoop: first off, have a look at my class in Berkeley. The slides for GP regression should be up shortly at http://alex.smola.org/teaching/berkeley2012 where I discuss this in more deta |
37,019 | Bias for kernel density estimator (periodic case) | A quick google brings up this, which indicates that when working with circular data you'll need a different definition of 'bias' for a start:
However, when using data on the circle, we cannot use distance in Euclidean space, so all differences θ − θi should be replaced by considering the angle between two vectors:
$d_... | Bias for kernel density estimator (periodic case) | A quick google brings up this, which indicates that when working with circular data you'll need a different definition of 'bias' for a start:
However, when using data on the circle, we cannot use dis | Bias for kernel density estimator (periodic case)
A quick google brings up this, which indicates that when working with circular data you'll need a different definition of 'bias' for a start:
However, when using data on the circle, we cannot use distance in Euclidean space, so all differences θ − θi should be replaced... | Bias for kernel density estimator (periodic case)
A quick google brings up this, which indicates that when working with circular data you'll need a different definition of 'bias' for a start:
However, when using data on the circle, we cannot use dis |
37,020 | Pitman's test of equality of variance and testing for regression to the mean: am I doing the right thing? | As far as I know, the Pitman test is formulated as :
$$F=\frac{SD_2}{SD_1} ~with~ SD_2 > SD_1$$
$$T=\frac{(F-1)\sqrt{n-2}}{2\sqrt{F(1-r^2)}} $$ with $r$ the correlation between the scores in sample 1 and sample 2. This is not equivalent to the formula you use and mentioned in the paper. I'm not positive about my formul... | Pitman's test of equality of variance and testing for regression to the mean: am I doing the right t | As far as I know, the Pitman test is formulated as :
$$F=\frac{SD_2}{SD_1} ~with~ SD_2 > SD_1$$
$$T=\frac{(F-1)\sqrt{n-2}}{2\sqrt{F(1-r^2)}} $$ with $r$ the correlation between the scores in sample 1 | Pitman's test of equality of variance and testing for regression to the mean: am I doing the right thing?
As far as I know, the Pitman test is formulated as :
$$F=\frac{SD_2}{SD_1} ~with~ SD_2 > SD_1$$
$$T=\frac{(F-1)\sqrt{n-2}}{2\sqrt{F(1-r^2)}} $$ with $r$ the correlation between the scores in sample 1 and sample 2. ... | Pitman's test of equality of variance and testing for regression to the mean: am I doing the right t
As far as I know, the Pitman test is formulated as :
$$F=\frac{SD_2}{SD_1} ~with~ SD_2 > SD_1$$
$$T=\frac{(F-1)\sqrt{n-2}}{2\sqrt{F(1-r^2)}} $$ with $r$ the correlation between the scores in sample 1 |
37,021 | Pitman's test of equality of variance and testing for regression to the mean: am I doing the right thing? | the short answer to your question is that with a sample size of n = 1280, a formal statistical test for equal variances is not really necessary. with that sample size, the variance 'estimates' you get are certainty-equivalents [almost certainly very close to the true population variances]. so you can just look at the t... | Pitman's test of equality of variance and testing for regression to the mean: am I doing the right t | the short answer to your question is that with a sample size of n = 1280, a formal statistical test for equal variances is not really necessary. with that sample size, the variance 'estimates' you get | Pitman's test of equality of variance and testing for regression to the mean: am I doing the right thing?
the short answer to your question is that with a sample size of n = 1280, a formal statistical test for equal variances is not really necessary. with that sample size, the variance 'estimates' you get are certainty... | Pitman's test of equality of variance and testing for regression to the mean: am I doing the right t
the short answer to your question is that with a sample size of n = 1280, a formal statistical test for equal variances is not really necessary. with that sample size, the variance 'estimates' you get |
37,022 | Is there a biglm equivalent for coxph? | Maybe take a look at the DatABEL package. I know it is used in genomic studies with large data that may be stored on the HD instead of RAM. From what I read in the help file, you can then apply different kind of model, including survival model. | Is there a biglm equivalent for coxph? | Maybe take a look at the DatABEL package. I know it is used in genomic studies with large data that may be stored on the HD instead of RAM. From what I read in the help file, you can then apply differ | Is there a biglm equivalent for coxph?
Maybe take a look at the DatABEL package. I know it is used in genomic studies with large data that may be stored on the HD instead of RAM. From what I read in the help file, you can then apply different kind of model, including survival model. | Is there a biglm equivalent for coxph?
Maybe take a look at the DatABEL package. I know it is used in genomic studies with large data that may be stored on the HD instead of RAM. From what I read in the help file, you can then apply differ |
37,023 | Fisher information and the "surface area of the typical set" | UPDATE
Tough crowd. :) For a concise account of connecting the trace of the Fisher matrix to surface area, please see section 4 ("Isoperimetric Inequalities") in the paper below. The crucial part is establishing the relation between differential entropy and the trace of the Fisher matrix, which the authors prove in ... | Fisher information and the "surface area of the typical set" | UPDATE
Tough crowd. :) For a concise account of connecting the trace of the Fisher matrix to surface area, please see section 4 ("Isoperimetric Inequalities") in the paper below. The crucial part i | Fisher information and the "surface area of the typical set"
UPDATE
Tough crowd. :) For a concise account of connecting the trace of the Fisher matrix to surface area, please see section 4 ("Isoperimetric Inequalities") in the paper below. The crucial part is establishing the relation between differential entropy an... | Fisher information and the "surface area of the typical set"
UPDATE
Tough crowd. :) For a concise account of connecting the trace of the Fisher matrix to surface area, please see section 4 ("Isoperimetric Inequalities") in the paper below. The crucial part i |
37,024 | Reliability in Elicitation Exercise | Maybe I misunderstood the question, but what you are describing sounds like a test-retest reliability study on your Q scores. You have a series of experts each going to assess a number of items or questions, at two occasions (presumably fixed in time). So, basically you can assess the temporal stability of the judgment... | Reliability in Elicitation Exercise | Maybe I misunderstood the question, but what you are describing sounds like a test-retest reliability study on your Q scores. You have a series of experts each going to assess a number of items or que | Reliability in Elicitation Exercise
Maybe I misunderstood the question, but what you are describing sounds like a test-retest reliability study on your Q scores. You have a series of experts each going to assess a number of items or questions, at two occasions (presumably fixed in time). So, basically you can assess th... | Reliability in Elicitation Exercise
Maybe I misunderstood the question, but what you are describing sounds like a test-retest reliability study on your Q scores. You have a series of experts each going to assess a number of items or que |
37,025 | Reliability in Elicitation Exercise | You would use Cronbach alpha if you do not know the true value but if you do know the true value then it seems a bit pointless to use Cronbach alpha. The use of Pearson correlation also seems a bit odd as you do not actually have a paired set of values. I would suggest using something like the Mean Squared Error (MSE).... | Reliability in Elicitation Exercise | You would use Cronbach alpha if you do not know the true value but if you do know the true value then it seems a bit pointless to use Cronbach alpha. The use of Pearson correlation also seems a bit od | Reliability in Elicitation Exercise
You would use Cronbach alpha if you do not know the true value but if you do know the true value then it seems a bit pointless to use Cronbach alpha. The use of Pearson correlation also seems a bit odd as you do not actually have a paired set of values. I would suggest using somethin... | Reliability in Elicitation Exercise
You would use Cronbach alpha if you do not know the true value but if you do know the true value then it seems a bit pointless to use Cronbach alpha. The use of Pearson correlation also seems a bit od |
37,026 | What do you think is the best goodness of fit test? | I have been told many times that the Anderson Darling (AD) test is much better than the Kolmogorov-Smirnov (KS) one because AD does a better job at fitting the tails of the distribution. KS is only good at fitting the mid-range of the distribution; but, is not better than AD even in this regard. I think the main adva... | What do you think is the best goodness of fit test? | I have been told many times that the Anderson Darling (AD) test is much better than the Kolmogorov-Smirnov (KS) one because AD does a better job at fitting the tails of the distribution. KS is only g | What do you think is the best goodness of fit test?
I have been told many times that the Anderson Darling (AD) test is much better than the Kolmogorov-Smirnov (KS) one because AD does a better job at fitting the tails of the distribution. KS is only good at fitting the mid-range of the distribution; but, is not better... | What do you think is the best goodness of fit test?
I have been told many times that the Anderson Darling (AD) test is much better than the Kolmogorov-Smirnov (KS) one because AD does a better job at fitting the tails of the distribution. KS is only g |
37,027 | What do you think is the best goodness of fit test? | I'm not sure about these tests, so this answer may be off-topic. Apologies if so. But, are you sure that you want a test? It really depends on what the purpose of the exercise is. Why are you fitting the distributions to the data, and what will you do with the fitted distributions afterward?
If you want to know ... | What do you think is the best goodness of fit test? | I'm not sure about these tests, so this answer may be off-topic. Apologies if so. But, are you sure that you want a test? It really depends on what the purpose of the exercise is. Why are you fitt | What do you think is the best goodness of fit test?
I'm not sure about these tests, so this answer may be off-topic. Apologies if so. But, are you sure that you want a test? It really depends on what the purpose of the exercise is. Why are you fitting the distributions to the data, and what will you do with the fit... | What do you think is the best goodness of fit test?
I'm not sure about these tests, so this answer may be off-topic. Apologies if so. But, are you sure that you want a test? It really depends on what the purpose of the exercise is. Why are you fitt |
37,028 | What do you think is the best goodness of fit test? | I think that my question is subsumed by this more general discussion: Motivation for Kolmogorov distance between distributions | What do you think is the best goodness of fit test? | I think that my question is subsumed by this more general discussion: Motivation for Kolmogorov distance between distributions | What do you think is the best goodness of fit test?
I think that my question is subsumed by this more general discussion: Motivation for Kolmogorov distance between distributions | What do you think is the best goodness of fit test?
I think that my question is subsumed by this more general discussion: Motivation for Kolmogorov distance between distributions |
37,029 | Why does degrees of freedom = $\frac{\operatorname{Tr}(H'H)^2}{\operatorname{Tr}(H'HH'H)}$? | There is a Wikipedia article that is more directly about the Welch-Satterthwaite approximation.
The Wikipedia article makes the following citations to original sources
Satterthwaite, F. E. (1946), "An Approximate Distribution of Estimates of Variance Components.", I Biometrics Bulletin, 2 (6): 110–114
Welch, B. L. (194... | Why does degrees of freedom = $\frac{\operatorname{Tr}(H'H)^2}{\operatorname{Tr}(H'HH'H)}$? | There is a Wikipedia article that is more directly about the Welch-Satterthwaite approximation.
The Wikipedia article makes the following citations to original sources
Satterthwaite, F. E. (1946), "An | Why does degrees of freedom = $\frac{\operatorname{Tr}(H'H)^2}{\operatorname{Tr}(H'HH'H)}$?
There is a Wikipedia article that is more directly about the Welch-Satterthwaite approximation.
The Wikipedia article makes the following citations to original sources
Satterthwaite, F. E. (1946), "An Approximate Distribution of... | Why does degrees of freedom = $\frac{\operatorname{Tr}(H'H)^2}{\operatorname{Tr}(H'HH'H)}$?
There is a Wikipedia article that is more directly about the Welch-Satterthwaite approximation.
The Wikipedia article makes the following citations to original sources
Satterthwaite, F. E. (1946), "An |
37,030 | Why multiple testing matters? | This is an interesting question, and I've thought about this as well. My current thinking is this: Hypothesis testing has to be seen within a wider research context. Generally testing a hypothesis cannot ultimately settle a scientific problem of interest. For sure 5% type I error probability is worryingly high if decis... | Why multiple testing matters? | This is an interesting question, and I've thought about this as well. My current thinking is this: Hypothesis testing has to be seen within a wider research context. Generally testing a hypothesis can | Why multiple testing matters?
This is an interesting question, and I've thought about this as well. My current thinking is this: Hypothesis testing has to be seen within a wider research context. Generally testing a hypothesis cannot ultimately settle a scientific problem of interest. For sure 5% type I error probabili... | Why multiple testing matters?
This is an interesting question, and I've thought about this as well. My current thinking is this: Hypothesis testing has to be seen within a wider research context. Generally testing a hypothesis can |
37,031 | Median of the sum vs. sum of the median for Gaussian variables | Here is a simple proof, by comparison with appropriate symmetric variables.
For a normal variable $X$, let $m$ be the median of $X^2$. The graphs show $X\sim N(4,1)$.
Now we take a variable $Y$ which is a version of $X^2$ but symmetrized from right to left about $m$:
In formulas:
$$
f_Y(y) =
\begin{cases}
f_{X^2}(y)\... | Median of the sum vs. sum of the median for Gaussian variables | Here is a simple proof, by comparison with appropriate symmetric variables.
For a normal variable $X$, let $m$ be the median of $X^2$. The graphs show $X\sim N(4,1)$.
Now we take a variable $Y$ which | Median of the sum vs. sum of the median for Gaussian variables
Here is a simple proof, by comparison with appropriate symmetric variables.
For a normal variable $X$, let $m$ be the median of $X^2$. The graphs show $X\sim N(4,1)$.
Now we take a variable $Y$ which is a version of $X^2$ but symmetrized from right to left ... | Median of the sum vs. sum of the median for Gaussian variables
Here is a simple proof, by comparison with appropriate symmetric variables.
For a normal variable $X$, let $m$ be the median of $X^2$. The graphs show $X\sim N(4,1)$.
Now we take a variable $Y$ which |
37,032 | Generalized Linear Model: Why do we insist on modeling the mean? | Generalized linear models are generalization of linear regression model. Linear regression minimizes squared error for fitting the regression line. Minimizing squared error is equivalent to maximizing Gaussian likelihood. GLMs generalize this idea to other distributions, so we are estimating conditional means of other ... | Generalized Linear Model: Why do we insist on modeling the mean? | Generalized linear models are generalization of linear regression model. Linear regression minimizes squared error for fitting the regression line. Minimizing squared error is equivalent to maximizing | Generalized Linear Model: Why do we insist on modeling the mean?
Generalized linear models are generalization of linear regression model. Linear regression minimizes squared error for fitting the regression line. Minimizing squared error is equivalent to maximizing Gaussian likelihood. GLMs generalize this idea to othe... | Generalized Linear Model: Why do we insist on modeling the mean?
Generalized linear models are generalization of linear regression model. Linear regression minimizes squared error for fitting the regression line. Minimizing squared error is equivalent to maximizing |
37,033 | Definition of heavy-tailed distribution | For a fixed $\lambda>0$, we can rewrite the product as a fraction, so we can see the complementary cdf (right tail) of $Y_\lambda\sim Exp(\lambda)$:
$$ P(Y_\lambda >x)=e^{-\lambda x}. $$
We have:
$$\lim_{x \to \infty} e^{\lambda x} \cdot P(X > x) = \lim_{x \rightarrow \infty} \frac{P(X > x)}{ P(Y_\lambda>x)} = \infty... | Definition of heavy-tailed distribution | For a fixed $\lambda>0$, we can rewrite the product as a fraction, so we can see the complementary cdf (right tail) of $Y_\lambda\sim Exp(\lambda)$:
$$ P(Y_\lambda >x)=e^{-\lambda x}. $$
We have:
$$\ | Definition of heavy-tailed distribution
For a fixed $\lambda>0$, we can rewrite the product as a fraction, so we can see the complementary cdf (right tail) of $Y_\lambda\sim Exp(\lambda)$:
$$ P(Y_\lambda >x)=e^{-\lambda x}. $$
We have:
$$\lim_{x \to \infty} e^{\lambda x} \cdot P(X > x) = \lim_{x \rightarrow \infty} \... | Definition of heavy-tailed distribution
For a fixed $\lambda>0$, we can rewrite the product as a fraction, so we can see the complementary cdf (right tail) of $Y_\lambda\sim Exp(\lambda)$:
$$ P(Y_\lambda >x)=e^{-\lambda x}. $$
We have:
$$\ |
37,034 | Mathematical notation for suppressing differentiation | I'm not aware of any standard notation for doing so on a "loss-level", in fact there's no stop_gradient function (in a mathematical sense) that would act as identity, but have a zero derivative. However, one can use standard substitution notation at the "gradient level":
$$
\left(\frac{d}{dt} f(x(s), t)\right) \Bigl|_{... | Mathematical notation for suppressing differentiation | I'm not aware of any standard notation for doing so on a "loss-level", in fact there's no stop_gradient function (in a mathematical sense) that would act as identity, but have a zero derivative. Howev | Mathematical notation for suppressing differentiation
I'm not aware of any standard notation for doing so on a "loss-level", in fact there's no stop_gradient function (in a mathematical sense) that would act as identity, but have a zero derivative. However, one can use standard substitution notation at the "gradient le... | Mathematical notation for suppressing differentiation
I'm not aware of any standard notation for doing so on a "loss-level", in fact there's no stop_gradient function (in a mathematical sense) that would act as identity, but have a zero derivative. Howev |
37,035 | How can I model negative binomial sampling with a maximum number of attempts until success? | This can be framed as a problem with censoring if you like. Given that it is possible to take a sample without finding the disease (even if it is there) you have two aspects to your inference problem: (a) estimating the probability that a person has the disease; and (b) estimating the probability that the disease is f... | How can I model negative binomial sampling with a maximum number of attempts until success? | This can be framed as a problem with censoring if you like. Given that it is possible to take a sample without finding the disease (even if it is there) you have two aspects to your inference problem | How can I model negative binomial sampling with a maximum number of attempts until success?
This can be framed as a problem with censoring if you like. Given that it is possible to take a sample without finding the disease (even if it is there) you have two aspects to your inference problem: (a) estimating the probabi... | How can I model negative binomial sampling with a maximum number of attempts until success?
This can be framed as a problem with censoring if you like. Given that it is possible to take a sample without finding the disease (even if it is there) you have two aspects to your inference problem |
37,036 | How support vectors is calculated on SVM example? | Let us use the figure on the wikipedia pare of SVM https://en.wikipedia.org/wiki/Support_vector_machine#/media/File:SVM_margin.png as a guide.
I understand that the $f(\phi(x))$ formula is for the separating hyperplane which is the red line in the figure. The formula is really $f(\phi(x))=0$.
Then $w = [4,9*4,4,0]$ a... | How support vectors is calculated on SVM example? | Let us use the figure on the wikipedia pare of SVM https://en.wikipedia.org/wiki/Support_vector_machine#/media/File:SVM_margin.png as a guide.
I understand that the $f(\phi(x))$ formula is for the se | How support vectors is calculated on SVM example?
Let us use the figure on the wikipedia pare of SVM https://en.wikipedia.org/wiki/Support_vector_machine#/media/File:SVM_margin.png as a guide.
I understand that the $f(\phi(x))$ formula is for the separating hyperplane which is the red line in the figure. The formula i... | How support vectors is calculated on SVM example?
Let us use the figure on the wikipedia pare of SVM https://en.wikipedia.org/wiki/Support_vector_machine#/media/File:SVM_margin.png as a guide.
I understand that the $f(\phi(x))$ formula is for the se |
37,037 | How support vectors is calculated on SVM example? | Regarding the comment in the question:
The real question is that: for detecting that some vector is support vector we should use which of them? 1) check the wx+b or 2) calculate Euclidean distance to hyperplane?
We can do both. This is because the support vectors, as has been already said by @Jacques Wainer, are the ... | How support vectors is calculated on SVM example? | Regarding the comment in the question:
The real question is that: for detecting that some vector is support vector we should use which of them? 1) check the wx+b or 2) calculate Euclidean distance to | How support vectors is calculated on SVM example?
Regarding the comment in the question:
The real question is that: for detecting that some vector is support vector we should use which of them? 1) check the wx+b or 2) calculate Euclidean distance to hyperplane?
We can do both. This is because the support vectors, as ... | How support vectors is calculated on SVM example?
Regarding the comment in the question:
The real question is that: for detecting that some vector is support vector we should use which of them? 1) check the wx+b or 2) calculate Euclidean distance to |
37,038 | AIC model averaging when models are correlated | To the best of my knowledge, such a modification of the weights in Bayesian Model Averaging to take the similarity (or other relations) between models into account does not exist in the literature. According to me, the main reason is that the problem your are raising (and that you nicely illustrated in your example) sh... | AIC model averaging when models are correlated | To the best of my knowledge, such a modification of the weights in Bayesian Model Averaging to take the similarity (or other relations) between models into account does not exist in the literature. Ac | AIC model averaging when models are correlated
To the best of my knowledge, such a modification of the weights in Bayesian Model Averaging to take the similarity (or other relations) between models into account does not exist in the literature. According to me, the main reason is that the problem your are raising (and ... | AIC model averaging when models are correlated
To the best of my knowledge, such a modification of the weights in Bayesian Model Averaging to take the similarity (or other relations) between models into account does not exist in the literature. Ac |
37,039 | Is AIC appropriate for comparing non linear models? | Yes, AIC (or AICc) are still "generally appropriate". We make the assumption of a Gaussian error (see below).
Avoid computing AIC manually if both models are of the same type, that said: Comparing across model types requires attention to detail to make sure that parameters are counted using similar rules, and that addi... | Is AIC appropriate for comparing non linear models? | Yes, AIC (or AICc) are still "generally appropriate". We make the assumption of a Gaussian error (see below).
Avoid computing AIC manually if both models are of the same type, that said: Comparing acr | Is AIC appropriate for comparing non linear models?
Yes, AIC (or AICc) are still "generally appropriate". We make the assumption of a Gaussian error (see below).
Avoid computing AIC manually if both models are of the same type, that said: Comparing across model types requires attention to detail to make sure that param... | Is AIC appropriate for comparing non linear models?
Yes, AIC (or AICc) are still "generally appropriate". We make the assumption of a Gaussian error (see below).
Avoid computing AIC manually if both models are of the same type, that said: Comparing acr |
37,040 | Understanding Bayesian Bootstrap theory | The $(K-1)$-variate Dirichlet distribution: What Rubin means here is that the Dirichlet distribution is giving a random probability vector $\boldsymbol{\pi} = (\pi_1,...,\pi_K)$ with $K$ elements, so only $K-1$ of these elements are "free variables". Since the probability values must sum to one you have the binding eq... | Understanding Bayesian Bootstrap theory | The $(K-1)$-variate Dirichlet distribution: What Rubin means here is that the Dirichlet distribution is giving a random probability vector $\boldsymbol{\pi} = (\pi_1,...,\pi_K)$ with $K$ elements, so | Understanding Bayesian Bootstrap theory
The $(K-1)$-variate Dirichlet distribution: What Rubin means here is that the Dirichlet distribution is giving a random probability vector $\boldsymbol{\pi} = (\pi_1,...,\pi_K)$ with $K$ elements, so only $K-1$ of these elements are "free variables". Since the probability values... | Understanding Bayesian Bootstrap theory
The $(K-1)$-variate Dirichlet distribution: What Rubin means here is that the Dirichlet distribution is giving a random probability vector $\boldsymbol{\pi} = (\pi_1,...,\pi_K)$ with $K$ elements, so |
37,041 | Conditional distribution of multivariate normal distribution | It looks like you're on the right track, but you're working way too hard on this one. Sooner or later algebraic and numerical mistakes will creep into even the best calculations. A good strategy is to minimize the amount calculation: the Principle of Mathematical Laziness. A key element of this principle is just-in-... | Conditional distribution of multivariate normal distribution | It looks like you're on the right track, but you're working way too hard on this one. Sooner or later algebraic and numerical mistakes will creep into even the best calculations. A good strategy is | Conditional distribution of multivariate normal distribution
It looks like you're on the right track, but you're working way too hard on this one. Sooner or later algebraic and numerical mistakes will creep into even the best calculations. A good strategy is to minimize the amount calculation: the Principle of Mathem... | Conditional distribution of multivariate normal distribution
It looks like you're on the right track, but you're working way too hard on this one. Sooner or later algebraic and numerical mistakes will creep into even the best calculations. A good strategy is |
37,042 | Correct way to combine 95% confidence interval bounds returned by a fitting routine with several measurements? | I imagine the 95% confidence intervals come from some assumptions on normality of data. Otherwise, please state how you got these CI. This implies you believe the mean of each slope (viewed as a RV) is $m_i$ with some variance $\sigma_i$
In this case you can average the slopes as you did and get the new variance of the... | Correct way to combine 95% confidence interval bounds returned by a fitting routine with several mea | I imagine the 95% confidence intervals come from some assumptions on normality of data. Otherwise, please state how you got these CI. This implies you believe the mean of each slope (viewed as a RV) i | Correct way to combine 95% confidence interval bounds returned by a fitting routine with several measurements?
I imagine the 95% confidence intervals come from some assumptions on normality of data. Otherwise, please state how you got these CI. This implies you believe the mean of each slope (viewed as a RV) is $m_i$ w... | Correct way to combine 95% confidence interval bounds returned by a fitting routine with several mea
I imagine the 95% confidence intervals come from some assumptions on normality of data. Otherwise, please state how you got these CI. This implies you believe the mean of each slope (viewed as a RV) i |
37,043 | Correct way to combine 95% confidence interval bounds returned by a fitting routine with several measurements? | I don't have much to offer in terms of methods, I think the ones presented here (esp inverse variance weighted approaches) are good ones. What I can add is a small simulation study to prove that under the assumption of Gaussian errors in the regression, this process has good enough coverage
set.seed(0)
library(tidyv... | Correct way to combine 95% confidence interval bounds returned by a fitting routine with several mea | I don't have much to offer in terms of methods, I think the ones presented here (esp inverse variance weighted approaches) are good ones. What I can add is a small simulation study to prove that unde | Correct way to combine 95% confidence interval bounds returned by a fitting routine with several measurements?
I don't have much to offer in terms of methods, I think the ones presented here (esp inverse variance weighted approaches) are good ones. What I can add is a small simulation study to prove that under the ass... | Correct way to combine 95% confidence interval bounds returned by a fitting routine with several mea
I don't have much to offer in terms of methods, I think the ones presented here (esp inverse variance weighted approaches) are good ones. What I can add is a small simulation study to prove that unde |
37,044 | Causality: Models, Reasoning and Inference, by Judea Pearl: Causal Bayesian Networks and the Truncated Factorization | $\newcommand{\doop}{\operatorname{do}}\newcommand{\op}[1]{\operatorname{#1}}$
I think your proof of the forward implication is correct. For the backward implication I may have something.
Suppose the Truncated Factorization: for all $v$ consistent with $x$,
$$P(v\mid \mathrm{do}(x))=\prod_{i\mid Vi\notin X}P(v_i \mid \m... | Causality: Models, Reasoning and Inference, by Judea Pearl: Causal Bayesian Networks and the Truncat | $\newcommand{\doop}{\operatorname{do}}\newcommand{\op}[1]{\operatorname{#1}}$
I think your proof of the forward implication is correct. For the backward implication I may have something.
Suppose the T | Causality: Models, Reasoning and Inference, by Judea Pearl: Causal Bayesian Networks and the Truncated Factorization
$\newcommand{\doop}{\operatorname{do}}\newcommand{\op}[1]{\operatorname{#1}}$
I think your proof of the forward implication is correct. For the backward implication I may have something.
Suppose the Trun... | Causality: Models, Reasoning and Inference, by Judea Pearl: Causal Bayesian Networks and the Truncat
$\newcommand{\doop}{\operatorname{do}}\newcommand{\op}[1]{\operatorname{#1}}$
I think your proof of the forward implication is correct. For the backward implication I may have something.
Suppose the T |
37,045 | Structure of Generative Adversarial Networks (GAN) for mapping a simulation model | So you want to learn a dynamics model of some system, and to improve the robustness of the model, you want to sample a wide range of inputs from a GAN.
While it's certainly possible to learn a dynamics model, I don't think it's a good idea to generate inputs exclusively with a GAN -- indeed, GANs are known for mode dro... | Structure of Generative Adversarial Networks (GAN) for mapping a simulation model | So you want to learn a dynamics model of some system, and to improve the robustness of the model, you want to sample a wide range of inputs from a GAN.
While it's certainly possible to learn a dynamic | Structure of Generative Adversarial Networks (GAN) for mapping a simulation model
So you want to learn a dynamics model of some system, and to improve the robustness of the model, you want to sample a wide range of inputs from a GAN.
While it's certainly possible to learn a dynamics model, I don't think it's a good ide... | Structure of Generative Adversarial Networks (GAN) for mapping a simulation model
So you want to learn a dynamics model of some system, and to improve the robustness of the model, you want to sample a wide range of inputs from a GAN.
While it's certainly possible to learn a dynamic |
37,046 | Gradient descent: Shouldn't step size be proportional to inverse of gradient of residual? | You want to head "downhill" -- in the direction in which the slope is decreasing most quickly. Considered in terms of the axes, the relative speeds of decrease will each be proportional to that derivative.
Which is to say, no the book is right.
How far you step in that direction is a slightly different matter.
[Howeve... | Gradient descent: Shouldn't step size be proportional to inverse of gradient of residual? | You want to head "downhill" -- in the direction in which the slope is decreasing most quickly. Considered in terms of the axes, the relative speeds of decrease will each be proportional to that deriva | Gradient descent: Shouldn't step size be proportional to inverse of gradient of residual?
You want to head "downhill" -- in the direction in which the slope is decreasing most quickly. Considered in terms of the axes, the relative speeds of decrease will each be proportional to that derivative.
Which is to say, no the ... | Gradient descent: Shouldn't step size be proportional to inverse of gradient of residual?
You want to head "downhill" -- in the direction in which the slope is decreasing most quickly. Considered in terms of the axes, the relative speeds of decrease will each be proportional to that deriva |
37,047 | Mixed Model Equations | Noting that $(A + BCD)^{-1} = A^{-1} - A^{-1}B(C^{-1} + DA^{-1}B)^{-1}DA^{-1}$ we can write
\begin{equation}
H^{-1} = (ZGZ^\top + R) = R^{-1} - R^{-1}Z(G^{-1} + Z^\top R^{-1}Z)^{-1}Z^\top R^{-1}.
\end{equation}
Using $H^{-1}$ we can write
\begin{equation}
\begin{aligned}
GZ^\top H^{-1}ZG &= GZ^\top R^{-1}ZG - GZ^\top... | Mixed Model Equations | Noting that $(A + BCD)^{-1} = A^{-1} - A^{-1}B(C^{-1} + DA^{-1}B)^{-1}DA^{-1}$ we can write
\begin{equation}
H^{-1} = (ZGZ^\top + R) = R^{-1} - R^{-1}Z(G^{-1} + Z^\top R^{-1}Z)^{-1}Z^\top R^{-1}.
\en | Mixed Model Equations
Noting that $(A + BCD)^{-1} = A^{-1} - A^{-1}B(C^{-1} + DA^{-1}B)^{-1}DA^{-1}$ we can write
\begin{equation}
H^{-1} = (ZGZ^\top + R) = R^{-1} - R^{-1}Z(G^{-1} + Z^\top R^{-1}Z)^{-1}Z^\top R^{-1}.
\end{equation}
Using $H^{-1}$ we can write
\begin{equation}
\begin{aligned}
GZ^\top H^{-1}ZG &= GZ^\... | Mixed Model Equations
Noting that $(A + BCD)^{-1} = A^{-1} - A^{-1}B(C^{-1} + DA^{-1}B)^{-1}DA^{-1}$ we can write
\begin{equation}
H^{-1} = (ZGZ^\top + R) = R^{-1} - R^{-1}Z(G^{-1} + Z^\top R^{-1}Z)^{-1}Z^\top R^{-1}.
\en |
37,048 | Conditioning in the definition of sufficient statistics | There is no impact on sufficiency of the fact that $\mathbb{P}_\theta(T=t)$ is not always positive. As indicated in my example, the statistic $$T=X_{(n)}=\max_{1\le i\le n} X_i$$ is sufficient when the $X_i$'s are uniform on $\{1,\ldots,\theta\}$. And $$\mathbb{P}_\theta(T=t)=0$$ when $\theta<t$. The joint probability ... | Conditioning in the definition of sufficient statistics | There is no impact on sufficiency of the fact that $\mathbb{P}_\theta(T=t)$ is not always positive. As indicated in my example, the statistic $$T=X_{(n)}=\max_{1\le i\le n} X_i$$ is sufficient when th | Conditioning in the definition of sufficient statistics
There is no impact on sufficiency of the fact that $\mathbb{P}_\theta(T=t)$ is not always positive. As indicated in my example, the statistic $$T=X_{(n)}=\max_{1\le i\le n} X_i$$ is sufficient when the $X_i$'s are uniform on $\{1,\ldots,\theta\}$. And $$\mathbb{P}... | Conditioning in the definition of sufficient statistics
There is no impact on sufficiency of the fact that $\mathbb{P}_\theta(T=t)$ is not always positive. As indicated in my example, the statistic $$T=X_{(n)}=\max_{1\le i\le n} X_i$$ is sufficient when th |
37,049 | Conditioning in the definition of sufficient statistics | In advanced probability texts, sufficiency is usually defined formally in terms of partitions on the sample space, and then we build up the standard definition as an implication of this for a parametric model. In any case, once you translate to the common definition, the requirement for sufficiency is that this condit... | Conditioning in the definition of sufficient statistics | In advanced probability texts, sufficiency is usually defined formally in terms of partitions on the sample space, and then we build up the standard definition as an implication of this for a parametr | Conditioning in the definition of sufficient statistics
In advanced probability texts, sufficiency is usually defined formally in terms of partitions on the sample space, and then we build up the standard definition as an implication of this for a parametric model. In any case, once you translate to the common definit... | Conditioning in the definition of sufficient statistics
In advanced probability texts, sufficiency is usually defined formally in terms of partitions on the sample space, and then we build up the standard definition as an implication of this for a parametr |
37,050 | How does random variable nesting in GAMs work (mgcv)? | There are several ways to specify this with mgcv, each yielding a different way to estimate the smoothing parameter and variance component. For the predicted random intercepts: these will not be identical between different specifications, but I do not expect substantial differences. If you are interested in interpretat... | How does random variable nesting in GAMs work (mgcv)? | There are several ways to specify this with mgcv, each yielding a different way to estimate the smoothing parameter and variance component. For the predicted random intercepts: these will not be ident | How does random variable nesting in GAMs work (mgcv)?
There are several ways to specify this with mgcv, each yielding a different way to estimate the smoothing parameter and variance component. For the predicted random intercepts: these will not be identical between different specifications, but I do not expect substan... | How does random variable nesting in GAMs work (mgcv)?
There are several ways to specify this with mgcv, each yielding a different way to estimate the smoothing parameter and variance component. For the predicted random intercepts: these will not be ident |
37,051 | Reconciling Langevin MC methods as one-step HMC versus as diffusion or brownian motion | The easiest way to understand why Langevin dynamics targets the "correct distribution" is to look at the corresponding Fokker-Planck equation.
Let me be more precise. Let us assume that our target distribution has the following density:
$\pi(x) = \frac1{Z} \exp(-U(x))$,
where $x \in \mathbb{R}^d$, $U$ is often called ... | Reconciling Langevin MC methods as one-step HMC versus as diffusion or brownian motion | The easiest way to understand why Langevin dynamics targets the "correct distribution" is to look at the corresponding Fokker-Planck equation.
Let me be more precise. Let us assume that our target dis | Reconciling Langevin MC methods as one-step HMC versus as diffusion or brownian motion
The easiest way to understand why Langevin dynamics targets the "correct distribution" is to look at the corresponding Fokker-Planck equation.
Let me be more precise. Let us assume that our target distribution has the following densi... | Reconciling Langevin MC methods as one-step HMC versus as diffusion or brownian motion
The easiest way to understand why Langevin dynamics targets the "correct distribution" is to look at the corresponding Fokker-Planck equation.
Let me be more precise. Let us assume that our target dis |
37,052 | Reconciling Langevin MC methods as one-step HMC versus as diffusion or brownian motion | Here is an elementary and not entirely rigorous "proof" that Langevin dynamics satisfies detailed balance. I only bothered considering the univariate case.
Let $\pi$ denote the distribution we are concerned with (and the distribution for which we'll prove detailed balance).
A single step of LMC is:
\begin{align}
x ... | Reconciling Langevin MC methods as one-step HMC versus as diffusion or brownian motion | Here is an elementary and not entirely rigorous "proof" that Langevin dynamics satisfies detailed balance. I only bothered considering the univariate case.
Let $\pi$ denote the distribution we are co | Reconciling Langevin MC methods as one-step HMC versus as diffusion or brownian motion
Here is an elementary and not entirely rigorous "proof" that Langevin dynamics satisfies detailed balance. I only bothered considering the univariate case.
Let $\pi$ denote the distribution we are concerned with (and the distributio... | Reconciling Langevin MC methods as one-step HMC versus as diffusion or brownian motion
Here is an elementary and not entirely rigorous "proof" that Langevin dynamics satisfies detailed balance. I only bothered considering the univariate case.
Let $\pi$ denote the distribution we are co |
37,053 | Get estimates about 'variable importance' across a large number of variables and their permutations | In my field (political science) conjoint experiments are very common. Typically the data come in the form of a forced-choice comparison — i.e., survey respondents are shown a series of comparisons between product profiles in which features of the products are randomized, and then respondents choose which product they p... | Get estimates about 'variable importance' across a large number of variables and their permutations | In my field (political science) conjoint experiments are very common. Typically the data come in the form of a forced-choice comparison — i.e., survey respondents are shown a series of comparisons bet | Get estimates about 'variable importance' across a large number of variables and their permutations
In my field (political science) conjoint experiments are very common. Typically the data come in the form of a forced-choice comparison — i.e., survey respondents are shown a series of comparisons between product profile... | Get estimates about 'variable importance' across a large number of variables and their permutations
In my field (political science) conjoint experiments are very common. Typically the data come in the form of a forced-choice comparison — i.e., survey respondents are shown a series of comparisons bet |
37,054 | Get estimates about 'variable importance' across a large number of variables and their permutations | Be cautious - most analysts do not use the bootstrap to get confidence intervals for variable importance measures, but when they do they are usually disappointed. The data, unless massive, do not contain sufficient information to tell you reliably which elements of the data are predictive, and do not have sufficient i... | Get estimates about 'variable importance' across a large number of variables and their permutations | Be cautious - most analysts do not use the bootstrap to get confidence intervals for variable importance measures, but when they do they are usually disappointed. The data, unless massive, do not con | Get estimates about 'variable importance' across a large number of variables and their permutations
Be cautious - most analysts do not use the bootstrap to get confidence intervals for variable importance measures, but when they do they are usually disappointed. The data, unless massive, do not contain sufficient info... | Get estimates about 'variable importance' across a large number of variables and their permutations
Be cautious - most analysts do not use the bootstrap to get confidence intervals for variable importance measures, but when they do they are usually disappointed. The data, unless massive, do not con |
37,055 | Get estimates about 'variable importance' across a large number of variables and their permutations | There are a whole host of options to determine variable importance. As you mentioned random forest has a variable importance metric built in. R package DALEX also has a model agnostic approach similar to the random forest importance but is model agnostic. When you say these don’t have effect sizes I would disagree and... | Get estimates about 'variable importance' across a large number of variables and their permutations | There are a whole host of options to determine variable importance. As you mentioned random forest has a variable importance metric built in. R package DALEX also has a model agnostic approach similar | Get estimates about 'variable importance' across a large number of variables and their permutations
There are a whole host of options to determine variable importance. As you mentioned random forest has a variable importance metric built in. R package DALEX also has a model agnostic approach similar to the random fores... | Get estimates about 'variable importance' across a large number of variables and their permutations
There are a whole host of options to determine variable importance. As you mentioned random forest has a variable importance metric built in. R package DALEX also has a model agnostic approach similar |
37,056 | Get estimates about 'variable importance' across a large number of variables and their permutations | Have you tried to look at Pareto-smoothed importance sampling (PSIS) by Vehtari, Gelman and Gabry?
Here's a fast recap: "Pareto-smoothed importance sampling (PSIS), a new procedure for regularizing importance weights". Look at the examples in the paper. It should be a good starting point. If it seems sounding to you th... | Get estimates about 'variable importance' across a large number of variables and their permutations | Have you tried to look at Pareto-smoothed importance sampling (PSIS) by Vehtari, Gelman and Gabry?
Here's a fast recap: "Pareto-smoothed importance sampling (PSIS), a new procedure for regularizing im | Get estimates about 'variable importance' across a large number of variables and their permutations
Have you tried to look at Pareto-smoothed importance sampling (PSIS) by Vehtari, Gelman and Gabry?
Here's a fast recap: "Pareto-smoothed importance sampling (PSIS), a new procedure for regularizing importance weights". L... | Get estimates about 'variable importance' across a large number of variables and their permutations
Have you tried to look at Pareto-smoothed importance sampling (PSIS) by Vehtari, Gelman and Gabry?
Here's a fast recap: "Pareto-smoothed importance sampling (PSIS), a new procedure for regularizing im |
37,057 | How to verify if a prediction performance improvement is significant better? | A few possibilities come to mind.
If you have multiple datasets, you could use a Diebold-Mariano test. The tag wiki contains information, as well as pointers to the original and a follow-up publication. The DM test is very commonly applied in the time series forecasting community, but there is nothing specifically "tim... | How to verify if a prediction performance improvement is significant better? | A few possibilities come to mind.
If you have multiple datasets, you could use a Diebold-Mariano test. The tag wiki contains information, as well as pointers to the original and a follow-up publicatio | How to verify if a prediction performance improvement is significant better?
A few possibilities come to mind.
If you have multiple datasets, you could use a Diebold-Mariano test. The tag wiki contains information, as well as pointers to the original and a follow-up publication. The DM test is very commonly applied in ... | How to verify if a prediction performance improvement is significant better?
A few possibilities come to mind.
If you have multiple datasets, you could use a Diebold-Mariano test. The tag wiki contains information, as well as pointers to the original and a follow-up publicatio |
37,058 | How to verify if a prediction performance improvement is significant better? | Dietterich (1998) proposes a 5x2 cross validation followed by a t-test to identify better models. When comparing models across multiple datasets, Demsar (2006) suggests the use of Wilcoxon Signed Rank test for comparing two models and Friedman test when comparing more than two models. | How to verify if a prediction performance improvement is significant better? | Dietterich (1998) proposes a 5x2 cross validation followed by a t-test to identify better models. When comparing models across multiple datasets, Demsar (2006) suggests the use of Wilcoxon Signed Rank | How to verify if a prediction performance improvement is significant better?
Dietterich (1998) proposes a 5x2 cross validation followed by a t-test to identify better models. When comparing models across multiple datasets, Demsar (2006) suggests the use of Wilcoxon Signed Rank test for comparing two models and Friedman... | How to verify if a prediction performance improvement is significant better?
Dietterich (1998) proposes a 5x2 cross validation followed by a t-test to identify better models. When comparing models across multiple datasets, Demsar (2006) suggests the use of Wilcoxon Signed Rank |
37,059 | Can MCMC algorithm estimate partition function (normalizing constant)? | While I produced a detailed answer to an earlier X validated question, let me recall here that there are many ways of approximating the normalising constant, besides importance sampling:
Chib's (1994) (or the candidate's) formula
Gelfand and Dey's (1995) representation, which includes the infamous harmonic mean estima... | Can MCMC algorithm estimate partition function (normalizing constant)? | While I produced a detailed answer to an earlier X validated question, let me recall here that there are many ways of approximating the normalising constant, besides importance sampling:
Chib's (1994 | Can MCMC algorithm estimate partition function (normalizing constant)?
While I produced a detailed answer to an earlier X validated question, let me recall here that there are many ways of approximating the normalising constant, besides importance sampling:
Chib's (1994) (or the candidate's) formula
Gelfand and Dey's ... | Can MCMC algorithm estimate partition function (normalizing constant)?
While I produced a detailed answer to an earlier X validated question, let me recall here that there are many ways of approximating the normalising constant, besides importance sampling:
Chib's (1994 |
37,060 | Recommended Mutual Information Estimator for Continuous Variable | I am not sure i understand why this should be a very hard problem, at least in such a low-dimensional setting as you describe. I am not active in the fields of those who have authored the articles you cite, but I do not see why this could not be framed as a relatively simple statistical problem. An idea you could (perh... | Recommended Mutual Information Estimator for Continuous Variable | I am not sure i understand why this should be a very hard problem, at least in such a low-dimensional setting as you describe. I am not active in the fields of those who have authored the articles you | Recommended Mutual Information Estimator for Continuous Variable
I am not sure i understand why this should be a very hard problem, at least in such a low-dimensional setting as you describe. I am not active in the fields of those who have authored the articles you cite, but I do not see why this could not be framed as... | Recommended Mutual Information Estimator for Continuous Variable
I am not sure i understand why this should be a very hard problem, at least in such a low-dimensional setting as you describe. I am not active in the fields of those who have authored the articles you |
37,061 | What is the definition of dataset (for Bonferroni purposes)? | The justification for control of multiple tests has to do with the family of tests. The family of tests can be mutually independent, which is often the case when they are drawn from different datasets; if so, Bonferroni is a good way to control for FWER. But in general, the concept of a dataset doesn't even enter the p... | What is the definition of dataset (for Bonferroni purposes)? | The justification for control of multiple tests has to do with the family of tests. The family of tests can be mutually independent, which is often the case when they are drawn from different datasets | What is the definition of dataset (for Bonferroni purposes)?
The justification for control of multiple tests has to do with the family of tests. The family of tests can be mutually independent, which is often the case when they are drawn from different datasets; if so, Bonferroni is a good way to control for FWER. But ... | What is the definition of dataset (for Bonferroni purposes)?
The justification for control of multiple tests has to do with the family of tests. The family of tests can be mutually independent, which is often the case when they are drawn from different datasets |
37,062 | What is the definition of dataset (for Bonferroni purposes)? | This is a much harder question than one would think and I doubt there are clear answers. The answer is relatively clear when we talk about clinical trials for regulatory purposes (whatever the regulatory authority says). I have the impression that this is an area of pragmatic traditions that have evolved in a kind of a... | What is the definition of dataset (for Bonferroni purposes)? | This is a much harder question than one would think and I doubt there are clear answers. The answer is relatively clear when we talk about clinical trials for regulatory purposes (whatever the regulat | What is the definition of dataset (for Bonferroni purposes)?
This is a much harder question than one would think and I doubt there are clear answers. The answer is relatively clear when we talk about clinical trials for regulatory purposes (whatever the regulatory authority says). I have the impression that this is an ... | What is the definition of dataset (for Bonferroni purposes)?
This is a much harder question than one would think and I doubt there are clear answers. The answer is relatively clear when we talk about clinical trials for regulatory purposes (whatever the regulat |
37,063 | Is "batch normalization" applied for output layer as well? | It's probably a bad idea to apply batch norm on the last layer. I haven't seen any rigorous explanation of why, but it's probably because it introduces so much variance / randomness / regularization in the final outputs that it hurts training. | Is "batch normalization" applied for output layer as well? | It's probably a bad idea to apply batch norm on the last layer. I haven't seen any rigorous explanation of why, but it's probably because it introduces so much variance / randomness / regularization i | Is "batch normalization" applied for output layer as well?
It's probably a bad idea to apply batch norm on the last layer. I haven't seen any rigorous explanation of why, but it's probably because it introduces so much variance / randomness / regularization in the final outputs that it hurts training. | Is "batch normalization" applied for output layer as well?
It's probably a bad idea to apply batch norm on the last layer. I haven't seen any rigorous explanation of why, but it's probably because it introduces so much variance / randomness / regularization i |
37,064 | Can a logistic regression have both continuous and discrete variables as covariates or as regressors? | There is absolutely no problem, just code your categorical predictor(s) as dummy variables, or some other form of categorical-encoding. This can be used with all form of regression models. It is usually the type of the response ($Y$) variable which "dictates" the type of regression model that can be used, not the predi... | Can a logistic regression have both continuous and discrete variables as covariates or as regressors | There is absolutely no problem, just code your categorical predictor(s) as dummy variables, or some other form of categorical-encoding. This can be used with all form of regression models. It is usual | Can a logistic regression have both continuous and discrete variables as covariates or as regressors?
There is absolutely no problem, just code your categorical predictor(s) as dummy variables, or some other form of categorical-encoding. This can be used with all form of regression models. It is usually the type of the... | Can a logistic regression have both continuous and discrete variables as covariates or as regressors
There is absolutely no problem, just code your categorical predictor(s) as dummy variables, or some other form of categorical-encoding. This can be used with all form of regression models. It is usual |
37,065 | How to choose best proposal distribution for importance sampling | This is definitely an interesting question, but there is no clear answer as far as I can tell. Indeed, first, one has to define a criterion to optimise. For instance, this could be the variance:
$$\min_{\mu,\sigma}\,\text{var}_{\mu,\sigma}\, h(X)f(X)\big/g_{\mu,\sigma}(X)$$or equivalently$$\min_{\mu,\sigma}\,\int \frac... | How to choose best proposal distribution for importance sampling | This is definitely an interesting question, but there is no clear answer as far as I can tell. Indeed, first, one has to define a criterion to optimise. For instance, this could be the variance:
$$\mi | How to choose best proposal distribution for importance sampling
This is definitely an interesting question, but there is no clear answer as far as I can tell. Indeed, first, one has to define a criterion to optimise. For instance, this could be the variance:
$$\min_{\mu,\sigma}\,\text{var}_{\mu,\sigma}\, h(X)f(X)\big/... | How to choose best proposal distribution for importance sampling
This is definitely an interesting question, but there is no clear answer as far as I can tell. Indeed, first, one has to define a criterion to optimise. For instance, this could be the variance:
$$\mi |
37,066 | How to choose best proposal distribution for importance sampling | According to the paper "The sample size
required in importance sampling" (Chatterjee & Diaconis, 2007, https://arxiv.org/pdf/1511.01437.pdf) the sample size required for importance sampling is approximately
$$
\exp (D_{KL}(|h(\cdot)|f(\cdot) \ || \ g))
$$
Therefore, $D_{KL}(|h(\cdot)|f(\cdot) \ || \ g)$ should be used ... | How to choose best proposal distribution for importance sampling | According to the paper "The sample size
required in importance sampling" (Chatterjee & Diaconis, 2007, https://arxiv.org/pdf/1511.01437.pdf) the sample size required for importance sampling is approxi | How to choose best proposal distribution for importance sampling
According to the paper "The sample size
required in importance sampling" (Chatterjee & Diaconis, 2007, https://arxiv.org/pdf/1511.01437.pdf) the sample size required for importance sampling is approximately
$$
\exp (D_{KL}(|h(\cdot)|f(\cdot) \ || \ g))
$$... | How to choose best proposal distribution for importance sampling
According to the paper "The sample size
required in importance sampling" (Chatterjee & Diaconis, 2007, https://arxiv.org/pdf/1511.01437.pdf) the sample size required for importance sampling is approxi |
37,067 | regression coefficient on sum of regressors | With a great deal of prodding (full credit to @whuber) I seem to have solved it:
We re-write $y = \beta_1X_1 + \beta_2X_2 + \epsilon$ as
$y = a(X_1 + X_2) + b(X_1 - X_2) + \epsilon$
$y = aX_1 + aX_2 + bX_1 - bX_2 + \epsilon$,
$y = (a + b)X_1 + (a - b)X_2 + \epsilon$, therefore
$\beta_1 = (a+b)$ and $\beta_2 = (a - b)$.... | regression coefficient on sum of regressors | With a great deal of prodding (full credit to @whuber) I seem to have solved it:
We re-write $y = \beta_1X_1 + \beta_2X_2 + \epsilon$ as
$y = a(X_1 + X_2) + b(X_1 - X_2) + \epsilon$
$y = aX_1 + aX_2 + | regression coefficient on sum of regressors
With a great deal of prodding (full credit to @whuber) I seem to have solved it:
We re-write $y = \beta_1X_1 + \beta_2X_2 + \epsilon$ as
$y = a(X_1 + X_2) + b(X_1 - X_2) + \epsilon$
$y = aX_1 + aX_2 + bX_1 - bX_2 + \epsilon$,
$y = (a + b)X_1 + (a - b)X_2 + \epsilon$, therefor... | regression coefficient on sum of regressors
With a great deal of prodding (full credit to @whuber) I seem to have solved it:
We re-write $y = \beta_1X_1 + \beta_2X_2 + \epsilon$ as
$y = a(X_1 + X_2) + b(X_1 - X_2) + \epsilon$
$y = aX_1 + aX_2 + |
37,068 | what is the intuition behind separate activation/memory paths in LSTM's? | In a vanilla RNN, there is only the activation path (also often referred to as the hidden-state $h_t$).
LSTM added on the cell memory $c_t$ as a way to store information over long time-spans in particular.
GRU, which was developed later, simplifies the LSTM by combining both the cell memory and the hidden-state.
Ther... | what is the intuition behind separate activation/memory paths in LSTM's? | In a vanilla RNN, there is only the activation path (also often referred to as the hidden-state $h_t$).
LSTM added on the cell memory $c_t$ as a way to store information over long time-spans in parti | what is the intuition behind separate activation/memory paths in LSTM's?
In a vanilla RNN, there is only the activation path (also often referred to as the hidden-state $h_t$).
LSTM added on the cell memory $c_t$ as a way to store information over long time-spans in particular.
GRU, which was developed later, simplifi... | what is the intuition behind separate activation/memory paths in LSTM's?
In a vanilla RNN, there is only the activation path (also often referred to as the hidden-state $h_t$).
LSTM added on the cell memory $c_t$ as a way to store information over long time-spans in parti |
37,069 | what is the intuition behind separate activation/memory paths in LSTM's? | To understand the intuition behind, one should probably know the evolution of the algorithms.
1. RNN
An RNN network initially wants to model the sequence of inputs. By sharing the network parameters between the individual inputs, and using the output of previous input as part of the input for next computation, the sequ... | what is the intuition behind separate activation/memory paths in LSTM's? | To understand the intuition behind, one should probably know the evolution of the algorithms.
1. RNN
An RNN network initially wants to model the sequence of inputs. By sharing the network parameters b | what is the intuition behind separate activation/memory paths in LSTM's?
To understand the intuition behind, one should probably know the evolution of the algorithms.
1. RNN
An RNN network initially wants to model the sequence of inputs. By sharing the network parameters between the individual inputs, and using the out... | what is the intuition behind separate activation/memory paths in LSTM's?
To understand the intuition behind, one should probably know the evolution of the algorithms.
1. RNN
An RNN network initially wants to model the sequence of inputs. By sharing the network parameters b |
37,070 | Choosing the number of clusters - clustering validation criterions vs domain theoretical considerations | The keys are finding meaningful clusters and what you value
in the resulting clusters.
Let me illustrate with a simple example.
The example is two Gaussian clusters that are pretty well separated. Using k-means to divide the data into either 2 or 3 clusters we get these partitions:
set.seed(1066)
x = c(rnorm(200,0,1),... | Choosing the number of clusters - clustering validation criterions vs domain theoretical considerati | The keys are finding meaningful clusters and what you value
in the resulting clusters.
Let me illustrate with a simple example.
The example is two Gaussian clusters that are pretty well separated. Us | Choosing the number of clusters - clustering validation criterions vs domain theoretical considerations
The keys are finding meaningful clusters and what you value
in the resulting clusters.
Let me illustrate with a simple example.
The example is two Gaussian clusters that are pretty well separated. Using k-means to d... | Choosing the number of clusters - clustering validation criterions vs domain theoretical considerati
The keys are finding meaningful clusters and what you value
in the resulting clusters.
Let me illustrate with a simple example.
The example is two Gaussian clusters that are pretty well separated. Us |
37,071 | Choosing the number of clusters - clustering validation criterions vs domain theoretical considerations | Note that, cross validation can be also used in clustering problem.
For example, in K means, increasing number of cluster will always decrease the objective we are fitting. An extreme case would be number of clusters equal to number of data points, and the objective is $0$. But that is an overfitted model and will fai... | Choosing the number of clusters - clustering validation criterions vs domain theoretical considerati | Note that, cross validation can be also used in clustering problem.
For example, in K means, increasing number of cluster will always decrease the objective we are fitting. An extreme case would be n | Choosing the number of clusters - clustering validation criterions vs domain theoretical considerations
Note that, cross validation can be also used in clustering problem.
For example, in K means, increasing number of cluster will always decrease the objective we are fitting. An extreme case would be number of cluster... | Choosing the number of clusters - clustering validation criterions vs domain theoretical considerati
Note that, cross validation can be also used in clustering problem.
For example, in K means, increasing number of cluster will always decrease the objective we are fitting. An extreme case would be n |
37,072 | Prediction intervals for the outcome of a logistic regression with binomial response | One way this should work without bootstrapping (which in practice may be the fastest thing tho implement), would be:
Assume that a normal approximation for the predicted log-odds ($x \hat{\beta}$) plus/minus its standard error works. Any logistic regression software will provide this.
The percentiles of this distribu... | Prediction intervals for the outcome of a logistic regression with binomial response | One way this should work without bootstrapping (which in practice may be the fastest thing tho implement), would be:
Assume that a normal approximation for the predicted log-odds ($x \hat{\beta}$) pl | Prediction intervals for the outcome of a logistic regression with binomial response
One way this should work without bootstrapping (which in practice may be the fastest thing tho implement), would be:
Assume that a normal approximation for the predicted log-odds ($x \hat{\beta}$) plus/minus its standard error works. ... | Prediction intervals for the outcome of a logistic regression with binomial response
One way this should work without bootstrapping (which in practice may be the fastest thing tho implement), would be:
Assume that a normal approximation for the predicted log-odds ($x \hat{\beta}$) pl |
37,073 | Difference between contextual anomaly and collective anomaly | That ECG timeseries is just one datastream, so it might not be the clearest example. If thinking about the presence of a pattern, I would say that missing a beat is just a regular anomaly. But can also be considered a contextual anomaly, the Y value is normal, but not at that particular point in time.
A collective anom... | Difference between contextual anomaly and collective anomaly | That ECG timeseries is just one datastream, so it might not be the clearest example. If thinking about the presence of a pattern, I would say that missing a beat is just a regular anomaly. But can als | Difference between contextual anomaly and collective anomaly
That ECG timeseries is just one datastream, so it might not be the clearest example. If thinking about the presence of a pattern, I would say that missing a beat is just a regular anomaly. But can also be considered a contextual anomaly, the Y value is normal... | Difference between contextual anomaly and collective anomaly
That ECG timeseries is just one datastream, so it might not be the clearest example. If thinking about the presence of a pattern, I would say that missing a beat is just a regular anomaly. But can als |
37,074 | Difference between contextual anomaly and collective anomaly | In the same paper you mentioned in your question author says right after definitions of types of anomalies:
It should be noted that while point anomalies can occur in any data set, collective
anomalies can occur only in data sets in which data instances are related. In
contrast, occurrence of contextual anomalies... | Difference between contextual anomaly and collective anomaly | In the same paper you mentioned in your question author says right after definitions of types of anomalies:
It should be noted that while point anomalies can occur in any data set, collective
anom | Difference between contextual anomaly and collective anomaly
In the same paper you mentioned in your question author says right after definitions of types of anomalies:
It should be noted that while point anomalies can occur in any data set, collective
anomalies can occur only in data sets in which data instances a... | Difference between contextual anomaly and collective anomaly
In the same paper you mentioned in your question author says right after definitions of types of anomalies:
It should be noted that while point anomalies can occur in any data set, collective
anom |
37,075 | Why is $x_t = x_{t-1}+w_t$ not wide-sense/second-order stationary? | We need to assume only that
There exists an index $t$ for which $\operatorname{Var}(X_{t})$ is finite.
$\sigma^2 = \sigma^2_w$ is nonzero.
The independence of $X_{t-1}$ and $w_t$ gives $$\operatorname{Var}(X_t) = \operatorname{Var}(X_{t-1}+w_t)=\operatorname{Var}(X_{t-1}) + \operatorname{Var}(w_t) = \operatorname{Var... | Why is $x_t = x_{t-1}+w_t$ not wide-sense/second-order stationary? | We need to assume only that
There exists an index $t$ for which $\operatorname{Var}(X_{t})$ is finite.
$\sigma^2 = \sigma^2_w$ is nonzero.
The independence of $X_{t-1}$ and $w_t$ gives $$\operatorna | Why is $x_t = x_{t-1}+w_t$ not wide-sense/second-order stationary?
We need to assume only that
There exists an index $t$ for which $\operatorname{Var}(X_{t})$ is finite.
$\sigma^2 = \sigma^2_w$ is nonzero.
The independence of $X_{t-1}$ and $w_t$ gives $$\operatorname{Var}(X_t) = \operatorname{Var}(X_{t-1}+w_t)=\opera... | Why is $x_t = x_{t-1}+w_t$ not wide-sense/second-order stationary?
We need to assume only that
There exists an index $t$ for which $\operatorname{Var}(X_{t})$ is finite.
$\sigma^2 = \sigma^2_w$ is nonzero.
The independence of $X_{t-1}$ and $w_t$ gives $$\operatorna |
37,076 | Why is $x_t = x_{t-1}+w_t$ not wide-sense/second-order stationary? | Now that I've finished a grad-level Time Series course, I can provide an answer. This is essentially exercise 2.8 in Brockwell and Davis' Introduction to Time Series and Forecasting, 3rd edition.
For the $\text{AR}(1)$ model indicated above, we have
$$x_t = \phi x_{t-1} + w_t\tag{*}$$
Using (*) above, for $n$ finite, w... | Why is $x_t = x_{t-1}+w_t$ not wide-sense/second-order stationary? | Now that I've finished a grad-level Time Series course, I can provide an answer. This is essentially exercise 2.8 in Brockwell and Davis' Introduction to Time Series and Forecasting, 3rd edition.
For | Why is $x_t = x_{t-1}+w_t$ not wide-sense/second-order stationary?
Now that I've finished a grad-level Time Series course, I can provide an answer. This is essentially exercise 2.8 in Brockwell and Davis' Introduction to Time Series and Forecasting, 3rd edition.
For the $\text{AR}(1)$ model indicated above, we have
$$x... | Why is $x_t = x_{t-1}+w_t$ not wide-sense/second-order stationary?
Now that I've finished a grad-level Time Series course, I can provide an answer. This is essentially exercise 2.8 in Brockwell and Davis' Introduction to Time Series and Forecasting, 3rd edition.
For |
37,077 | Variance and expectation of dot product | \begin{equation}
\begin{aligned}
E(\sum_{i=1}^{n} a_{i} b_{i})
&=\sum_{i=1}^{n}E( a_{i} b_{i}) \text{, due to linearity}\\
&= n E(XY) \text{ , due to i.i.d}\\
\end{aligned}
\end{equation}
Note that variance of sum of independent variables is equal to the sum of their variance.
\begin{equation}
\begin{al... | Variance and expectation of dot product | \begin{equation}
\begin{aligned}
E(\sum_{i=1}^{n} a_{i} b_{i})
&=\sum_{i=1}^{n}E( a_{i} b_{i}) \text{, due to linearity}\\
&= n E(XY) \text{ , due to i.i.d}\\
\end{aligned}
\end{equation}
| Variance and expectation of dot product
\begin{equation}
\begin{aligned}
E(\sum_{i=1}^{n} a_{i} b_{i})
&=\sum_{i=1}^{n}E( a_{i} b_{i}) \text{, due to linearity}\\
&= n E(XY) \text{ , due to i.i.d}\\
\end{aligned}
\end{equation}
Note that variance of sum of independent variables is equal to the sum of their ... | Variance and expectation of dot product
\begin{equation}
\begin{aligned}
E(\sum_{i=1}^{n} a_{i} b_{i})
&=\sum_{i=1}^{n}E( a_{i} b_{i}) \text{, due to linearity}\\
&= n E(XY) \text{ , due to i.i.d}\\
\end{aligned}
\end{equation}
|
37,078 | Epochs in keras meaning? [closed] | Since deep learning often separates training data into smaller batches when training, it is important to know when all the training examples have been processed a single time. This is called an epoch.
There is a more detailed answer here: https://stackoverflow.com/a/31157729/7082163 | Epochs in keras meaning? [closed] | Since deep learning often separates training data into smaller batches when training, it is important to know when all the training examples have been processed a single time. This is called an epoch. | Epochs in keras meaning? [closed]
Since deep learning often separates training data into smaller batches when training, it is important to know when all the training examples have been processed a single time. This is called an epoch.
There is a more detailed answer here: https://stackoverflow.com/a/31157729/7082163 | Epochs in keras meaning? [closed]
Since deep learning often separates training data into smaller batches when training, it is important to know when all the training examples have been processed a single time. This is called an epoch. |
37,079 | Epochs in keras meaning? [closed] | What is your batch size, training set size (num imgs)? But essentially, from this I can tell that you have finished the first epoch on a fit keras call with verbose=1. You have 7200 steps per epochs, which will mean your model will see (7200*batch_size) imgs. This may or may not be your entire training set, for steps p... | Epochs in keras meaning? [closed] | What is your batch size, training set size (num imgs)? But essentially, from this I can tell that you have finished the first epoch on a fit keras call with verbose=1. You have 7200 steps per epochs, | Epochs in keras meaning? [closed]
What is your batch size, training set size (num imgs)? But essentially, from this I can tell that you have finished the first epoch on a fit keras call with verbose=1. You have 7200 steps per epochs, which will mean your model will see (7200*batch_size) imgs. This may or may not be you... | Epochs in keras meaning? [closed]
What is your batch size, training set size (num imgs)? But essentially, from this I can tell that you have finished the first epoch on a fit keras call with verbose=1. You have 7200 steps per epochs, |
37,080 | Rate of convergence of EM algorithm? | In the general case you need to verify that your problem setup satisfies certain properties for the EM algorithm to converge to a stationary point that is a local maximum. Further requirements are needed for global maximum. Assuming these criteria are met you can then quantify the rate of the convergence toward the glo... | Rate of convergence of EM algorithm? | In the general case you need to verify that your problem setup satisfies certain properties for the EM algorithm to converge to a stationary point that is a local maximum. Further requirements are nee | Rate of convergence of EM algorithm?
In the general case you need to verify that your problem setup satisfies certain properties for the EM algorithm to converge to a stationary point that is a local maximum. Further requirements are needed for global maximum. Assuming these criteria are met you can then quantify the r... | Rate of convergence of EM algorithm?
In the general case you need to verify that your problem setup satisfies certain properties for the EM algorithm to converge to a stationary point that is a local maximum. Further requirements are nee |
37,081 | How to show that a sufficient statistic is NOT minimal sufficient? | As you stated:
If there exist $x1,x2∈X$ such that $f(x1)=f(x2)$ but $g(x1)≠g(x2)$, then $g$ can not be written as a function of $f$, i.e. there exists no function $h$ with $g=h∘f$.
So, for example, in the case where $X_1, ...., X_n$ are independent Bernoulli random variables. We can prove that $(x_1, ...., x_n)$ is n... | How to show that a sufficient statistic is NOT minimal sufficient? | As you stated:
If there exist $x1,x2∈X$ such that $f(x1)=f(x2)$ but $g(x1)≠g(x2)$, then $g$ can not be written as a function of $f$, i.e. there exists no function $h$ with $g=h∘f$.
So, for example, | How to show that a sufficient statistic is NOT minimal sufficient?
As you stated:
If there exist $x1,x2∈X$ such that $f(x1)=f(x2)$ but $g(x1)≠g(x2)$, then $g$ can not be written as a function of $f$, i.e. there exists no function $h$ with $g=h∘f$.
So, for example, in the case where $X_1, ...., X_n$ are independent Be... | How to show that a sufficient statistic is NOT minimal sufficient?
As you stated:
If there exist $x1,x2∈X$ such that $f(x1)=f(x2)$ but $g(x1)≠g(x2)$, then $g$ can not be written as a function of $f$, i.e. there exists no function $h$ with $g=h∘f$.
So, for example, |
37,082 | How to show that a sufficient statistic is NOT minimal sufficient? | I have been thinking about this problem some more recently, and here is what I have come up with.
Let $\Omega$ be a probability space, then a random variable $X$ is a measurable function $X: \Omega \to \mathcal{X}$, where $\mathcal{X}$ is a measurable space ($\mathcal{X}$ has a designated $\sigma$-algebra, and $X$ is m... | How to show that a sufficient statistic is NOT minimal sufficient? | I have been thinking about this problem some more recently, and here is what I have come up with.
Let $\Omega$ be a probability space, then a random variable $X$ is a measurable function $X: \Omega \t | How to show that a sufficient statistic is NOT minimal sufficient?
I have been thinking about this problem some more recently, and here is what I have come up with.
Let $\Omega$ be a probability space, then a random variable $X$ is a measurable function $X: \Omega \to \mathcal{X}$, where $\mathcal{X}$ is a measurable s... | How to show that a sufficient statistic is NOT minimal sufficient?
I have been thinking about this problem some more recently, and here is what I have come up with.
Let $\Omega$ be a probability space, then a random variable $X$ is a measurable function $X: \Omega \t |
37,083 | AUC score less than 0.5 for logistic regression | UPDATE: Sycorax posted the following link in the comments: Can a random forest be used for feature selection in multiple linear regression? deals with this problem and describes why this might not work too well.
Similar explanation: your data/model might suffer from the Curse of dimensionality, as logistic regression i... | AUC score less than 0.5 for logistic regression | UPDATE: Sycorax posted the following link in the comments: Can a random forest be used for feature selection in multiple linear regression? deals with this problem and describes why this might not wor | AUC score less than 0.5 for logistic regression
UPDATE: Sycorax posted the following link in the comments: Can a random forest be used for feature selection in multiple linear regression? deals with this problem and describes why this might not work too well.
Similar explanation: your data/model might suffer from the C... | AUC score less than 0.5 for logistic regression
UPDATE: Sycorax posted the following link in the comments: Can a random forest be used for feature selection in multiple linear regression? deals with this problem and describes why this might not wor |
37,084 | How to do permutation test on model coefficients when including an interaction term? | As I'm just starting with permutation tests, I though a question was a good idea. Indeed, thanks to comments by @Glen_b and @user43849, I perceived many misunderstandings and inconsistencies of the theory from my part. For one, I was thinking about testing the magnitude of the coefficient instead of the effect, which i... | How to do permutation test on model coefficients when including an interaction term? | As I'm just starting with permutation tests, I though a question was a good idea. Indeed, thanks to comments by @Glen_b and @user43849, I perceived many misunderstandings and inconsistencies of the th | How to do permutation test on model coefficients when including an interaction term?
As I'm just starting with permutation tests, I though a question was a good idea. Indeed, thanks to comments by @Glen_b and @user43849, I perceived many misunderstandings and inconsistencies of the theory from my part. For one, I was t... | How to do permutation test on model coefficients when including an interaction term?
As I'm just starting with permutation tests, I though a question was a good idea. Indeed, thanks to comments by @Glen_b and @user43849, I perceived many misunderstandings and inconsistencies of the th |
37,085 | Are more features always better? | It depends on what you want your model to achieve. Are these features necessary to your model? For example, gene expression dataset usually have 10000 features (one for each gene), but they are all usually necessary to help determine significant genetic pathways. If the features are not helpful, then a small feature... | Are more features always better? | It depends on what you want your model to achieve. Are these features necessary to your model? For example, gene expression dataset usually have 10000 features (one for each gene), but they are all | Are more features always better?
It depends on what you want your model to achieve. Are these features necessary to your model? For example, gene expression dataset usually have 10000 features (one for each gene), but they are all usually necessary to help determine significant genetic pathways. If the features are ... | Are more features always better?
It depends on what you want your model to achieve. Are these features necessary to your model? For example, gene expression dataset usually have 10000 features (one for each gene), but they are all |
37,086 | Are more features always better? | The probability of a spurious feature-target correlation in the training set is small, but nonzero. So, we use a test set. The probability of a spurious feature-target correlation in the training set that also holds in the test set is even smaller. However, with each new feature (hypothesis), you increase the risk th... | Are more features always better? | The probability of a spurious feature-target correlation in the training set is small, but nonzero. So, we use a test set. The probability of a spurious feature-target correlation in the training set | Are more features always better?
The probability of a spurious feature-target correlation in the training set is small, but nonzero. So, we use a test set. The probability of a spurious feature-target correlation in the training set that also holds in the test set is even smaller. However, with each new feature (hypo... | Are more features always better?
The probability of a spurious feature-target correlation in the training set is small, but nonzero. So, we use a test set. The probability of a spurious feature-target correlation in the training set |
37,087 | Use Fisher's Exact Test or a Hypergeometric Test? | Hypergeometric test assesses the extremeness of observing x or more of "good" cases (overlap) and thus same as a one-sided Fisher's exact test (where the alternative hypothesis is "greater" in R jargon). If you do not care about the directionality then you can use two-sided Fisher's exact test. | Use Fisher's Exact Test or a Hypergeometric Test? | Hypergeometric test assesses the extremeness of observing x or more of "good" cases (overlap) and thus same as a one-sided Fisher's exact test (where the alternative hypothesis is "greater" in R jargo | Use Fisher's Exact Test or a Hypergeometric Test?
Hypergeometric test assesses the extremeness of observing x or more of "good" cases (overlap) and thus same as a one-sided Fisher's exact test (where the alternative hypothesis is "greater" in R jargon). If you do not care about the directionality then you can use two-s... | Use Fisher's Exact Test or a Hypergeometric Test?
Hypergeometric test assesses the extremeness of observing x or more of "good" cases (overlap) and thus same as a one-sided Fisher's exact test (where the alternative hypothesis is "greater" in R jargo |
37,088 | Use Fisher's Exact Test or a Hypergeometric Test? | Just a comment to show the result of R commands, both approaches being the same.
> phyper(15, 150, 400-150, 50)
[1] 0.1549789
> fisher.test(matrix(c(15, 50-15, 150-15, 400 - 50 - 150 + 15), nr = 2),
alternative = "less")$p.value
[1] 0.1549789 | Use Fisher's Exact Test or a Hypergeometric Test? | Just a comment to show the result of R commands, both approaches being the same.
> phyper(15, 150, 400-150, 50)
[1] 0.1549789
> fisher.test(matrix(c(15, 50-15, 150-15, 400 - 50 - 150 + 15), nr = 2),
| Use Fisher's Exact Test or a Hypergeometric Test?
Just a comment to show the result of R commands, both approaches being the same.
> phyper(15, 150, 400-150, 50)
[1] 0.1549789
> fisher.test(matrix(c(15, 50-15, 150-15, 400 - 50 - 150 + 15), nr = 2),
alternative = "less")$p.value
[1] 0.1549789 | Use Fisher's Exact Test or a Hypergeometric Test?
Just a comment to show the result of R commands, both approaches being the same.
> phyper(15, 150, 400-150, 50)
[1] 0.1549789
> fisher.test(matrix(c(15, 50-15, 150-15, 400 - 50 - 150 + 15), nr = 2),
|
37,089 | Repeated measures ANOVA vs. factorial ANOVA with subject factor: understanding "error strata" and Error() term in aov | ... two-way ANOVA tests the effect of A by comparing SS of A with the
residual SS, while RM-ANOVA tests the effect of A by comparing SS of A
with the A⋅subject interaction SS.
1) Does this difference automatically follow from the repeated-measures structure of the data, or is it some convention?
It follows from th... | Repeated measures ANOVA vs. factorial ANOVA with subject factor: understanding "error strata" and Er | ... two-way ANOVA tests the effect of A by comparing SS of A with the
residual SS, while RM-ANOVA tests the effect of A by comparing SS of A
with the A⋅subject interaction SS.
1) Does this differ | Repeated measures ANOVA vs. factorial ANOVA with subject factor: understanding "error strata" and Error() term in aov
... two-way ANOVA tests the effect of A by comparing SS of A with the
residual SS, while RM-ANOVA tests the effect of A by comparing SS of A
with the A⋅subject interaction SS.
1) Does this differen... | Repeated measures ANOVA vs. factorial ANOVA with subject factor: understanding "error strata" and Er
... two-way ANOVA tests the effect of A by comparing SS of A with the
residual SS, while RM-ANOVA tests the effect of A by comparing SS of A
with the A⋅subject interaction SS.
1) Does this differ |
37,090 | Repeated measures ANOVA vs. factorial ANOVA with subject factor: understanding "error strata" and Error() term in aov | This note depends on results contained in Moser's Linear Models: A Mean Model Approach. I'll cite some results from this book in what follows. When I saw your question, I started looking through the book: this note is just the way my thoughts were organized after.
Let $y \sim \mathcal{N}_n(\mu, \Sigma)$ be the response... | Repeated measures ANOVA vs. factorial ANOVA with subject factor: understanding "error strata" and Er | This note depends on results contained in Moser's Linear Models: A Mean Model Approach. I'll cite some results from this book in what follows. When I saw your question, I started looking through the b | Repeated measures ANOVA vs. factorial ANOVA with subject factor: understanding "error strata" and Error() term in aov
This note depends on results contained in Moser's Linear Models: A Mean Model Approach. I'll cite some results from this book in what follows. When I saw your question, I started looking through the boo... | Repeated measures ANOVA vs. factorial ANOVA with subject factor: understanding "error strata" and Er
This note depends on results contained in Moser's Linear Models: A Mean Model Approach. I'll cite some results from this book in what follows. When I saw your question, I started looking through the b |
37,091 | Generative models for time series simulation | For time series, you want to use LSTM or other recurrent neural network instead of unsupervised generative models in deep learning. You can also convert it to a supervised model which uses $x_t$ to predict $x_{t+1}$ with whichever machine learning algorithm you like.
As you can see in the blog post and in wikipedia of ... | Generative models for time series simulation | For time series, you want to use LSTM or other recurrent neural network instead of unsupervised generative models in deep learning. You can also convert it to a supervised model which uses $x_t$ to pr | Generative models for time series simulation
For time series, you want to use LSTM or other recurrent neural network instead of unsupervised generative models in deep learning. You can also convert it to a supervised model which uses $x_t$ to predict $x_{t+1}$ with whichever machine learning algorithm you like.
As you ... | Generative models for time series simulation
For time series, you want to use LSTM or other recurrent neural network instead of unsupervised generative models in deep learning. You can also convert it to a supervised model which uses $x_t$ to pr |
37,092 | Implication of grouped and ungrouped data for Poisson Regression | Since the sums of counts by combination of factors in the model together with the anti-logged offsets are the sufficient statistics for a Poisson distribution, there should be no difference between the two analyses. Any differing analysis results are due to software-usage errors.
In this case, the problem is that the ... | Implication of grouped and ungrouped data for Poisson Regression | Since the sums of counts by combination of factors in the model together with the anti-logged offsets are the sufficient statistics for a Poisson distribution, there should be no difference between th | Implication of grouped and ungrouped data for Poisson Regression
Since the sums of counts by combination of factors in the model together with the anti-logged offsets are the sufficient statistics for a Poisson distribution, there should be no difference between the two analyses. Any differing analysis results are due ... | Implication of grouped and ungrouped data for Poisson Regression
Since the sums of counts by combination of factors in the model together with the anti-logged offsets are the sufficient statistics for a Poisson distribution, there should be no difference between th |
37,093 | Implication of grouped and ungrouped data for Poisson Regression | My expectation would be that the aggregated model will always have a lower deviance as it does not have to explain the variance within the groups. In the 1-dimensional linear case this is easier to see, as grouped data will usually have higher correlation. See: "Better fit" using aggregated data in comparison to disagg... | Implication of grouped and ungrouped data for Poisson Regression | My expectation would be that the aggregated model will always have a lower deviance as it does not have to explain the variance within the groups. In the 1-dimensional linear case this is easier to se | Implication of grouped and ungrouped data for Poisson Regression
My expectation would be that the aggregated model will always have a lower deviance as it does not have to explain the variance within the groups. In the 1-dimensional linear case this is easier to see, as grouped data will usually have higher correlation... | Implication of grouped and ungrouped data for Poisson Regression
My expectation would be that the aggregated model will always have a lower deviance as it does not have to explain the variance within the groups. In the 1-dimensional linear case this is easier to se |
37,094 | Multi-Seasonal Time Series function in Python [closed] | If you have two (or n) specific seasonalities you want to adjust for, you could get each of them separately from statsmodels by calling sm.decompose() twice with the different frequency= parameters. Then you could subtract both those seasonal components from your raw signal.
For an example where you have hourly data a... | Multi-Seasonal Time Series function in Python [closed] | If you have two (or n) specific seasonalities you want to adjust for, you could get each of them separately from statsmodels by calling sm.decompose() twice with the different frequency= parameters. T | Multi-Seasonal Time Series function in Python [closed]
If you have two (or n) specific seasonalities you want to adjust for, you could get each of them separately from statsmodels by calling sm.decompose() twice with the different frequency= parameters. Then you could subtract both those seasonal components from your r... | Multi-Seasonal Time Series function in Python [closed]
If you have two (or n) specific seasonalities you want to adjust for, you could get each of them separately from statsmodels by calling sm.decompose() twice with the different frequency= parameters. T |
37,095 | Policy and value iteration algorithm convergence conditions | To answer your question, let me first write out some important (in)equalities.
Bellman optimality equation:
\begin{align} v_∗(s) &= \max_{a} \mathbb{E}[R_{t+1} + \gamma v_*
(S_{t+1}) \mid S_t =s, A_t =a] \\ &= \max_{a} \sum_{s'}p(s'\mid s, a)
\biggl[r(s, a, s') + \gamma v_∗(s')\biggl]
\end{align}
where $v_*(.)$ is... | Policy and value iteration algorithm convergence conditions | To answer your question, let me first write out some important (in)equalities.
Bellman optimality equation:
\begin{align} v_∗(s) &= \max_{a} \mathbb{E}[R_{t+1} + \gamma v_*
(S_{t+1}) \mid S_t =s, A | Policy and value iteration algorithm convergence conditions
To answer your question, let me first write out some important (in)equalities.
Bellman optimality equation:
\begin{align} v_∗(s) &= \max_{a} \mathbb{E}[R_{t+1} + \gamma v_*
(S_{t+1}) \mid S_t =s, A_t =a] \\ &= \max_{a} \sum_{s'}p(s'\mid s, a)
\biggl[r(s, a... | Policy and value iteration algorithm convergence conditions
To answer your question, let me first write out some important (in)equalities.
Bellman optimality equation:
\begin{align} v_∗(s) &= \max_{a} \mathbb{E}[R_{t+1} + \gamma v_*
(S_{t+1}) \mid S_t =s, A |
37,096 | Policy and value iteration algorithm convergence conditions | You're correct: either the current value function estimate or the current policy estimate can completely describe the state of the algorithm. Each one implies a unique next choice for the other. From the paper linked below,
"Policy iteration continues until $V_{n+1} = V_n, α_{n+1} = α_n$."
https://editorialexpress.com... | Policy and value iteration algorithm convergence conditions | You're correct: either the current value function estimate or the current policy estimate can completely describe the state of the algorithm. Each one implies a unique next choice for the other. From | Policy and value iteration algorithm convergence conditions
You're correct: either the current value function estimate or the current policy estimate can completely describe the state of the algorithm. Each one implies a unique next choice for the other. From the paper linked below,
"Policy iteration continues until $... | Policy and value iteration algorithm convergence conditions
You're correct: either the current value function estimate or the current policy estimate can completely describe the state of the algorithm. Each one implies a unique next choice for the other. From |
37,097 | Pooling levels of categorical variables for regression trees | I have implemented my solution to this. I wrote two functions:
prox_matrix(df, target, features, cluster_dimension,trees = 10)
Parameters
df: Input dataframe
target: Dependant variable you are trying to predict with the random forrest
features: List of independent variables
cluster_dimension: Dimension you would li... | Pooling levels of categorical variables for regression trees | I have implemented my solution to this. I wrote two functions:
prox_matrix(df, target, features, cluster_dimension,trees = 10)
Parameters
df: Input dataframe
target: Dependant variable you are try | Pooling levels of categorical variables for regression trees
I have implemented my solution to this. I wrote two functions:
prox_matrix(df, target, features, cluster_dimension,trees = 10)
Parameters
df: Input dataframe
target: Dependant variable you are trying to predict with the random forrest
features: List of in... | Pooling levels of categorical variables for regression trees
I have implemented my solution to this. I wrote two functions:
prox_matrix(df, target, features, cluster_dimension,trees = 10)
Parameters
df: Input dataframe
target: Dependant variable you are try |
37,098 | Why is the sum of the sample autocorrelations of a stationary series equal to -1/2? | Let's start by representing the sum $S$ using the definition of the autocorrelation function:
\begin{equation}
S = \sum_{h=1}^{n-1} \hat{\rho}(h) = \sum_{h=1}^{n-1} \left(\frac{\frac{1}{n}\sum_{t=1}^{n-h}(X_t-\bar{X})(X_{t+h}-\bar{X})}{\frac{1}{n}\sum_{t=1}^{n}(X_t-\bar{X})^2}\right)
\end{equation}
Denominator does not... | Why is the sum of the sample autocorrelations of a stationary series equal to -1/2? | Let's start by representing the sum $S$ using the definition of the autocorrelation function:
\begin{equation}
S = \sum_{h=1}^{n-1} \hat{\rho}(h) = \sum_{h=1}^{n-1} \left(\frac{\frac{1}{n}\sum_{t=1}^{ | Why is the sum of the sample autocorrelations of a stationary series equal to -1/2?
Let's start by representing the sum $S$ using the definition of the autocorrelation function:
\begin{equation}
S = \sum_{h=1}^{n-1} \hat{\rho}(h) = \sum_{h=1}^{n-1} \left(\frac{\frac{1}{n}\sum_{t=1}^{n-h}(X_t-\bar{X})(X_{t+h}-\bar{X})}{... | Why is the sum of the sample autocorrelations of a stationary series equal to -1/2?
Let's start by representing the sum $S$ using the definition of the autocorrelation function:
\begin{equation}
S = \sum_{h=1}^{n-1} \hat{\rho}(h) = \sum_{h=1}^{n-1} \left(\frac{\frac{1}{n}\sum_{t=1}^{ |
37,099 | How to compare clinical trial data to a natural history control | Years later, I've arrived at a satisfactory answer. This did indeed turn out to be to use a Cox proportional hazards counting model, which allows you to account for different left-truncation times (ages at which you started following the individuals) in addition to different right-censoring times. As noted in the quest... | How to compare clinical trial data to a natural history control | Years later, I've arrived at a satisfactory answer. This did indeed turn out to be to use a Cox proportional hazards counting model, which allows you to account for different left-truncation times (ag | How to compare clinical trial data to a natural history control
Years later, I've arrived at a satisfactory answer. This did indeed turn out to be to use a Cox proportional hazards counting model, which allows you to account for different left-truncation times (ages at which you started following the individuals) in ad... | How to compare clinical trial data to a natural history control
Years later, I've arrived at a satisfactory answer. This did indeed turn out to be to use a Cox proportional hazards counting model, which allows you to account for different left-truncation times (ag |
37,100 | How to compare clinical trial data to a natural history control | Eric, broadly speaking, your problem sounds severe enough that a search for off-the-shelf solutions seems misguided. Rather, you almost surely need recourse to bespoke modeling to exploit your special domain knowledge about the pathophysiology of the disease. Unless you use a modeling approach that enables you to bring... | How to compare clinical trial data to a natural history control | Eric, broadly speaking, your problem sounds severe enough that a search for off-the-shelf solutions seems misguided. Rather, you almost surely need recourse to bespoke modeling to exploit your special | How to compare clinical trial data to a natural history control
Eric, broadly speaking, your problem sounds severe enough that a search for off-the-shelf solutions seems misguided. Rather, you almost surely need recourse to bespoke modeling to exploit your special domain knowledge about the pathophysiology of the disea... | How to compare clinical trial data to a natural history control
Eric, broadly speaking, your problem sounds severe enough that a search for off-the-shelf solutions seems misguided. Rather, you almost surely need recourse to bespoke modeling to exploit your special |
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