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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40,501 | When does the sd stay the same, even after values in the sample were changed? | The question is about the data, not random variables.
Let $X = (x_1, x_2, \ldots, x_n)$ be the data and $Y = (y_1, y_2, \ldots, y_n)$ be additive changes to the data so that the new values are $(x_1+y_1, \ldots, x_n+y_n)$. From
$$\text{Var}(X) = \text{Var}(X+Y) = \text{Var}(X) + 2 \text{Cov}(X,Y) + \text{Var}(Y)$$
we ... | When does the sd stay the same, even after values in the sample were changed? | The question is about the data, not random variables.
Let $X = (x_1, x_2, \ldots, x_n)$ be the data and $Y = (y_1, y_2, \ldots, y_n)$ be additive changes to the data so that the new values are $(x_1+y | When does the sd stay the same, even after values in the sample were changed?
The question is about the data, not random variables.
Let $X = (x_1, x_2, \ldots, x_n)$ be the data and $Y = (y_1, y_2, \ldots, y_n)$ be additive changes to the data so that the new values are $(x_1+y_1, \ldots, x_n+y_n)$. From
$$\text{Var}(... | When does the sd stay the same, even after values in the sample were changed?
The question is about the data, not random variables.
Let $X = (x_1, x_2, \ldots, x_n)$ be the data and $Y = (y_1, y_2, \ldots, y_n)$ be additive changes to the data so that the new values are $(x_1+y |
40,502 | When does the sd stay the same, even after values in the sample were changed? | Suppose that you have a random variable $X$ and you wish to find the set of transformations $Y=f(X)$ such that the standard deviation of $Y$ is the identical to the standard deviation of $X$.
Consider first the set of linear transformations:
$Y = a X + b$ where $a, b$ are constants.
It is clear that:
$Var(Y) = a^2 Var(... | When does the sd stay the same, even after values in the sample were changed? | Suppose that you have a random variable $X$ and you wish to find the set of transformations $Y=f(X)$ such that the standard deviation of $Y$ is the identical to the standard deviation of $X$.
Consider | When does the sd stay the same, even after values in the sample were changed?
Suppose that you have a random variable $X$ and you wish to find the set of transformations $Y=f(X)$ such that the standard deviation of $Y$ is the identical to the standard deviation of $X$.
Consider first the set of linear transformations:
... | When does the sd stay the same, even after values in the sample were changed?
Suppose that you have a random variable $X$ and you wish to find the set of transformations $Y=f(X)$ such that the standard deviation of $Y$ is the identical to the standard deviation of $X$.
Consider |
40,503 | Forecasting unemployment rate | The Arellano-Bond estimator has been designed for precisely this type of problems.
You will find a short non-technical paper with a examples here. In a nutshell, it combines the information embedded in the large number of cross-section to make up for the small number of points in each series. This estimator is widely u... | Forecasting unemployment rate | The Arellano-Bond estimator has been designed for precisely this type of problems.
You will find a short non-technical paper with a examples here. In a nutshell, it combines the information embedded i | Forecasting unemployment rate
The Arellano-Bond estimator has been designed for precisely this type of problems.
You will find a short non-technical paper with a examples here. In a nutshell, it combines the information embedded in the large number of cross-section to make up for the small number of points in each seri... | Forecasting unemployment rate
The Arellano-Bond estimator has been designed for precisely this type of problems.
You will find a short non-technical paper with a examples here. In a nutshell, it combines the information embedded i |
40,504 | Forecasting unemployment rate | Given the nature of your data I would suggest you investigate the use of exponential smoothing as well as fitting ARIMA type models, especially due to the temporal constraints within your data. Although I wouldn't doubt spatial dependencies exist, I would be abit skeptical about their usefulness in forecasting (in what... | Forecasting unemployment rate | Given the nature of your data I would suggest you investigate the use of exponential smoothing as well as fitting ARIMA type models, especially due to the temporal constraints within your data. Althou | Forecasting unemployment rate
Given the nature of your data I would suggest you investigate the use of exponential smoothing as well as fitting ARIMA type models, especially due to the temporal constraints within your data. Although I wouldn't doubt spatial dependencies exist, I would be abit skeptical about their usef... | Forecasting unemployment rate
Given the nature of your data I would suggest you investigate the use of exponential smoothing as well as fitting ARIMA type models, especially due to the temporal constraints within your data. Althou |
40,505 | Pseudo-random orthogonal matrix generation | It's in the Test Matrix Toolbox, not the Matrix Computation Toolbox. The M-file qmult.m (premultiplication by a Haar-distributed pseudorandom orthogonal matrix) can be found here or here. | Pseudo-random orthogonal matrix generation | It's in the Test Matrix Toolbox, not the Matrix Computation Toolbox. The M-file qmult.m (premultiplication by a Haar-distributed pseudorandom orthogonal matrix) can be found here or here. | Pseudo-random orthogonal matrix generation
It's in the Test Matrix Toolbox, not the Matrix Computation Toolbox. The M-file qmult.m (premultiplication by a Haar-distributed pseudorandom orthogonal matrix) can be found here or here. | Pseudo-random orthogonal matrix generation
It's in the Test Matrix Toolbox, not the Matrix Computation Toolbox. The M-file qmult.m (premultiplication by a Haar-distributed pseudorandom orthogonal matrix) can be found here or here. |
40,506 | Predicting proportions from time with a discontinuity | Sounds to me that Y(X) is a sigmiodal process. Thus logistic regression should be suitable for this data. If you model this in R with:
glm(Y~X,family=binomial)
you will find that the "sharpness" of the transition is determined by the magnitude of the X coefficient, and the point of transition (technically the mid-poin... | Predicting proportions from time with a discontinuity | Sounds to me that Y(X) is a sigmiodal process. Thus logistic regression should be suitable for this data. If you model this in R with:
glm(Y~X,family=binomial)
you will find that the "sharpness" of t | Predicting proportions from time with a discontinuity
Sounds to me that Y(X) is a sigmiodal process. Thus logistic regression should be suitable for this data. If you model this in R with:
glm(Y~X,family=binomial)
you will find that the "sharpness" of the transition is determined by the magnitude of the X coefficient,... | Predicting proportions from time with a discontinuity
Sounds to me that Y(X) is a sigmiodal process. Thus logistic regression should be suitable for this data. If you model this in R with:
glm(Y~X,family=binomial)
you will find that the "sharpness" of t |
40,507 | Predicting proportions from time with a discontinuity | Ignoring the "change point" your description suggests to me a (non-linear) mixed effects model of the following form:
g(E(Yi)) = Xi*beta + Zi*U
Where The betas are fixed effects, the U's are random effects, g(E(yi)) is the (logit) link to a binomial mean.
This will deal with logitudinal correlation of data and the non... | Predicting proportions from time with a discontinuity | Ignoring the "change point" your description suggests to me a (non-linear) mixed effects model of the following form:
g(E(Yi)) = Xi*beta + Zi*U
Where The betas are fixed effects, the U's are random e | Predicting proportions from time with a discontinuity
Ignoring the "change point" your description suggests to me a (non-linear) mixed effects model of the following form:
g(E(Yi)) = Xi*beta + Zi*U
Where The betas are fixed effects, the U's are random effects, g(E(yi)) is the (logit) link to a binomial mean.
This will... | Predicting proportions from time with a discontinuity
Ignoring the "change point" your description suggests to me a (non-linear) mixed effects model of the following form:
g(E(Yi)) = Xi*beta + Zi*U
Where The betas are fixed effects, the U's are random e |
40,508 | Predicting proportions from time with a discontinuity | James' approach looks good: each observation, according to your description, might have a Binomial(n[i], p[i]) distribution where n[i] is known and--to be fully general--p[i] is a completely unknown function that rises from near 0 to near 1 as i increases. A logistic regression (GLM with binomial response and logistic... | Predicting proportions from time with a discontinuity | James' approach looks good: each observation, according to your description, might have a Binomial(n[i], p[i]) distribution where n[i] is known and--to be fully general--p[i] is a completely unknown f | Predicting proportions from time with a discontinuity
James' approach looks good: each observation, according to your description, might have a Binomial(n[i], p[i]) distribution where n[i] is known and--to be fully general--p[i] is a completely unknown function that rises from near 0 to near 1 as i increases. A logist... | Predicting proportions from time with a discontinuity
James' approach looks good: each observation, according to your description, might have a Binomial(n[i], p[i]) distribution where n[i] is known and--to be fully general--p[i] is a completely unknown f |
40,509 | Estimating beta-binomial distribution | To fit the model you can use JAGS or Winbugs. In fact if you look at the week 3 of the lecture notes at Paul Hewson's webpage, the rats JAGS example is a beta binomial model. He puts gamma priors on alpha and beta. | Estimating beta-binomial distribution | To fit the model you can use JAGS or Winbugs. In fact if you look at the week 3 of the lecture notes at Paul Hewson's webpage, the rats JAGS example is a beta binomial model. He puts gamma priors on a | Estimating beta-binomial distribution
To fit the model you can use JAGS or Winbugs. In fact if you look at the week 3 of the lecture notes at Paul Hewson's webpage, the rats JAGS example is a beta binomial model. He puts gamma priors on alpha and beta. | Estimating beta-binomial distribution
To fit the model you can use JAGS or Winbugs. In fact if you look at the week 3 of the lecture notes at Paul Hewson's webpage, the rats JAGS example is a beta binomial model. He puts gamma priors on a |
40,510 | Estimating beta-binomial distribution | You don't necessarily have to go Bayesian on your model, plain maximum likelihood estimation works just fine (though has no explicit solution). Multiple R packages (eg. aod or VGAM) will fit the distribution for you.
Alternatively, you can use the quasi-likelihood based overdispersed binomial model that does not assume... | Estimating beta-binomial distribution | You don't necessarily have to go Bayesian on your model, plain maximum likelihood estimation works just fine (though has no explicit solution). Multiple R packages (eg. aod or VGAM) will fit the distr | Estimating beta-binomial distribution
You don't necessarily have to go Bayesian on your model, plain maximum likelihood estimation works just fine (though has no explicit solution). Multiple R packages (eg. aod or VGAM) will fit the distribution for you.
Alternatively, you can use the quasi-likelihood based overdispers... | Estimating beta-binomial distribution
You don't necessarily have to go Bayesian on your model, plain maximum likelihood estimation works just fine (though has no explicit solution). Multiple R packages (eg. aod or VGAM) will fit the distr |
40,511 | Estimating beta-binomial distribution | You have a hierarchical bayesian model. Brief details below:
Likelihood Function:
$$f(n_i | p_i, t_i) = (t_i n_i) p_i^{n_i} (1-p_i)^{(t_i - n_i)}$$
Priors on $p_i, \alpha, \beta$:
$$pi \sim Beta(\alpha, \beta)$$
$$\alpha ~ N(\alpha_{mean}, \alpha_{var}) I(\alpha > 0)$$
$$\beta ~ N(\beta_{mean}, \beta_{var}) I(\beta > ... | Estimating beta-binomial distribution | You have a hierarchical bayesian model. Brief details below:
Likelihood Function:
$$f(n_i | p_i, t_i) = (t_i n_i) p_i^{n_i} (1-p_i)^{(t_i - n_i)}$$
Priors on $p_i, \alpha, \beta$:
$$pi \sim Beta(\alp | Estimating beta-binomial distribution
You have a hierarchical bayesian model. Brief details below:
Likelihood Function:
$$f(n_i | p_i, t_i) = (t_i n_i) p_i^{n_i} (1-p_i)^{(t_i - n_i)}$$
Priors on $p_i, \alpha, \beta$:
$$pi \sim Beta(\alpha, \beta)$$
$$\alpha ~ N(\alpha_{mean}, \alpha_{var}) I(\alpha > 0)$$
$$\beta ~ N... | Estimating beta-binomial distribution
You have a hierarchical bayesian model. Brief details below:
Likelihood Function:
$$f(n_i | p_i, t_i) = (t_i n_i) p_i^{n_i} (1-p_i)^{(t_i - n_i)}$$
Priors on $p_i, \alpha, \beta$:
$$pi \sim Beta(\alp |
40,512 | Is the class of models for which the MLE exists also the one for which flat priors are permissible? | No, these are somewhat different problems. If you have an improper flat prior and you don't have a unique MLE, you will often not have a unique posterior mode, so neither MLE nor MAP estimation will be useful without some additional thought/constraints. But you can easily have a proper posterior.
Some examples:
Mixt... | Is the class of models for which the MLE exists also the one for which flat priors are permissible? | No, these are somewhat different problems. If you have an improper flat prior and you don't have a unique MLE, you will often not have a unique posterior mode, so neither MLE nor MAP estimation will | Is the class of models for which the MLE exists also the one for which flat priors are permissible?
No, these are somewhat different problems. If you have an improper flat prior and you don't have a unique MLE, you will often not have a unique posterior mode, so neither MLE nor MAP estimation will be useful without so... | Is the class of models for which the MLE exists also the one for which flat priors are permissible?
No, these are somewhat different problems. If you have an improper flat prior and you don't have a unique MLE, you will often not have a unique posterior mode, so neither MLE nor MAP estimation will |
40,513 | Is the class of models for which the MLE exists also the one for which flat priors are permissible? | If the prior is uniform then
$$f(\theta|x) \propto \frac{\mathcal{L}(\theta,x)}{\int_{\theta \in \Theta} \mathcal{L}(\theta,x) d\theta}$$
And this is a proper distribution when the integral of the likelihood function in the denominator is finite.
An simple example where this is not gonna work is when for a particular o... | Is the class of models for which the MLE exists also the one for which flat priors are permissible? | If the prior is uniform then
$$f(\theta|x) \propto \frac{\mathcal{L}(\theta,x)}{\int_{\theta \in \Theta} \mathcal{L}(\theta,x) d\theta}$$
And this is a proper distribution when the integral of the lik | Is the class of models for which the MLE exists also the one for which flat priors are permissible?
If the prior is uniform then
$$f(\theta|x) \propto \frac{\mathcal{L}(\theta,x)}{\int_{\theta \in \Theta} \mathcal{L}(\theta,x) d\theta}$$
And this is a proper distribution when the integral of the likelihood function in ... | Is the class of models for which the MLE exists also the one for which flat priors are permissible?
If the prior is uniform then
$$f(\theta|x) \propto \frac{\mathcal{L}(\theta,x)}{\int_{\theta \in \Theta} \mathcal{L}(\theta,x) d\theta}$$
And this is a proper distribution when the integral of the lik |
40,514 | Property of two independent Beta distribution | It does not seem to be a correct conjecture.
It seems your condition is that the mode for $X$ is greater than the mode for $Y$. Since in non-symmetric Beta distributions, the mode is not equal to the mean, it should be possible to find a counterexample.
One way is to take a non-symmetric case where $\frac{a_1}{a_1+b_1... | Property of two independent Beta distribution | It does not seem to be a correct conjecture.
It seems your condition is that the mode for $X$ is greater than the mode for $Y$. Since in non-symmetric Beta distributions, the mode is not equal to the | Property of two independent Beta distribution
It does not seem to be a correct conjecture.
It seems your condition is that the mode for $X$ is greater than the mode for $Y$. Since in non-symmetric Beta distributions, the mode is not equal to the mean, it should be possible to find a counterexample.
One way is to take ... | Property of two independent Beta distribution
It does not seem to be a correct conjecture.
It seems your condition is that the mode for $X$ is greater than the mode for $Y$. Since in non-symmetric Beta distributions, the mode is not equal to the |
40,515 | How to perform a 2 sided binomial test with the alternative being larger | Let $X_1,\ldots,X_k$ be the number of successes in group $A$, and $Y_1,\ldots,Y_l$ for group $B$. By assumption
$$X_i \sim \text{Binomial}(n_i, \theta_A),\quad i=1,\ldots,k,$$
$$Y_j \sim \text{Binomial}(m_i, \theta_B), \quad j=1,\ldots,l$$
and $X_i$'s are independent, $Y_j$'s are independent and $X_i,Y_j,$ are also ind... | How to perform a 2 sided binomial test with the alternative being larger | Let $X_1,\ldots,X_k$ be the number of successes in group $A$, and $Y_1,\ldots,Y_l$ for group $B$. By assumption
$$X_i \sim \text{Binomial}(n_i, \theta_A),\quad i=1,\ldots,k,$$
$$Y_j \sim \text{Binomia | How to perform a 2 sided binomial test with the alternative being larger
Let $X_1,\ldots,X_k$ be the number of successes in group $A$, and $Y_1,\ldots,Y_l$ for group $B$. By assumption
$$X_i \sim \text{Binomial}(n_i, \theta_A),\quad i=1,\ldots,k,$$
$$Y_j \sim \text{Binomial}(m_i, \theta_B), \quad j=1,\ldots,l$$
and $X_... | How to perform a 2 sided binomial test with the alternative being larger
Let $X_1,\ldots,X_k$ be the number of successes in group $A$, and $Y_1,\ldots,Y_l$ for group $B$. By assumption
$$X_i \sim \text{Binomial}(n_i, \theta_A),\quad i=1,\ldots,k,$$
$$Y_j \sim \text{Binomia |
40,516 | Do Spline Models Have The Same Properties Of Standard Regression Models? | If you’re comfortable with the usual assumptions (e.g., Gauss-Markov conditions), then yes.
Spline regressions just apply linear regression to spline-transformed variables. In many regards, that is no different from including a quadratic or a $\log$ as a feature, but instead you do a spline transformation.
Once you hav... | Do Spline Models Have The Same Properties Of Standard Regression Models? | If you’re comfortable with the usual assumptions (e.g., Gauss-Markov conditions), then yes.
Spline regressions just apply linear regression to spline-transformed variables. In many regards, that is no | Do Spline Models Have The Same Properties Of Standard Regression Models?
If you’re comfortable with the usual assumptions (e.g., Gauss-Markov conditions), then yes.
Spline regressions just apply linear regression to spline-transformed variables. In many regards, that is no different from including a quadratic or a $\lo... | Do Spline Models Have The Same Properties Of Standard Regression Models?
If you’re comfortable with the usual assumptions (e.g., Gauss-Markov conditions), then yes.
Spline regressions just apply linear regression to spline-transformed variables. In many regards, that is no |
40,517 | How to simulate data for an interaction? | I see it as a question of how to generate data from a (linear?) model with interaction. See below for the R code where y is the response, x is the covariate and z is the binary moderator.
set.seed(12)
n <- 1000
# simulate a covariate
x <- rnorm(n)
# simualte a binary moderator
z <- sample(c(0,1),size = n, replace = ... | How to simulate data for an interaction? | I see it as a question of how to generate data from a (linear?) model with interaction. See below for the R code where y is the response, x is the covariate and z is the binary moderator.
set.seed(12) | How to simulate data for an interaction?
I see it as a question of how to generate data from a (linear?) model with interaction. See below for the R code where y is the response, x is the covariate and z is the binary moderator.
set.seed(12)
n <- 1000
# simulate a covariate
x <- rnorm(n)
# simualte a binary moderato... | How to simulate data for an interaction?
I see it as a question of how to generate data from a (linear?) model with interaction. See below for the R code where y is the response, x is the covariate and z is the binary moderator.
set.seed(12) |
40,518 | What does "parameterized by" mean? | “Parametrized by” means that the function $f$ of $x$ has additional parameters $\Theta$, so we can write it as $f(x;\Theta)$. In such a case, we call the function by $x$ given a fixed value of $\Theta$.
Likelihood has two meanings, the traditional on, and Bayesian. Traditionally, likelihood is written as
$$
L(\theta|x)... | What does "parameterized by" mean? | “Parametrized by” means that the function $f$ of $x$ has additional parameters $\Theta$, so we can write it as $f(x;\Theta)$. In such a case, we call the function by $x$ given a fixed value of $\Theta | What does "parameterized by" mean?
“Parametrized by” means that the function $f$ of $x$ has additional parameters $\Theta$, so we can write it as $f(x;\Theta)$. In such a case, we call the function by $x$ given a fixed value of $\Theta$.
Likelihood has two meanings, the traditional on, and Bayesian. Traditionally, like... | What does "parameterized by" mean?
“Parametrized by” means that the function $f$ of $x$ has additional parameters $\Theta$, so we can write it as $f(x;\Theta)$. In such a case, we call the function by $x$ given a fixed value of $\Theta |
40,519 | What does "parameterized by" mean? | What is parameter? What is a parametric model?
Definition $1.$ Let $(\Omega,\mathfrak F) $ be a probability space. The set of probability measures $\{ \mathbb P_\theta:~{\boldsymbol \theta\in\Theta}\}$ indexed by a parameter $\boldsymbol\theta$ is a parametric family if and only if $\Theta\in\mathbb R^n,~n\in\mathbb Z^... | What does "parameterized by" mean? | What is parameter? What is a parametric model?
Definition $1.$ Let $(\Omega,\mathfrak F) $ be a probability space. The set of probability measures $\{ \mathbb P_\theta:~{\boldsymbol \theta\in\Theta}\} | What does "parameterized by" mean?
What is parameter? What is a parametric model?
Definition $1.$ Let $(\Omega,\mathfrak F) $ be a probability space. The set of probability measures $\{ \mathbb P_\theta:~{\boldsymbol \theta\in\Theta}\}$ indexed by a parameter $\boldsymbol\theta$ is a parametric family if and only if $\... | What does "parameterized by" mean?
What is parameter? What is a parametric model?
Definition $1.$ Let $(\Omega,\mathfrak F) $ be a probability space. The set of probability measures $\{ \mathbb P_\theta:~{\boldsymbol \theta\in\Theta}\} |
40,520 | How to report a P-value? | When is it appropriate to give the exact P-value, instead of writing e.g. P<0.05 (also in case of non-significant P-values)?
As a general guideline you want to convey as much information about your results as possible. When you report a p-value as p=0.016 instead of as p<0.02 then the readers will have a more precise ... | How to report a P-value? | When is it appropriate to give the exact P-value, instead of writing e.g. P<0.05 (also in case of non-significant P-values)?
As a general guideline you want to convey as much information about your r | How to report a P-value?
When is it appropriate to give the exact P-value, instead of writing e.g. P<0.05 (also in case of non-significant P-values)?
As a general guideline you want to convey as much information about your results as possible. When you report a p-value as p=0.016 instead of as p<0.02 then the readers ... | How to report a P-value?
When is it appropriate to give the exact P-value, instead of writing e.g. P<0.05 (also in case of non-significant P-values)?
As a general guideline you want to convey as much information about your r |
40,521 | How to report a P-value? | In practice, people report the exact p-value then declare whether the hypothesis test is statistically significant. Most statisticians don't advocate this. if the goal is hypothesis testing then reporting the p-value is moot because it doesn't make the test more significant, on the other hand the exact p-value matters ... | How to report a P-value? | In practice, people report the exact p-value then declare whether the hypothesis test is statistically significant. Most statisticians don't advocate this. if the goal is hypothesis testing then repor | How to report a P-value?
In practice, people report the exact p-value then declare whether the hypothesis test is statistically significant. Most statisticians don't advocate this. if the goal is hypothesis testing then reporting the p-value is moot because it doesn't make the test more significant, on the other hand t... | How to report a P-value?
In practice, people report the exact p-value then declare whether the hypothesis test is statistically significant. Most statisticians don't advocate this. if the goal is hypothesis testing then repor |
40,522 | Why exactly does a classifier need the same prevalence in the train and test sets? | "Why exactly does a classifier need the same prevalence in the train and test sets?"
Perhaps my answer to a related question on the DS SE might help
Doesn't over(/under)sampling an imbalanced dataset cause issues?
Yes, the classifier will expect the relative class frequencies in
operation to be the same as those in th... | Why exactly does a classifier need the same prevalence in the train and test sets? | "Why exactly does a classifier need the same prevalence in the train and test sets?"
Perhaps my answer to a related question on the DS SE might help
Doesn't over(/under)sampling an imbalanced dataset | Why exactly does a classifier need the same prevalence in the train and test sets?
"Why exactly does a classifier need the same prevalence in the train and test sets?"
Perhaps my answer to a related question on the DS SE might help
Doesn't over(/under)sampling an imbalanced dataset cause issues?
Yes, the classifier wi... | Why exactly does a classifier need the same prevalence in the train and test sets?
"Why exactly does a classifier need the same prevalence in the train and test sets?"
Perhaps my answer to a related question on the DS SE might help
Doesn't over(/under)sampling an imbalanced dataset |
40,523 | Why exactly does a classifier need the same prevalence in the train and test sets? | I would proceed with caution. The only time I would rebalance would be if I knew the characteristics of the original population, so that when I did rebalanced it would reflect the proportion of the true population. In other words treat it as a true sample, and weight it. But, arbitrarily adding or subtracting from gro... | Why exactly does a classifier need the same prevalence in the train and test sets? | I would proceed with caution. The only time I would rebalance would be if I knew the characteristics of the original population, so that when I did rebalanced it would reflect the proportion of the tr | Why exactly does a classifier need the same prevalence in the train and test sets?
I would proceed with caution. The only time I would rebalance would be if I knew the characteristics of the original population, so that when I did rebalanced it would reflect the proportion of the true population. In other words treat ... | Why exactly does a classifier need the same prevalence in the train and test sets?
I would proceed with caution. The only time I would rebalance would be if I knew the characteristics of the original population, so that when I did rebalanced it would reflect the proportion of the tr |
40,524 | Is correlation a percentage? | It’s wrong, and if a reviewer wants to tell you to change it, you have no argument. I would not, however, consider that to be more than a typo (minor revision), even if I said it should be changed.
I see an argument that it’s just a slang that perhaps has no place in formal writing like a scientific article but is fine... | Is correlation a percentage? | It’s wrong, and if a reviewer wants to tell you to change it, you have no argument. I would not, however, consider that to be more than a typo (minor revision), even if I said it should be changed.
I | Is correlation a percentage?
It’s wrong, and if a reviewer wants to tell you to change it, you have no argument. I would not, however, consider that to be more than a typo (minor revision), even if I said it should be changed.
I see an argument that it’s just a slang that perhaps has no place in formal writing like a s... | Is correlation a percentage?
It’s wrong, and if a reviewer wants to tell you to change it, you have no argument. I would not, however, consider that to be more than a typo (minor revision), even if I said it should be changed.
I |
40,525 | Is correlation a percentage? | Informally, you can do whatever works for you/your group. I can see a lot of reason why it can be easy to visualize a set of positive correlations as proportions towards two ideal states (0 and 1).
However, formally, I think this is wrong on many conceptual levels. Most important: correlations measures are (usually) no... | Is correlation a percentage? | Informally, you can do whatever works for you/your group. I can see a lot of reason why it can be easy to visualize a set of positive correlations as proportions towards two ideal states (0 and 1).
Ho | Is correlation a percentage?
Informally, you can do whatever works for you/your group. I can see a lot of reason why it can be easy to visualize a set of positive correlations as proportions towards two ideal states (0 and 1).
However, formally, I think this is wrong on many conceptual levels. Most important: correlati... | Is correlation a percentage?
Informally, you can do whatever works for you/your group. I can see a lot of reason why it can be easy to visualize a set of positive correlations as proportions towards two ideal states (0 and 1).
Ho |
40,526 | Covariance of random sums | Law of total covariance (wiki) helps here. What you found is when $N$ and $N'$ are given:
$$\operatorname{cov}(X,Y|N,N')=\operatorname{var}(a)\min(N,N')$$
$$\mathbb E[X|N,N']=N\mathbb E[a], \ \ E[Y|N,N']=N'\mathbb E[a]$$
Plugging in gives us:
$$\operatorname{cov}(X,Y)=\operatorname{var}(a)\mathbb E[\min(N,N')]+\mathbb ... | Covariance of random sums | Law of total covariance (wiki) helps here. What you found is when $N$ and $N'$ are given:
$$\operatorname{cov}(X,Y|N,N')=\operatorname{var}(a)\min(N,N')$$
$$\mathbb E[X|N,N']=N\mathbb E[a], \ \ E[Y|N, | Covariance of random sums
Law of total covariance (wiki) helps here. What you found is when $N$ and $N'$ are given:
$$\operatorname{cov}(X,Y|N,N')=\operatorname{var}(a)\min(N,N')$$
$$\mathbb E[X|N,N']=N\mathbb E[a], \ \ E[Y|N,N']=N'\mathbb E[a]$$
Plugging in gives us:
$$\operatorname{cov}(X,Y)=\operatorname{var}(a)\mat... | Covariance of random sums
Law of total covariance (wiki) helps here. What you found is when $N$ and $N'$ are given:
$$\operatorname{cov}(X,Y|N,N')=\operatorname{var}(a)\min(N,N')$$
$$\mathbb E[X|N,N']=N\mathbb E[a], \ \ E[Y|N, |
40,527 | Statistics question: Why is the standard error, which is calculated from 1 sample, a good approximation for the spread of many hypothetical means? | As I understand it, the standard error is the spread of many sample means in an attempt to gauge how precise (not accurate) our estimate of the population mean is, but what if there's just the one sample?
Very short
A sample is not just one sample but contains many individual observations. Each of the observations can... | Statistics question: Why is the standard error, which is calculated from 1 sample, a good approximat | As I understand it, the standard error is the spread of many sample means in an attempt to gauge how precise (not accurate) our estimate of the population mean is, but what if there's just the one sam | Statistics question: Why is the standard error, which is calculated from 1 sample, a good approximation for the spread of many hypothetical means?
As I understand it, the standard error is the spread of many sample means in an attempt to gauge how precise (not accurate) our estimate of the population mean is, but what ... | Statistics question: Why is the standard error, which is calculated from 1 sample, a good approximat
As I understand it, the standard error is the spread of many sample means in an attempt to gauge how precise (not accurate) our estimate of the population mean is, but what if there's just the one sam |
40,528 | Statistics question: Why is the standard error, which is calculated from 1 sample, a good approximation for the spread of many hypothetical means? | I propose to put some visuals/intuition to your question... using an empirical approach (bootstrapping) to make it more concrete, especially in reference to the following:
Usually experiments can't or just aren't repeated and only have 1 sample from a population
As you highlighted it, we are talking about the standar... | Statistics question: Why is the standard error, which is calculated from 1 sample, a good approximat | I propose to put some visuals/intuition to your question... using an empirical approach (bootstrapping) to make it more concrete, especially in reference to the following:
Usually experiments can't o | Statistics question: Why is the standard error, which is calculated from 1 sample, a good approximation for the spread of many hypothetical means?
I propose to put some visuals/intuition to your question... using an empirical approach (bootstrapping) to make it more concrete, especially in reference to the following:
... | Statistics question: Why is the standard error, which is calculated from 1 sample, a good approximat
I propose to put some visuals/intuition to your question... using an empirical approach (bootstrapping) to make it more concrete, especially in reference to the following:
Usually experiments can't o |
40,529 | Statistics question: Why is the standard error, which is calculated from 1 sample, a good approximation for the spread of many hypothetical means? | Frequentist methods have a concept called a confidence procedure. A confidence interval is an example of such a thing. It is the procedure that we have confidence in, not so much the specific interval or point estimate.
If you were to perform an experiment very many times over the sample space, you would get many dif... | Statistics question: Why is the standard error, which is calculated from 1 sample, a good approximat | Frequentist methods have a concept called a confidence procedure. A confidence interval is an example of such a thing. It is the procedure that we have confidence in, not so much the specific interv | Statistics question: Why is the standard error, which is calculated from 1 sample, a good approximation for the spread of many hypothetical means?
Frequentist methods have a concept called a confidence procedure. A confidence interval is an example of such a thing. It is the procedure that we have confidence in, not ... | Statistics question: Why is the standard error, which is calculated from 1 sample, a good approximat
Frequentist methods have a concept called a confidence procedure. A confidence interval is an example of such a thing. It is the procedure that we have confidence in, not so much the specific interv |
40,530 | What is the parameter in Spearman's $\rho$ | Suppose $(X_1,Y_1),(X_2,Y_2),\ldots,(X_n,Y_n)$ are i.i.d random vectors with a continuous distribution. Let $R_i =\operatorname{Rank}(X_i)$ among $X_1,X_2,\ldots,X_n$ and $Q_i=\operatorname{Rank}(Y_i)$ among $Y_1,Y_2,\ldots,Y_n$, $\,i=1,2,\ldots,n$.
Spearman's rank correlation coefficient is then the sample quantity
$$... | What is the parameter in Spearman's $\rho$ | Suppose $(X_1,Y_1),(X_2,Y_2),\ldots,(X_n,Y_n)$ are i.i.d random vectors with a continuous distribution. Let $R_i =\operatorname{Rank}(X_i)$ among $X_1,X_2,\ldots,X_n$ and $Q_i=\operatorname{Rank}(Y_i) | What is the parameter in Spearman's $\rho$
Suppose $(X_1,Y_1),(X_2,Y_2),\ldots,(X_n,Y_n)$ are i.i.d random vectors with a continuous distribution. Let $R_i =\operatorname{Rank}(X_i)$ among $X_1,X_2,\ldots,X_n$ and $Q_i=\operatorname{Rank}(Y_i)$ among $Y_1,Y_2,\ldots,Y_n$, $\,i=1,2,\ldots,n$.
Spearman's rank correlation... | What is the parameter in Spearman's $\rho$
Suppose $(X_1,Y_1),(X_2,Y_2),\ldots,(X_n,Y_n)$ are i.i.d random vectors with a continuous distribution. Let $R_i =\operatorname{Rank}(X_i)$ among $X_1,X_2,\ldots,X_n$ and $Q_i=\operatorname{Rank}(Y_i) |
40,531 | How to estimate the (approximate) variance of the weighted mean? | You can get general answer to this question (and the specific answer) from just considering the variances of sums. Suppose there are $N$ individuals in the population and you sample $n$ of them. The $X_i$ are fixed (Bob's opinion is whatever it is, whether you measure it or not) but the sampling indicators are random ... | How to estimate the (approximate) variance of the weighted mean? | You can get general answer to this question (and the specific answer) from just considering the variances of sums. Suppose there are $N$ individuals in the population and you sample $n$ of them. The | How to estimate the (approximate) variance of the weighted mean?
You can get general answer to this question (and the specific answer) from just considering the variances of sums. Suppose there are $N$ individuals in the population and you sample $n$ of them. The $X_i$ are fixed (Bob's opinion is whatever it is, wheth... | How to estimate the (approximate) variance of the weighted mean?
You can get general answer to this question (and the specific answer) from just considering the variances of sums. Suppose there are $N$ individuals in the population and you sample $n$ of them. The |
40,532 | How to estimate the (approximate) variance of the weighted mean? | A bit nonrigorous, but define $z_i = w_i y_i$. Calculate the variance for the total, $\sum_i z_i$, using the usual formula. Then replace $z_i$ with $w_i y_i$ and notice that the total of $z_i$ is the weighted mean. This is the fastest/most intuitive way to get it -- you're just treating $w_i y_i$ as a single random var... | How to estimate the (approximate) variance of the weighted mean? | A bit nonrigorous, but define $z_i = w_i y_i$. Calculate the variance for the total, $\sum_i z_i$, using the usual formula. Then replace $z_i$ with $w_i y_i$ and notice that the total of $z_i$ is the | How to estimate the (approximate) variance of the weighted mean?
A bit nonrigorous, but define $z_i = w_i y_i$. Calculate the variance for the total, $\sum_i z_i$, using the usual formula. Then replace $z_i$ with $w_i y_i$ and notice that the total of $z_i$ is the weighted mean. This is the fastest/most intuitive way t... | How to estimate the (approximate) variance of the weighted mean?
A bit nonrigorous, but define $z_i = w_i y_i$. Calculate the variance for the total, $\sum_i z_i$, using the usual formula. Then replace $z_i$ with $w_i y_i$ and notice that the total of $z_i$ is the |
40,533 | Variance of Bernoulli when success probability varies | In general, you solve problems like this using the 'law of iterated variance'.
Let $Y|X \sim \text{Bern}(X)$ and use your stipulated prior mean and variance for the success probability $X$. Using the law of iterated variance, you get:
$$\begin{align}
\mathbb{V}(Y)
&= \mathbb{V}(\mathbb{E}(Y|X)) + \mathbb{E}(\mathbb{V... | Variance of Bernoulli when success probability varies | In general, you solve problems like this using the 'law of iterated variance'.
Let $Y|X \sim \text{Bern}(X)$ and use your stipulated prior mean and variance for the success probability $X$. Using the | Variance of Bernoulli when success probability varies
In general, you solve problems like this using the 'law of iterated variance'.
Let $Y|X \sim \text{Bern}(X)$ and use your stipulated prior mean and variance for the success probability $X$. Using the law of iterated variance, you get:
$$\begin{align}
\mathbb{V}(Y) ... | Variance of Bernoulli when success probability varies
In general, you solve problems like this using the 'law of iterated variance'.
Let $Y|X \sim \text{Bern}(X)$ and use your stipulated prior mean and variance for the success probability $X$. Using the |
40,534 | Is the emmeans R package performing causal inference G-computation? | Re-reading your question, my understanding is that you are asking if emmeans() does G-computation as part of what it ordinarily does. And based on my very limited understanding of causal models and G-computation, I would say the answer is NO. That is simply because we don't treat covariates in any special way. For a nu... | Is the emmeans R package performing causal inference G-computation? | Re-reading your question, my understanding is that you are asking if emmeans() does G-computation as part of what it ordinarily does. And based on my very limited understanding of causal models and G- | Is the emmeans R package performing causal inference G-computation?
Re-reading your question, my understanding is that you are asking if emmeans() does G-computation as part of what it ordinarily does. And based on my very limited understanding of causal models and G-computation, I would say the answer is NO. That is s... | Is the emmeans R package performing causal inference G-computation?
Re-reading your question, my understanding is that you are asking if emmeans() does G-computation as part of what it ordinarily does. And based on my very limited understanding of causal models and G- |
40,535 | How to convert a list of integers into a probability distribution such that the smaller the integer the larger the probability value? | A simple way would be to use the first $n$ terms of the geometric series: if $0<r<1$, then
$$ 1+r+r^2+\dots+r^{n-1} = \frac{1-r^n}{1-r}. $$
So if you assign
$$ p_k:=\frac{1-r}{1-r^n}r^k \text{ for }k=0, \dots, n-1, $$
then these probabilities will sum to $1$. The parameter $r$ lets you tune how quickly the probabilitie... | How to convert a list of integers into a probability distribution such that the smaller the integer | A simple way would be to use the first $n$ terms of the geometric series: if $0<r<1$, then
$$ 1+r+r^2+\dots+r^{n-1} = \frac{1-r^n}{1-r}. $$
So if you assign
$$ p_k:=\frac{1-r}{1-r^n}r^k \text{ for }k= | How to convert a list of integers into a probability distribution such that the smaller the integer the larger the probability value?
A simple way would be to use the first $n$ terms of the geometric series: if $0<r<1$, then
$$ 1+r+r^2+\dots+r^{n-1} = \frac{1-r^n}{1-r}. $$
So if you assign
$$ p_k:=\frac{1-r}{1-r^n}r^k ... | How to convert a list of integers into a probability distribution such that the smaller the integer
A simple way would be to use the first $n$ terms of the geometric series: if $0<r<1$, then
$$ 1+r+r^2+\dots+r^{n-1} = \frac{1-r^n}{1-r}. $$
So if you assign
$$ p_k:=\frac{1-r}{1-r^n}r^k \text{ for }k= |
40,536 | $R^2$ on out-sample data set | It makes sense. It is more common however to keep the training and test sets separated. So that you train your model on the train set, and then predict on the test set alone. From there you can calculate the prediction error, and a $R^2_{pred}$ if you like. (train on $n$ data points, evaluate on $p$ data points, in you... | $R^2$ on out-sample data set | It makes sense. It is more common however to keep the training and test sets separated. So that you train your model on the train set, and then predict on the test set alone. From there you can calcul | $R^2$ on out-sample data set
It makes sense. It is more common however to keep the training and test sets separated. So that you train your model on the train set, and then predict on the test set alone. From there you can calculate the prediction error, and a $R^2_{pred}$ if you like. (train on $n$ data points, evalua... | $R^2$ on out-sample data set
It makes sense. It is more common however to keep the training and test sets separated. So that you train your model on the train set, and then predict on the test set alone. From there you can calcul |
40,537 | $R^2$ on out-sample data set | It makes sense to apply $R^2= 1-{\sum(y_i-\hat y_i)^2}/{\sum(y_i-\bar y)^2}$ to test set directly. It's a measure of the size of squared residuals compared to the variance of true values.
Alternatively, if you adopt the notion of deviance (see this answer), then you might use the null model from training data instead:
... | $R^2$ on out-sample data set | It makes sense to apply $R^2= 1-{\sum(y_i-\hat y_i)^2}/{\sum(y_i-\bar y)^2}$ to test set directly. It's a measure of the size of squared residuals compared to the variance of true values.
Alternativel | $R^2$ on out-sample data set
It makes sense to apply $R^2= 1-{\sum(y_i-\hat y_i)^2}/{\sum(y_i-\bar y)^2}$ to test set directly. It's a measure of the size of squared residuals compared to the variance of true values.
Alternatively, if you adopt the notion of deviance (see this answer), then you might use the null model... | $R^2$ on out-sample data set
It makes sense to apply $R^2= 1-{\sum(y_i-\hat y_i)^2}/{\sum(y_i-\bar y)^2}$ to test set directly. It's a measure of the size of squared residuals compared to the variance of true values.
Alternativel |
40,538 | $R^2$ on out-sample data set | There are several definitions of $R^2$ that are equivalent for in-sample OLS linear regression.
The squared correlation between the $x$ and $y$ in simple linear regression
The squared correlation between the predictions $\hat y$ and truth $y$
The proportion of variance explained
A comparison of the model performanc... | $R^2$ on out-sample data set | There are several definitions of $R^2$ that are equivalent for in-sample OLS linear regression.
The squared correlation between the $x$ and $y$ in simple linear regression
The squared correlation be | $R^2$ on out-sample data set
There are several definitions of $R^2$ that are equivalent for in-sample OLS linear regression.
The squared correlation between the $x$ and $y$ in simple linear regression
The squared correlation between the predictions $\hat y$ and truth $y$
The proportion of variance explained
A compa... | $R^2$ on out-sample data set
There are several definitions of $R^2$ that are equivalent for in-sample OLS linear regression.
The squared correlation between the $x$ and $y$ in simple linear regression
The squared correlation be |
40,539 | How can a cross-level interaction be random in a 2-level mixed-model? | It's important to remember that an interaction is the product of the two variables.
For example if we have
sch.id ses sector
1 1 1
1 2 1
2 3 2
2 4 2
..then the interaction for ses:sector will be:
sch.id ses sector ses:sector
1 1 1 1
1 2 1 ... | How can a cross-level interaction be random in a 2-level mixed-model? | It's important to remember that an interaction is the product of the two variables.
For example if we have
sch.id ses sector
1 1 1
1 2 1
2 3 2
2 4 2
..th | How can a cross-level interaction be random in a 2-level mixed-model?
It's important to remember that an interaction is the product of the two variables.
For example if we have
sch.id ses sector
1 1 1
1 2 1
2 3 2
2 4 2
..then the interaction for ses:sector will be:
sch.id ... | How can a cross-level interaction be random in a 2-level mixed-model?
It's important to remember that an interaction is the product of the two variables.
For example if we have
sch.id ses sector
1 1 1
1 2 1
2 3 2
2 4 2
..th |
40,540 | Why functions sampled from a linear kernel Gaussian Process are guaranteed to be a linear function? | Observe that you're covariance is degenerate, as it has rank 1. Sampling from this distribution does not give you as much "randomness" as expected as it can be written as the push-forward of a random variable on a lower dimensional space. Let's make this explicit!
Notice that we can write your covariance matrix as
$$\b... | Why functions sampled from a linear kernel Gaussian Process are guaranteed to be a linear function? | Observe that you're covariance is degenerate, as it has rank 1. Sampling from this distribution does not give you as much "randomness" as expected as it can be written as the push-forward of a random | Why functions sampled from a linear kernel Gaussian Process are guaranteed to be a linear function?
Observe that you're covariance is degenerate, as it has rank 1. Sampling from this distribution does not give you as much "randomness" as expected as it can be written as the push-forward of a random variable on a lower ... | Why functions sampled from a linear kernel Gaussian Process are guaranteed to be a linear function?
Observe that you're covariance is degenerate, as it has rank 1. Sampling from this distribution does not give you as much "randomness" as expected as it can be written as the push-forward of a random |
40,541 | Why functions sampled from a linear kernel Gaussian Process are guaranteed to be a linear function? | Refer to Drawing values from the distribution section in Wikipedia. Let me use that to prove the linearity of functions with basic linear algebra.
We will use the following fact,
If we can write the covariance matrix $\Sigma$ of a multivariate normal
distribution as $\Sigma=AA^T$ for any real $A$, then
$\mathbf{y} = \... | Why functions sampled from a linear kernel Gaussian Process are guaranteed to be a linear function? | Refer to Drawing values from the distribution section in Wikipedia. Let me use that to prove the linearity of functions with basic linear algebra.
We will use the following fact,
If we can write the | Why functions sampled from a linear kernel Gaussian Process are guaranteed to be a linear function?
Refer to Drawing values from the distribution section in Wikipedia. Let me use that to prove the linearity of functions with basic linear algebra.
We will use the following fact,
If we can write the covariance matrix $\... | Why functions sampled from a linear kernel Gaussian Process are guaranteed to be a linear function?
Refer to Drawing values from the distribution section in Wikipedia. Let me use that to prove the linearity of functions with basic linear algebra.
We will use the following fact,
If we can write the |
40,542 | Why is ReLU so popular despite being NOT zero-centered | ReLU's non-zero centering is an issue. ReLUs are popular because it is simple and fast. On the other hand, if the only problem you're finding with ReLU is that the optimization is slow, training the network longer is a reasonable solution.
However, it's more common for state-of-the-art papers to use more complex activa... | Why is ReLU so popular despite being NOT zero-centered | ReLU's non-zero centering is an issue. ReLUs are popular because it is simple and fast. On the other hand, if the only problem you're finding with ReLU is that the optimization is slow, training the n | Why is ReLU so popular despite being NOT zero-centered
ReLU's non-zero centering is an issue. ReLUs are popular because it is simple and fast. On the other hand, if the only problem you're finding with ReLU is that the optimization is slow, training the network longer is a reasonable solution.
However, it's more common... | Why is ReLU so popular despite being NOT zero-centered
ReLU's non-zero centering is an issue. ReLUs are popular because it is simple and fast. On the other hand, if the only problem you're finding with ReLU is that the optimization is slow, training the n |
40,543 | Does using a random train-test split lead to data leakage? | Yes, random train-test splits can lead to data leakage, and if traditional k-fold and leave-one-out CV are the default procedures being followed, data leakage will happen. Leakage is the major reason why traditional CV is not appropriate for time series. Using these cross-validators when you shouldn't will inflate perf... | Does using a random train-test split lead to data leakage? | Yes, random train-test splits can lead to data leakage, and if traditional k-fold and leave-one-out CV are the default procedures being followed, data leakage will happen. Leakage is the major reason | Does using a random train-test split lead to data leakage?
Yes, random train-test splits can lead to data leakage, and if traditional k-fold and leave-one-out CV are the default procedures being followed, data leakage will happen. Leakage is the major reason why traditional CV is not appropriate for time series. Using ... | Does using a random train-test split lead to data leakage?
Yes, random train-test splits can lead to data leakage, and if traditional k-fold and leave-one-out CV are the default procedures being followed, data leakage will happen. Leakage is the major reason |
40,544 | Defining extremeness of test statistic and defining $p$-value for a two-sided test | Defining extremeness of test statistic and defining p-value for a
two-sided test...
I would suggest that an appropriate perspective here is that, when one has the "right" statistic, the statistic itself tells you what "extremeness" means for the test problem at hand---one-sided or two-sided. The more basic question is... | Defining extremeness of test statistic and defining $p$-value for a two-sided test | Defining extremeness of test statistic and defining p-value for a
two-sided test...
I would suggest that an appropriate perspective here is that, when one has the "right" statistic, the statistic its | Defining extremeness of test statistic and defining $p$-value for a two-sided test
Defining extremeness of test statistic and defining p-value for a
two-sided test...
I would suggest that an appropriate perspective here is that, when one has the "right" statistic, the statistic itself tells you what "extremeness" mean... | Defining extremeness of test statistic and defining $p$-value for a two-sided test
Defining extremeness of test statistic and defining p-value for a
two-sided test...
I would suggest that an appropriate perspective here is that, when one has the "right" statistic, the statistic its |
40,545 | Defining extremeness of test statistic and defining $p$-value for a two-sided test | In addition to scenarios in two-sided tests, this question arises in a less avoidable way in group sequential clinical trials.
In a group sequential trial there are a set of analysis times, and a stopping boundary specifying thresholds at each analysis for the trial to stop. In calculating $p$-values or confidence int... | Defining extremeness of test statistic and defining $p$-value for a two-sided test | In addition to scenarios in two-sided tests, this question arises in a less avoidable way in group sequential clinical trials.
In a group sequential trial there are a set of analysis times, and a stop | Defining extremeness of test statistic and defining $p$-value for a two-sided test
In addition to scenarios in two-sided tests, this question arises in a less avoidable way in group sequential clinical trials.
In a group sequential trial there are a set of analysis times, and a stopping boundary specifying thresholds a... | Defining extremeness of test statistic and defining $p$-value for a two-sided test
In addition to scenarios in two-sided tests, this question arises in a less avoidable way in group sequential clinical trials.
In a group sequential trial there are a set of analysis times, and a stop |
40,546 | Defining extremeness of test statistic and defining $p$-value for a two-sided test | The answer to this question is what defines the particular test
But how do we define what is more extreme?
This choice is really the essence of what defines the particular hypothesis test under use. Indeed, a classical hypothesis test can be reduced to a specification of a total order $\preceq$ on the set of possibl... | Defining extremeness of test statistic and defining $p$-value for a two-sided test | The answer to this question is what defines the particular test
But how do we define what is more extreme?
This choice is really the essence of what defines the particular hypothesis test under use. | Defining extremeness of test statistic and defining $p$-value for a two-sided test
The answer to this question is what defines the particular test
But how do we define what is more extreme?
This choice is really the essence of what defines the particular hypothesis test under use. Indeed, a classical hypothesis test... | Defining extremeness of test statistic and defining $p$-value for a two-sided test
The answer to this question is what defines the particular test
But how do we define what is more extreme?
This choice is really the essence of what defines the particular hypothesis test under use. |
40,547 | Defining extremeness of test statistic and defining $p$-value for a two-sided test | Not taking it further than likelihood ratios ...
$\renewcommand\vec{\boldsymbol}$
The generalized likelihood ratio, for a test of the null hypothesis $\vec\theta \in \Theta_0$ vs the alternative $\vec\theta \in \Theta_\mathrm{A}$ is
$$
\renewcommand\vec{\boldsymbol}
\frac{\sup_{\vec{\theta} \in \Theta_\mathrm{A}} \left... | Defining extremeness of test statistic and defining $p$-value for a two-sided test | Not taking it further than likelihood ratios ...
$\renewcommand\vec{\boldsymbol}$
The generalized likelihood ratio, for a test of the null hypothesis $\vec\theta \in \Theta_0$ vs the alternative $\vec | Defining extremeness of test statistic and defining $p$-value for a two-sided test
Not taking it further than likelihood ratios ...
$\renewcommand\vec{\boldsymbol}$
The generalized likelihood ratio, for a test of the null hypothesis $\vec\theta \in \Theta_0$ vs the alternative $\vec\theta \in \Theta_\mathrm{A}$ is
$$
\... | Defining extremeness of test statistic and defining $p$-value for a two-sided test
Not taking it further than likelihood ratios ...
$\renewcommand\vec{\boldsymbol}$
The generalized likelihood ratio, for a test of the null hypothesis $\vec\theta \in \Theta_0$ vs the alternative $\vec |
40,548 | LRT comparing a random effects model and nested logistic regression model | Yes, they are nested: the mixed model reduces to the simpler model if $\sigma^2_1=\sigma^2_{x_2}=0$. (This is the same as $G=0$, because the covariances must be zero if the variances are, but stating it in terms of a joint condition on $\{\sigma^2_1, \sigma^2_{x_2}\}$ is probably easier to understand.)
The likelihood r... | LRT comparing a random effects model and nested logistic regression model | Yes, they are nested: the mixed model reduces to the simpler model if $\sigma^2_1=\sigma^2_{x_2}=0$. (This is the same as $G=0$, because the covariances must be zero if the variances are, but stating | LRT comparing a random effects model and nested logistic regression model
Yes, they are nested: the mixed model reduces to the simpler model if $\sigma^2_1=\sigma^2_{x_2}=0$. (This is the same as $G=0$, because the covariances must be zero if the variances are, but stating it in terms of a joint condition on $\{\sigma^... | LRT comparing a random effects model and nested logistic regression model
Yes, they are nested: the mixed model reduces to the simpler model if $\sigma^2_1=\sigma^2_{x_2}=0$. (This is the same as $G=0$, because the covariances must be zero if the variances are, but stating |
40,549 | Standard errors in LME4 linear mixed models | By default in R, treatment contrasts are used for factors. This means that what you get in the output from summary(mod) are the differences from the reference level for treatment. E.g., 37.4 is the difference between treatment B and treatment A.
If you want to get the mean for treatment B, you will need to add the coef... | Standard errors in LME4 linear mixed models | By default in R, treatment contrasts are used for factors. This means that what you get in the output from summary(mod) are the differences from the reference level for treatment. E.g., 37.4 is the di | Standard errors in LME4 linear mixed models
By default in R, treatment contrasts are used for factors. This means that what you get in the output from summary(mod) are the differences from the reference level for treatment. E.g., 37.4 is the difference between treatment B and treatment A.
If you want to get the mean fo... | Standard errors in LME4 linear mixed models
By default in R, treatment contrasts are used for factors. This means that what you get in the output from summary(mod) are the differences from the reference level for treatment. E.g., 37.4 is the di |
40,550 | Is my proof that relative entropy is never negative correct? | I think you have introduced good ideas, but some care is needed to make sense of all this.
The unifying concept is of absolutely continuous measure. Given two measures $\nu$ and $\mu$ on the same measure space, $\nu$ is said to be absolutely continuous with respect to $\mu$ provided $\nu$ never assigns a nonzero value... | Is my proof that relative entropy is never negative correct? | I think you have introduced good ideas, but some care is needed to make sense of all this.
The unifying concept is of absolutely continuous measure. Given two measures $\nu$ and $\mu$ on the same mea | Is my proof that relative entropy is never negative correct?
I think you have introduced good ideas, but some care is needed to make sense of all this.
The unifying concept is of absolutely continuous measure. Given two measures $\nu$ and $\mu$ on the same measure space, $\nu$ is said to be absolutely continuous with ... | Is my proof that relative entropy is never negative correct?
I think you have introduced good ideas, but some care is needed to make sense of all this.
The unifying concept is of absolutely continuous measure. Given two measures $\nu$ and $\mu$ on the same mea |
40,551 | Smoothing splines with a boundary constraint | As pointed in the comments, the pc argument of the s() function included in the mgcv package does not allow for multiple constraint points. This is unfortunate but I think should not be too complicated to achieve the objective outside the realm of the specific package.
Intro
I think we can obtain the result we wish usi... | Smoothing splines with a boundary constraint | As pointed in the comments, the pc argument of the s() function included in the mgcv package does not allow for multiple constraint points. This is unfortunate but I think should not be too complicate | Smoothing splines with a boundary constraint
As pointed in the comments, the pc argument of the s() function included in the mgcv package does not allow for multiple constraint points. This is unfortunate but I think should not be too complicated to achieve the objective outside the realm of the specific package.
Intro... | Smoothing splines with a boundary constraint
As pointed in the comments, the pc argument of the s() function included in the mgcv package does not allow for multiple constraint points. This is unfortunate but I think should not be too complicate |
40,552 | Smoothing splines with a boundary constraint | This might be a bit long so I will reply in another answer.
Following the comment to my previous answer I will attempt a solution to the following problem: fit an additive model with a shared smooth trend effect subject to boundary constraint and a random Id intercept.
Extra penalty in matrix augmentation form
In the c... | Smoothing splines with a boundary constraint | This might be a bit long so I will reply in another answer.
Following the comment to my previous answer I will attempt a solution to the following problem: fit an additive model with a shared smooth t | Smoothing splines with a boundary constraint
This might be a bit long so I will reply in another answer.
Following the comment to my previous answer I will attempt a solution to the following problem: fit an additive model with a shared smooth trend effect subject to boundary constraint and a random Id intercept.
Extra... | Smoothing splines with a boundary constraint
This might be a bit long so I will reply in another answer.
Following the comment to my previous answer I will attempt a solution to the following problem: fit an additive model with a shared smooth t |
40,553 | What is an induced probability function? | Not quite. The setting is a probability space $(\Omega,\mathfrak{F},\mathbb{P})$ and a measurable function $X$ whose domain is $\Omega$ and whose codomain usually is $\mathbb{R}$ with its Borel sigma-algebra $\mathfrak{B}$ (but generally could be any measurable space).
$X$ induces a probability distribution $\mathbb{P... | What is an induced probability function? | Not quite. The setting is a probability space $(\Omega,\mathfrak{F},\mathbb{P})$ and a measurable function $X$ whose domain is $\Omega$ and whose codomain usually is $\mathbb{R}$ with its Borel sigma | What is an induced probability function?
Not quite. The setting is a probability space $(\Omega,\mathfrak{F},\mathbb{P})$ and a measurable function $X$ whose domain is $\Omega$ and whose codomain usually is $\mathbb{R}$ with its Borel sigma-algebra $\mathfrak{B}$ (but generally could be any measurable space).
$X$ indu... | What is an induced probability function?
Not quite. The setting is a probability space $(\Omega,\mathfrak{F},\mathbb{P})$ and a measurable function $X$ whose domain is $\Omega$ and whose codomain usually is $\mathbb{R}$ with its Borel sigma |
40,554 | AIC can recommend an overfitting model? | AIC can most definitely select an overfit model, because you e.g.
only assess overfit models (so one of those gets selected), or
you offer up an overfit model vs. an inappropriate model (seems to be your example)/a very overfit model/an underfit model, or
you compare more than one model via AIC (the more models the w... | AIC can recommend an overfitting model? | AIC can most definitely select an overfit model, because you e.g.
only assess overfit models (so one of those gets selected), or
you offer up an overfit model vs. an inappropriate model (seems to be | AIC can recommend an overfitting model?
AIC can most definitely select an overfit model, because you e.g.
only assess overfit models (so one of those gets selected), or
you offer up an overfit model vs. an inappropriate model (seems to be your example)/a very overfit model/an underfit model, or
you compare more than ... | AIC can recommend an overfitting model?
AIC can most definitely select an overfit model, because you e.g.
only assess overfit models (so one of those gets selected), or
you offer up an overfit model vs. an inappropriate model (seems to be |
40,555 | Can random forest detect squared terms? | Indeed it can. Here are some simulated data with a squared relationship between the predictor and the response, and the fit from a Random Forest:
R code:
nn <- 1e4
set.seed(1)
xx <- runif(nn)
yy <- xx^2+rnorm(nn,0,0.1)
plot(xx,yy,pch=19,cex=0.6,col="lightgray")
library(randomForest)
model <- randomForest(yy~xx)
xx_... | Can random forest detect squared terms? | Indeed it can. Here are some simulated data with a squared relationship between the predictor and the response, and the fit from a Random Forest:
R code:
nn <- 1e4
set.seed(1)
xx <- runif(nn)
yy <- x | Can random forest detect squared terms?
Indeed it can. Here are some simulated data with a squared relationship between the predictor and the response, and the fit from a Random Forest:
R code:
nn <- 1e4
set.seed(1)
xx <- runif(nn)
yy <- xx^2+rnorm(nn,0,0.1)
plot(xx,yy,pch=19,cex=0.6,col="lightgray")
library(randomF... | Can random forest detect squared terms?
Indeed it can. Here are some simulated data with a squared relationship between the predictor and the response, and the fit from a Random Forest:
R code:
nn <- 1e4
set.seed(1)
xx <- runif(nn)
yy <- x |
40,556 | Specifying model in glmer() - interaction terms | First, note that A*B is just shorthand for A + B + A:B and it does not make sense to specify a model with only the interaction term, as in your last model. That is, when including an interaction, as a general rule you also need to include the main effects for each variable involved in the interaction. In other words y... | Specifying model in glmer() - interaction terms | First, note that A*B is just shorthand for A + B + A:B and it does not make sense to specify a model with only the interaction term, as in your last model. That is, when including an interaction, as | Specifying model in glmer() - interaction terms
First, note that A*B is just shorthand for A + B + A:B and it does not make sense to specify a model with only the interaction term, as in your last model. That is, when including an interaction, as a general rule you also need to include the main effects for each variab... | Specifying model in glmer() - interaction terms
First, note that A*B is just shorthand for A + B + A:B and it does not make sense to specify a model with only the interaction term, as in your last model. That is, when including an interaction, as |
40,557 | Plausibility, Possibility, and Probability | Possibility and probability can be examined together
This answer is based on material in the paper O'Neill (2014) which gives a detailed exposition of the relationship between probability and possibility for the assistance of statistics students and practitioners. Some of the material in this answer is copied from tha... | Plausibility, Possibility, and Probability | Possibility and probability can be examined together
This answer is based on material in the paper O'Neill (2014) which gives a detailed exposition of the relationship between probability and possibil | Plausibility, Possibility, and Probability
Possibility and probability can be examined together
This answer is based on material in the paper O'Neill (2014) which gives a detailed exposition of the relationship between probability and possibility for the assistance of statistics students and practitioners. Some of the... | Plausibility, Possibility, and Probability
Possibility and probability can be examined together
This answer is based on material in the paper O'Neill (2014) which gives a detailed exposition of the relationship between probability and possibil |
40,558 | Dense vs Sequential Layers in Keras | In Keras, "dense" usually refers to a single layer, whereas "sequential" usually refers to an entire model, not just one layer. So I'm not sure the comparison between "Dense vs. Sequential" makes sense.
Sequential refers to the way you build models in Keras using the sequential api (from keras.models import Sequential... | Dense vs Sequential Layers in Keras | In Keras, "dense" usually refers to a single layer, whereas "sequential" usually refers to an entire model, not just one layer. So I'm not sure the comparison between "Dense vs. Sequential" makes sens | Dense vs Sequential Layers in Keras
In Keras, "dense" usually refers to a single layer, whereas "sequential" usually refers to an entire model, not just one layer. So I'm not sure the comparison between "Dense vs. Sequential" makes sense.
Sequential refers to the way you build models in Keras using the sequential api ... | Dense vs Sequential Layers in Keras
In Keras, "dense" usually refers to a single layer, whereas "sequential" usually refers to an entire model, not just one layer. So I'm not sure the comparison between "Dense vs. Sequential" makes sens |
40,559 | Dense vs Sequential Layers in Keras | Sequential is not a layer, it is a model. In sequential models, you stack up multiple same/or different layers where one's output goes into another ahead. This is the default structure with neural nets. Dense is a layer type (fully connected layer). There are others such as Convolutional, Pooling, LSTM etc. | Dense vs Sequential Layers in Keras | Sequential is not a layer, it is a model. In sequential models, you stack up multiple same/or different layers where one's output goes into another ahead. This is the default structure with neural net | Dense vs Sequential Layers in Keras
Sequential is not a layer, it is a model. In sequential models, you stack up multiple same/or different layers where one's output goes into another ahead. This is the default structure with neural nets. Dense is a layer type (fully connected layer). There are others such as Convoluti... | Dense vs Sequential Layers in Keras
Sequential is not a layer, it is a model. In sequential models, you stack up multiple same/or different layers where one's output goes into another ahead. This is the default structure with neural net |
40,560 | How can one implement PCA using gradient descent? | We can pose PCA as a variance maximization problem. These are some of the hints:
The objective is to find the directions in which the variance, $\Bbb E(\vec X \vec X^T)$, is maximum.
Let $\vec w$ denote the unit vector direction along which the variance is maximum. The variance along this direction is given by:
\begi... | How can one implement PCA using gradient descent? | We can pose PCA as a variance maximization problem. These are some of the hints:
The objective is to find the directions in which the variance, $\Bbb E(\vec X \vec X^T)$, is maximum.
Let $\vec w$ den | How can one implement PCA using gradient descent?
We can pose PCA as a variance maximization problem. These are some of the hints:
The objective is to find the directions in which the variance, $\Bbb E(\vec X \vec X^T)$, is maximum.
Let $\vec w$ denote the unit vector direction along which the variance is maximum. The... | How can one implement PCA using gradient descent?
We can pose PCA as a variance maximization problem. These are some of the hints:
The objective is to find the directions in which the variance, $\Bbb E(\vec X \vec X^T)$, is maximum.
Let $\vec w$ den |
40,561 | Reference request - Computer Vision Book | To the best of my knowledge there is not yet a comprehensive academic computer vision textbook (as of 2019) that has been written that incorporates the deep learning.
It's useful to separate the discussion and formulation of a problem from algorithms deployed to solve it. @shimao makes good point that often earlier me... | Reference request - Computer Vision Book | To the best of my knowledge there is not yet a comprehensive academic computer vision textbook (as of 2019) that has been written that incorporates the deep learning.
It's useful to separate the disc | Reference request - Computer Vision Book
To the best of my knowledge there is not yet a comprehensive academic computer vision textbook (as of 2019) that has been written that incorporates the deep learning.
It's useful to separate the discussion and formulation of a problem from algorithms deployed to solve it. @shim... | Reference request - Computer Vision Book
To the best of my knowledge there is not yet a comprehensive academic computer vision textbook (as of 2019) that has been written that incorporates the deep learning.
It's useful to separate the disc |
40,562 | Reference request - Computer Vision Book | To add to @MachineEpsilon's answer. Deep learning has become the de-facto tool for tasks like segmentation and object detection. And is starting to take over in domains like 3D reconstruction. Nevertheless, it is still an ongoing process.
You still need to have a good knowledge of projective geometry (Hartley and Zimme... | Reference request - Computer Vision Book | To add to @MachineEpsilon's answer. Deep learning has become the de-facto tool for tasks like segmentation and object detection. And is starting to take over in domains like 3D reconstruction. Neverth | Reference request - Computer Vision Book
To add to @MachineEpsilon's answer. Deep learning has become the de-facto tool for tasks like segmentation and object detection. And is starting to take over in domains like 3D reconstruction. Nevertheless, it is still an ongoing process.
You still need to have a good knowledge ... | Reference request - Computer Vision Book
To add to @MachineEpsilon's answer. Deep learning has become the de-facto tool for tasks like segmentation and object detection. And is starting to take over in domains like 3D reconstruction. Neverth |
40,563 | Name and interpretation of "$h(x)$" in exponential family | The $h(x)$ function in the exponential family is known as the "underlying measure." It serves to ensure $x$ is in the right space. For many functions, this correction is unnecessary (i.e. it is set to $1$ or $1/\sqrt{2}$). It does play a strong role in defining many functions, however. Since the role is function-specif... | Name and interpretation of "$h(x)$" in exponential family | The $h(x)$ function in the exponential family is known as the "underlying measure." It serves to ensure $x$ is in the right space. For many functions, this correction is unnecessary (i.e. it is set to | Name and interpretation of "$h(x)$" in exponential family
The $h(x)$ function in the exponential family is known as the "underlying measure." It serves to ensure $x$ is in the right space. For many functions, this correction is unnecessary (i.e. it is set to $1$ or $1/\sqrt{2}$). It does play a strong role in defining ... | Name and interpretation of "$h(x)$" in exponential family
The $h(x)$ function in the exponential family is known as the "underlying measure." It serves to ensure $x$ is in the right space. For many functions, this correction is unnecessary (i.e. it is set to |
40,564 | How to include an interaction with a quadratic term? [closed] | It's the same formula (meaning that the models are equivalent), just the R notation is different.
Here is an example with random data:
x1 <- rnorm(100)
x2 <- rnorm(100)
y <- x1 + x2 + x2**2 + x1*x2 + rnorm(100)
fit <- lm(y ~ x1 + x2 + I(x2^2) + x1:x2 + x1:I(x2^2))
fit <- lm(y ~ x1 + x2 + I(x2^2) + x1:(x2 + I(x2^2)))
... | How to include an interaction with a quadratic term? [closed] | It's the same formula (meaning that the models are equivalent), just the R notation is different.
Here is an example with random data:
x1 <- rnorm(100)
x2 <- rnorm(100)
y <- x1 + x2 + x2**2 + x1*x2 + | How to include an interaction with a quadratic term? [closed]
It's the same formula (meaning that the models are equivalent), just the R notation is different.
Here is an example with random data:
x1 <- rnorm(100)
x2 <- rnorm(100)
y <- x1 + x2 + x2**2 + x1*x2 + rnorm(100)
fit <- lm(y ~ x1 + x2 + I(x2^2) + x1:x2 + x1:I... | How to include an interaction with a quadratic term? [closed]
It's the same formula (meaning that the models are equivalent), just the R notation is different.
Here is an example with random data:
x1 <- rnorm(100)
x2 <- rnorm(100)
y <- x1 + x2 + x2**2 + x1*x2 + |
40,565 | Difference between Granger causality and Instantaneous causality? | I was looking for the answer to this same question and I found it on the book Introduction to Modern Time Series Analysis (second edition) by Gebhard Kirchgassner, Jurgen Wolters and Uwe Hassler on page 97.
Granger Causality: x granger causes y if a model that uses current and past values of x and current and past valu... | Difference between Granger causality and Instantaneous causality? | I was looking for the answer to this same question and I found it on the book Introduction to Modern Time Series Analysis (second edition) by Gebhard Kirchgassner, Jurgen Wolters and Uwe Hassler on pa | Difference between Granger causality and Instantaneous causality?
I was looking for the answer to this same question and I found it on the book Introduction to Modern Time Series Analysis (second edition) by Gebhard Kirchgassner, Jurgen Wolters and Uwe Hassler on page 97.
Granger Causality: x granger causes y if a mode... | Difference between Granger causality and Instantaneous causality?
I was looking for the answer to this same question and I found it on the book Introduction to Modern Time Series Analysis (second edition) by Gebhard Kirchgassner, Jurgen Wolters and Uwe Hassler on pa |
40,566 | What makes a Random Forest random besides bootstrapping and random sampling of features? | If we set aside the discrepancies arising from roundoff error, the remaining differences originate in the treatment of ties. Class sklearn.ensemble.RandomForestClassifier is composed of many instances of sklearn.tree.DecisionTreeClassifier (you can verify this by reading the source). If we read the documentation for sk... | What makes a Random Forest random besides bootstrapping and random sampling of features? | If we set aside the discrepancies arising from roundoff error, the remaining differences originate in the treatment of ties. Class sklearn.ensemble.RandomForestClassifier is composed of many instances | What makes a Random Forest random besides bootstrapping and random sampling of features?
If we set aside the discrepancies arising from roundoff error, the remaining differences originate in the treatment of ties. Class sklearn.ensemble.RandomForestClassifier is composed of many instances of sklearn.tree.DecisionTreeCl... | What makes a Random Forest random besides bootstrapping and random sampling of features?
If we set aside the discrepancies arising from roundoff error, the remaining differences originate in the treatment of ties. Class sklearn.ensemble.RandomForestClassifier is composed of many instances |
40,567 | Var(XY), if X and Y are independent random variables [duplicate] | You can follow Henry's comments to arrive at the answer. However, another way to come to the answer is to use the fact that if $X$ and $Y$ are independent, then $Y | X = Y$ and $X |Y = X$.
By iterated expectations and variance expressions
\begin{align*}
\text{Var}(XY) & = \text{Var}[\,\text{E}(XY|X)\,] + \text{E}[\,\te... | Var(XY), if X and Y are independent random variables [duplicate] | You can follow Henry's comments to arrive at the answer. However, another way to come to the answer is to use the fact that if $X$ and $Y$ are independent, then $Y | X = Y$ and $X |Y = X$.
By iterated | Var(XY), if X and Y are independent random variables [duplicate]
You can follow Henry's comments to arrive at the answer. However, another way to come to the answer is to use the fact that if $X$ and $Y$ are independent, then $Y | X = Y$ and $X |Y = X$.
By iterated expectations and variance expressions
\begin{align*}
\... | Var(XY), if X and Y are independent random variables [duplicate]
You can follow Henry's comments to arrive at the answer. However, another way to come to the answer is to use the fact that if $X$ and $Y$ are independent, then $Y | X = Y$ and $X |Y = X$.
By iterated |
40,568 | Understanding semi supervised technique called mean teachers | I'll answer it for in case someone ever stumbles upon the need to understand the same issues. I mailed Antti Tarvainen, the author of the paper and this is what he responded:
Hi,
Thanks for the questions. I perhaps tried to pack too much into that figure so it may be hard to interpret our meaning. I will try to clarify... | Understanding semi supervised technique called mean teachers | I'll answer it for in case someone ever stumbles upon the need to understand the same issues. I mailed Antti Tarvainen, the author of the paper and this is what he responded:
Hi,
Thanks for the questi | Understanding semi supervised technique called mean teachers
I'll answer it for in case someone ever stumbles upon the need to understand the same issues. I mailed Antti Tarvainen, the author of the paper and this is what he responded:
Hi,
Thanks for the questions. I perhaps tried to pack too much into that figure so i... | Understanding semi supervised technique called mean teachers
I'll answer it for in case someone ever stumbles upon the need to understand the same issues. I mailed Antti Tarvainen, the author of the paper and this is what he responded:
Hi,
Thanks for the questi |
40,569 | Regression - variance of predictions much lower than variance of target | My answer will focus on the baseline OLS case, but the mechanics are similar for techniques like Lasso (although I'll admit that I do not know how $R^2$ is computed for such methods). Also, my answer relates to in-sample fit.
Recall that $R^2$ is defined as (also recall that the mean of the fitted values equals the mea... | Regression - variance of predictions much lower than variance of target | My answer will focus on the baseline OLS case, but the mechanics are similar for techniques like Lasso (although I'll admit that I do not know how $R^2$ is computed for such methods). Also, my answer | Regression - variance of predictions much lower than variance of target
My answer will focus on the baseline OLS case, but the mechanics are similar for techniques like Lasso (although I'll admit that I do not know how $R^2$ is computed for such methods). Also, my answer relates to in-sample fit.
Recall that $R^2$ is d... | Regression - variance of predictions much lower than variance of target
My answer will focus on the baseline OLS case, but the mechanics are similar for techniques like Lasso (although I'll admit that I do not know how $R^2$ is computed for such methods). Also, my answer |
40,570 | What is the best point forecast for lognormally distributed data? | It is a standard result from introductory statistics that the expectation of a distribution is the one number summary that will minimize the expected squared error. The expectation of the lognormal distribution with log-mean $\mu$ and log-variance $\sigma^2$ is $\exp\big(\mu+\frac{\sigma^2}{2}\big)$.
It is almost as we... | What is the best point forecast for lognormally distributed data? | It is a standard result from introductory statistics that the expectation of a distribution is the one number summary that will minimize the expected squared error. The expectation of the lognormal di | What is the best point forecast for lognormally distributed data?
It is a standard result from introductory statistics that the expectation of a distribution is the one number summary that will minimize the expected squared error. The expectation of the lognormal distribution with log-mean $\mu$ and log-variance $\sigm... | What is the best point forecast for lognormally distributed data?
It is a standard result from introductory statistics that the expectation of a distribution is the one number summary that will minimize the expected squared error. The expectation of the lognormal di |
40,571 | What is the best point forecast for lognormally distributed data? | My answers are for a distribution with known parameters. It's not specific to lognormal, but obviously applies to it too. The integrals use the lower limit of integration 0, but you can change them to $-\infty$ then they'll work for any distribution with finite mean, not only the ones with positive domain such as logno... | What is the best point forecast for lognormally distributed data? | My answers are for a distribution with known parameters. It's not specific to lognormal, but obviously applies to it too. The integrals use the lower limit of integration 0, but you can change them to | What is the best point forecast for lognormally distributed data?
My answers are for a distribution with known parameters. It's not specific to lognormal, but obviously applies to it too. The integrals use the lower limit of integration 0, but you can change them to $-\infty$ then they'll work for any distribution with... | What is the best point forecast for lognormally distributed data?
My answers are for a distribution with known parameters. It's not specific to lognormal, but obviously applies to it too. The integrals use the lower limit of integration 0, but you can change them to |
40,572 | The distribution of the product of a Bernoulli & an exponential random variable | I don't know why you call this an invalid CDF.
You have a mixed random variable $Y$:
$$Y=XZ=\begin{cases}0&,\text{ if }Z=0\\X&,\text{ if }Z=1\end{cases}$$
So the distribution function of $Y$ must be
\begin{align}
P(Y\le y)&=P(XZ\le y)
\\\\&=P(XZ\le y\mid Z=0)P(Z=0)+P(XZ\le y\mid Z=1)P(Z=1)
\\\\&=\begin{cases}P(Z=0)+P... | The distribution of the product of a Bernoulli & an exponential random variable | I don't know why you call this an invalid CDF.
You have a mixed random variable $Y$:
$$Y=XZ=\begin{cases}0&,\text{ if }Z=0\\X&,\text{ if }Z=1\end{cases}$$
So the distribution function of $Y$ must be | The distribution of the product of a Bernoulli & an exponential random variable
I don't know why you call this an invalid CDF.
You have a mixed random variable $Y$:
$$Y=XZ=\begin{cases}0&,\text{ if }Z=0\\X&,\text{ if }Z=1\end{cases}$$
So the distribution function of $Y$ must be
\begin{align}
P(Y\le y)&=P(XZ\le y)
\\\... | The distribution of the product of a Bernoulli & an exponential random variable
I don't know why you call this an invalid CDF.
You have a mixed random variable $Y$:
$$Y=XZ=\begin{cases}0&,\text{ if }Z=0\\X&,\text{ if }Z=1\end{cases}$$
So the distribution function of $Y$ must be |
40,573 | The distribution of the product of a Bernoulli & an exponential random variable | The first three central moments of $Y$ are as follows:
The expectation of $Y$ can be written as:
$$\displaystyle \mu_{{1}}\, = \, E(Y) \, = \, 0 \, (1-p) \, + \, \int_{0}^{\infty }\!\, p \, y \, c \,{{\rm e}^{- c \,y}}\,{\rm d}y \,= \, \frac {p}{ c }$$
The variance of $Y$ can be writt
en as:
$$\displaystyle \mu_{{... | The distribution of the product of a Bernoulli & an exponential random variable | The first three central moments of $Y$ are as follows:
The expectation of $Y$ can be written as:
$$\displaystyle \mu_{{1}}\, = \, E(Y) \, = \, 0 \, (1-p) \, + \, \int_{0}^{\infty }\!\, p \, y \, c | The distribution of the product of a Bernoulli & an exponential random variable
The first three central moments of $Y$ are as follows:
The expectation of $Y$ can be written as:
$$\displaystyle \mu_{{1}}\, = \, E(Y) \, = \, 0 \, (1-p) \, + \, \int_{0}^{\infty }\!\, p \, y \, c \,{{\rm e}^{- c \,y}}\,{\rm d}y \,= \,... | The distribution of the product of a Bernoulli & an exponential random variable
The first three central moments of $Y$ are as follows:
The expectation of $Y$ can be written as:
$$\displaystyle \mu_{{1}}\, = \, E(Y) \, = \, 0 \, (1-p) \, + \, \int_{0}^{\infty }\!\, p \, y \, c |
40,574 | Can anyone help explain this basic example of posterior | Sorry for being confusing! The joint posterior distribution on $(\xi_1,\xi_2)$ is
$$\pi(\xi_1,\xi_2|x)\propto \exp\{-(x-\xi_1)^2/2\}\pi_1(2\xi_1)\pi_2(2\xi_2)$$
Therefore the marginal posterior on $\xi_2$ is given by the marginal of the above, up to a constant, that is,
$$\pi(\xi_2|x)\propto \int\exp\{-(x-\xi_1)^2/2\}\... | Can anyone help explain this basic example of posterior | Sorry for being confusing! The joint posterior distribution on $(\xi_1,\xi_2)$ is
$$\pi(\xi_1,\xi_2|x)\propto \exp\{-(x-\xi_1)^2/2\}\pi_1(2\xi_1)\pi_2(2\xi_2)$$
Therefore the marginal posterior on $\x | Can anyone help explain this basic example of posterior
Sorry for being confusing! The joint posterior distribution on $(\xi_1,\xi_2)$ is
$$\pi(\xi_1,\xi_2|x)\propto \exp\{-(x-\xi_1)^2/2\}\pi_1(2\xi_1)\pi_2(2\xi_2)$$
Therefore the marginal posterior on $\xi_2$ is given by the marginal of the above, up to a constant, th... | Can anyone help explain this basic example of posterior
Sorry for being confusing! The joint posterior distribution on $(\xi_1,\xi_2)$ is
$$\pi(\xi_1,\xi_2|x)\propto \exp\{-(x-\xi_1)^2/2\}\pi_1(2\xi_1)\pi_2(2\xi_2)$$
Therefore the marginal posterior on $\x |
40,575 | Can anyone help explain this basic example of posterior | I am not providing another answer to the question, but instead another example of the same sort as presented in the question. The example I present came up in the course of my research (as part of a Gibbs sampler within a Dirichlet process mixture model); in that sense it is not "contrived."
There is one observation $... | Can anyone help explain this basic example of posterior | I am not providing another answer to the question, but instead another example of the same sort as presented in the question. The example I present came up in the course of my research (as part of a G | Can anyone help explain this basic example of posterior
I am not providing another answer to the question, but instead another example of the same sort as presented in the question. The example I present came up in the course of my research (as part of a Gibbs sampler within a Dirichlet process mixture model); in that ... | Can anyone help explain this basic example of posterior
I am not providing another answer to the question, but instead another example of the same sort as presented in the question. The example I present came up in the course of my research (as part of a G |
40,576 | Reparameterization trick for gamma distribution | The answer is "yes" in one sense and "no" in another sense.
Suppose $X \sim \operatorname{Gamma}(\alpha,\beta)$. Let $F_{\alpha,\beta}$ denote the cdf of a Gamma distribution. Then define $\epsilon = \Phi^{-1}[F_{\alpha,\beta}(X)]$. Then if you simulate $\epsilon \sim N(0,1)$ you can get the relevant gamma distributio... | Reparameterization trick for gamma distribution | The answer is "yes" in one sense and "no" in another sense.
Suppose $X \sim \operatorname{Gamma}(\alpha,\beta)$. Let $F_{\alpha,\beta}$ denote the cdf of a Gamma distribution. Then define $\epsilon = | Reparameterization trick for gamma distribution
The answer is "yes" in one sense and "no" in another sense.
Suppose $X \sim \operatorname{Gamma}(\alpha,\beta)$. Let $F_{\alpha,\beta}$ denote the cdf of a Gamma distribution. Then define $\epsilon = \Phi^{-1}[F_{\alpha,\beta}(X)]$. Then if you simulate $\epsilon \sim N(... | Reparameterization trick for gamma distribution
The answer is "yes" in one sense and "no" in another sense.
Suppose $X \sim \operatorname{Gamma}(\alpha,\beta)$. Let $F_{\alpha,\beta}$ denote the cdf of a Gamma distribution. Then define $\epsilon = |
40,577 | Reparameterization trick for gamma distribution | In terms of the strategy from the VAE paper, no there's not a straightforward way of using a gamma distribution (as guy explained). However, there's a lot of research that's gone into other strategies for reparameterizing gammas to use in the VAE type framework:
This paper by Knowles https://arxiv.org/pdf/1509.01631.p... | Reparameterization trick for gamma distribution | In terms of the strategy from the VAE paper, no there's not a straightforward way of using a gamma distribution (as guy explained). However, there's a lot of research that's gone into other strategies | Reparameterization trick for gamma distribution
In terms of the strategy from the VAE paper, no there's not a straightforward way of using a gamma distribution (as guy explained). However, there's a lot of research that's gone into other strategies for reparameterizing gammas to use in the VAE type framework:
This pap... | Reparameterization trick for gamma distribution
In terms of the strategy from the VAE paper, no there's not a straightforward way of using a gamma distribution (as guy explained). However, there's a lot of research that's gone into other strategies |
40,578 | MAP estimation as regularisation of MLE | Maximum likelihood method aims at finding model parameters that best match some data:
$$
\theta_{ML}=\mathrm{argmax}_\theta \,p(x|y,\theta)
$$
Maximum likelihood does not use any prior knowledge about the expected distribution of the parameters $\theta$ and thus may overfit to the particular data $x$, $y$.
Maximum a-po... | MAP estimation as regularisation of MLE | Maximum likelihood method aims at finding model parameters that best match some data:
$$
\theta_{ML}=\mathrm{argmax}_\theta \,p(x|y,\theta)
$$
Maximum likelihood does not use any prior knowledge about | MAP estimation as regularisation of MLE
Maximum likelihood method aims at finding model parameters that best match some data:
$$
\theta_{ML}=\mathrm{argmax}_\theta \,p(x|y,\theta)
$$
Maximum likelihood does not use any prior knowledge about the expected distribution of the parameters $\theta$ and thus may overfit to th... | MAP estimation as regularisation of MLE
Maximum likelihood method aims at finding model parameters that best match some data:
$$
\theta_{ML}=\mathrm{argmax}_\theta \,p(x|y,\theta)
$$
Maximum likelihood does not use any prior knowledge about |
40,579 | How to you measure the accuracy of a model that gives quantile forecasts or distributions of forecasts? | The standard approach is to use probability scoring. See Gneiting and Katzfuss (2014) for some of the mathematical background.
One example of a probability scoring measure is quantile scoring based on the pinball loss function. For each time period throughout the forecast horizon, you compute the $0.01, 0.02, \dots, 0.... | How to you measure the accuracy of a model that gives quantile forecasts or distributions of forecas | The standard approach is to use probability scoring. See Gneiting and Katzfuss (2014) for some of the mathematical background.
One example of a probability scoring measure is quantile scoring based on | How to you measure the accuracy of a model that gives quantile forecasts or distributions of forecasts?
The standard approach is to use probability scoring. See Gneiting and Katzfuss (2014) for some of the mathematical background.
One example of a probability scoring measure is quantile scoring based on the pinball los... | How to you measure the accuracy of a model that gives quantile forecasts or distributions of forecas
The standard approach is to use probability scoring. See Gneiting and Katzfuss (2014) for some of the mathematical background.
One example of a probability scoring measure is quantile scoring based on |
40,580 | Puzzled by definition of sufficient statistics | All those interpretations seem to be a variation of expressing the same thing:
The independence of the distribution of the sample $X$ on a true parameter $\theta$ and the statistic $T$.
Which means that the sample $X$, conditional on $T$, does not tell any more information (beyond the information from the statistic $T$... | Puzzled by definition of sufficient statistics | All those interpretations seem to be a variation of expressing the same thing:
The independence of the distribution of the sample $X$ on a true parameter $\theta$ and the statistic $T$.
Which means th | Puzzled by definition of sufficient statistics
All those interpretations seem to be a variation of expressing the same thing:
The independence of the distribution of the sample $X$ on a true parameter $\theta$ and the statistic $T$.
Which means that the sample $X$, conditional on $T$, does not tell any more information... | Puzzled by definition of sufficient statistics
All those interpretations seem to be a variation of expressing the same thing:
The independence of the distribution of the sample $X$ on a true parameter $\theta$ and the statistic $T$.
Which means th |
40,581 | Puzzled by definition of sufficient statistics | It's a somewhat stronger condition to state that the joint probability distribution function (aka $F_X = P(X_1 < x_1, \ldots, X_n < x_n)$) doesn't depend on $\theta$. I could for instance have a modified normal ($\mu$,1) distribution whose probability density function is $f_x = \phi(x)$ if $x \ne \mu$, and $1,000$ when... | Puzzled by definition of sufficient statistics | It's a somewhat stronger condition to state that the joint probability distribution function (aka $F_X = P(X_1 < x_1, \ldots, X_n < x_n)$) doesn't depend on $\theta$. I could for instance have a modif | Puzzled by definition of sufficient statistics
It's a somewhat stronger condition to state that the joint probability distribution function (aka $F_X = P(X_1 < x_1, \ldots, X_n < x_n)$) doesn't depend on $\theta$. I could for instance have a modified normal ($\mu$,1) distribution whose probability density function is $... | Puzzled by definition of sufficient statistics
It's a somewhat stronger condition to state that the joint probability distribution function (aka $F_X = P(X_1 < x_1, \ldots, X_n < x_n)$) doesn't depend on $\theta$. I could for instance have a modif |
40,582 | Deterministic or stochastic universe in frequentist statistics? | I respectfully disagree with the answer by Frans; there is nothing in the frequentist methodology that takes any position on whether data-generating processes, as modeled by statistics, are deterministic or not. (While Wikipedia is a useful source for some statistical material, this unreferenced sentence does not hold... | Deterministic or stochastic universe in frequentist statistics? | I respectfully disagree with the answer by Frans; there is nothing in the frequentist methodology that takes any position on whether data-generating processes, as modeled by statistics, are determinis | Deterministic or stochastic universe in frequentist statistics?
I respectfully disagree with the answer by Frans; there is nothing in the frequentist methodology that takes any position on whether data-generating processes, as modeled by statistics, are deterministic or not. (While Wikipedia is a useful source for som... | Deterministic or stochastic universe in frequentist statistics?
I respectfully disagree with the answer by Frans; there is nothing in the frequentist methodology that takes any position on whether data-generating processes, as modeled by statistics, are determinis |
40,583 | Deterministic or stochastic universe in frequentist statistics? | Interesting question!
I would say from a statistical modelling perspective, all data are assumed to come from a combination of systematic components and a stochastic component, which means that data-generating processes as modeled by statistics, are assumed to be non-deterministic in nature. Wikipedia even states that... | Deterministic or stochastic universe in frequentist statistics? | Interesting question!
I would say from a statistical modelling perspective, all data are assumed to come from a combination of systematic components and a stochastic component, which means that data- | Deterministic or stochastic universe in frequentist statistics?
Interesting question!
I would say from a statistical modelling perspective, all data are assumed to come from a combination of systematic components and a stochastic component, which means that data-generating processes as modeled by statistics, are assum... | Deterministic or stochastic universe in frequentist statistics?
Interesting question!
I would say from a statistical modelling perspective, all data are assumed to come from a combination of systematic components and a stochastic component, which means that data- |
40,584 | How to overcome the computational cost of the KNN algorithm? | There's a large literature on speeding up nearest neighbor search, as well as numerous software libraries (you might consider using one of these instead of re-implementing from scratch). I'll describe some general classes of solutions below. Most of these methods aim to reduce the amount of computation by exploiting so... | How to overcome the computational cost of the KNN algorithm? | There's a large literature on speeding up nearest neighbor search, as well as numerous software libraries (you might consider using one of these instead of re-implementing from scratch). I'll describe | How to overcome the computational cost of the KNN algorithm?
There's a large literature on speeding up nearest neighbor search, as well as numerous software libraries (you might consider using one of these instead of re-implementing from scratch). I'll describe some general classes of solutions below. Most of these met... | How to overcome the computational cost of the KNN algorithm?
There's a large literature on speeding up nearest neighbor search, as well as numerous software libraries (you might consider using one of these instead of re-implementing from scratch). I'll describe |
40,585 | what does it mean by more "efficient" estimator | I am just wondering, when comparing two estimator says T1
and T2, what does it mean by saying T1 is more efficient than T2
https://en.wikipedia.org/wiki/Efficiency_(statistics)
For an unbiased estimator, efficiency is the precision of the estimator (reciprocal of the variance) divided by the upper bound of the preci... | what does it mean by more "efficient" estimator | I am just wondering, when comparing two estimator says T1
and T2, what does it mean by saying T1 is more efficient than T2
https://en.wikipedia.org/wiki/Efficiency_(statistics)
For an unbiased esti | what does it mean by more "efficient" estimator
I am just wondering, when comparing two estimator says T1
and T2, what does it mean by saying T1 is more efficient than T2
https://en.wikipedia.org/wiki/Efficiency_(statistics)
For an unbiased estimator, efficiency is the precision of the estimator (reciprocal of the v... | what does it mean by more "efficient" estimator
I am just wondering, when comparing two estimator says T1
and T2, what does it mean by saying T1 is more efficient than T2
https://en.wikipedia.org/wiki/Efficiency_(statistics)
For an unbiased esti |
40,586 | Reasonable hyperparameter range for Latent Dirichlet Allocation? | Choice of $\alpha$ and $\beta$ is indeed tricky, since it impacts the topic modeling results. The Gibbs sampling paper by Griffiths et al. gives some insight into this:
The value of $\beta$ thus affects the granularity of the model: a
corpus of documents can be sensibly factorized into a set of topics at
several ... | Reasonable hyperparameter range for Latent Dirichlet Allocation? | Choice of $\alpha$ and $\beta$ is indeed tricky, since it impacts the topic modeling results. The Gibbs sampling paper by Griffiths et al. gives some insight into this:
The value of $\beta$ thus aff | Reasonable hyperparameter range for Latent Dirichlet Allocation?
Choice of $\alpha$ and $\beta$ is indeed tricky, since it impacts the topic modeling results. The Gibbs sampling paper by Griffiths et al. gives some insight into this:
The value of $\beta$ thus affects the granularity of the model: a
corpus of docume... | Reasonable hyperparameter range for Latent Dirichlet Allocation?
Choice of $\alpha$ and $\beta$ is indeed tricky, since it impacts the topic modeling results. The Gibbs sampling paper by Griffiths et al. gives some insight into this:
The value of $\beta$ thus aff |
40,587 | GAM with categorical variables - interpretation | In a factor by variable smooth, like other simple smooths, the bases for the smooths are subject to identifiability constraints. If you just naively computed the basis of the required dimension, and given the defaults for s(), you'd get 2 basis functions that are in the null space of the smoothness penalty:
a flat, ho... | GAM with categorical variables - interpretation | In a factor by variable smooth, like other simple smooths, the bases for the smooths are subject to identifiability constraints. If you just naively computed the basis of the required dimension, and g | GAM with categorical variables - interpretation
In a factor by variable smooth, like other simple smooths, the bases for the smooths are subject to identifiability constraints. If you just naively computed the basis of the required dimension, and given the defaults for s(), you'd get 2 basis functions that are in the n... | GAM with categorical variables - interpretation
In a factor by variable smooth, like other simple smooths, the bases for the smooths are subject to identifiability constraints. If you just naively computed the basis of the required dimension, and g |
40,588 | Implicit hypothesis testing: mean greater than variance and Delta Method | What distinguishes the first order test statistic (normal distribution) from the second order test statistic (Chi-squared distribution)
With the first order approximation you approximate the function $g(\theta)$ as a linear function of $\theta$. But this works only when there is actually a slope, that is when $g(\thet... | Implicit hypothesis testing: mean greater than variance and Delta Method | What distinguishes the first order test statistic (normal distribution) from the second order test statistic (Chi-squared distribution)
With the first order approximation you approximate the function | Implicit hypothesis testing: mean greater than variance and Delta Method
What distinguishes the first order test statistic (normal distribution) from the second order test statistic (Chi-squared distribution)
With the first order approximation you approximate the function $g(\theta)$ as a linear function of $\theta$. ... | Implicit hypothesis testing: mean greater than variance and Delta Method
What distinguishes the first order test statistic (normal distribution) from the second order test statistic (Chi-squared distribution)
With the first order approximation you approximate the function |
40,589 | Implicit hypothesis testing: mean greater than variance and Delta Method | Few notes:
1)In last eqaution $(\hat \mu - \hat \sigma)$ should be squared.
2)Cov. matrix $\Sigma(\mu, \sigma^2) $ shouldn't depend on $n$, it is asymptotic entity. So as $\Gamma$ which in this case is equal $\Gamma = \frac{3 \sigma^2}{2}$.
So the final answer is $T_n = \frac{2n^2 (\hat \mu - \hat \sigma)^2}{3 \hat \... | Implicit hypothesis testing: mean greater than variance and Delta Method | Few notes:
1)In last eqaution $(\hat \mu - \hat \sigma)$ should be squared.
2)Cov. matrix $\Sigma(\mu, \sigma^2) $ shouldn't depend on $n$, it is asymptotic entity. So as $\Gamma$ which in this case | Implicit hypothesis testing: mean greater than variance and Delta Method
Few notes:
1)In last eqaution $(\hat \mu - \hat \sigma)$ should be squared.
2)Cov. matrix $\Sigma(\mu, \sigma^2) $ shouldn't depend on $n$, it is asymptotic entity. So as $\Gamma$ which in this case is equal $\Gamma = \frac{3 \sigma^2}{2}$.
So th... | Implicit hypothesis testing: mean greater than variance and Delta Method
Few notes:
1)In last eqaution $(\hat \mu - \hat \sigma)$ should be squared.
2)Cov. matrix $\Sigma(\mu, \sigma^2) $ shouldn't depend on $n$, it is asymptotic entity. So as $\Gamma$ which in this case |
40,590 | How does lifelines calculate baseline hazard in CoxPHFitter? | I'm the author of lifelines and can help you.
Does it mean that I can specify it as any functions and select the best one during experiments?
Not normally, no. (In fact, if you could, that would make the Cox model fully parametric, and not semi parametric). It being "unspecified" means it is non-parametric. If you st... | How does lifelines calculate baseline hazard in CoxPHFitter? | I'm the author of lifelines and can help you.
Does it mean that I can specify it as any functions and select the best one during experiments?
Not normally, no. (In fact, if you could, that would mak | How does lifelines calculate baseline hazard in CoxPHFitter?
I'm the author of lifelines and can help you.
Does it mean that I can specify it as any functions and select the best one during experiments?
Not normally, no. (In fact, if you could, that would make the Cox model fully parametric, and not semi parametric).... | How does lifelines calculate baseline hazard in CoxPHFitter?
I'm the author of lifelines and can help you.
Does it mean that I can specify it as any functions and select the best one during experiments?
Not normally, no. (In fact, if you could, that would mak |
40,591 | What is the probability that a student gets a better score than another on a test with randomly selected questions? | A dynamic program will make short work of this.
Suppose we administer all questions to the students and then randomly select a subset $\mathcal{I}$ of $k=10$ out of all $n=100$ questions. Let's define a random variable $X_i$ to compare the two students on question $i:$ set it to $1$ if student A is correct and studen... | What is the probability that a student gets a better score than another on a test with randomly sele | A dynamic program will make short work of this.
Suppose we administer all questions to the students and then randomly select a subset $\mathcal{I}$ of $k=10$ out of all $n=100$ questions. Let's defi | What is the probability that a student gets a better score than another on a test with randomly selected questions?
A dynamic program will make short work of this.
Suppose we administer all questions to the students and then randomly select a subset $\mathcal{I}$ of $k=10$ out of all $n=100$ questions. Let's define a... | What is the probability that a student gets a better score than another on a test with randomly sele
A dynamic program will make short work of this.
Suppose we administer all questions to the students and then randomly select a subset $\mathcal{I}$ of $k=10$ out of all $n=100$ questions. Let's defi |
40,592 | What is the probability that a student gets a better score than another on a test with randomly selected questions? | At each iteration,
$P(A>B)=P(A)∗(1−P(B))$
because A is only better when A is right and B is wrong. So for a particular question, if A is right 90% of the time and B is right 80% of the time, then the joint probability that A is right and B is wrong is
$ P(A,B) = 0.9 ∗ 0.2 = 0.18 $
Now you could write some code that goe... | What is the probability that a student gets a better score than another on a test with randomly sele | At each iteration,
$P(A>B)=P(A)∗(1−P(B))$
because A is only better when A is right and B is wrong. So for a particular question, if A is right 90% of the time and B is right 80% of the time, then the | What is the probability that a student gets a better score than another on a test with randomly selected questions?
At each iteration,
$P(A>B)=P(A)∗(1−P(B))$
because A is only better when A is right and B is wrong. So for a particular question, if A is right 90% of the time and B is right 80% of the time, then the join... | What is the probability that a student gets a better score than another on a test with randomly sele
At each iteration,
$P(A>B)=P(A)∗(1−P(B))$
because A is only better when A is right and B is wrong. So for a particular question, if A is right 90% of the time and B is right 80% of the time, then the |
40,593 | For creating the joint posterior distribution for multiple variables, are the associated Bayesian priors usually assume independent of each other? | Your expression is only correct when you assume independent priors. Otherwise, the expression would become
$$
p(\theta_1, \theta_2 | D) \propto p(D|\theta_1, \theta_2) p(\theta_1, \theta_2)
$$
In this expression, you may need further assumptions to work with $p(\theta_1, \theta_2)$ or you can work with it as is. If you... | For creating the joint posterior distribution for multiple variables, are the associated Bayesian pr | Your expression is only correct when you assume independent priors. Otherwise, the expression would become
$$
p(\theta_1, \theta_2 | D) \propto p(D|\theta_1, \theta_2) p(\theta_1, \theta_2)
$$
In this | For creating the joint posterior distribution for multiple variables, are the associated Bayesian priors usually assume independent of each other?
Your expression is only correct when you assume independent priors. Otherwise, the expression would become
$$
p(\theta_1, \theta_2 | D) \propto p(D|\theta_1, \theta_2) p(\th... | For creating the joint posterior distribution for multiple variables, are the associated Bayesian pr
Your expression is only correct when you assume independent priors. Otherwise, the expression would become
$$
p(\theta_1, \theta_2 | D) \propto p(D|\theta_1, \theta_2) p(\theta_1, \theta_2)
$$
In this |
40,594 | For creating the joint posterior distribution for multiple variables, are the associated Bayesian priors usually assume independent of each other? | As Maurits M mentions in another answer whether independence makes sense is really problem specific. The OP question asked:
"I am wondering why most set-ups assume that the priors above are independent. What happens if we do not have it?"
which is really 2 questions.
(1) Why most set-ups assume the priors are indepen... | For creating the joint posterior distribution for multiple variables, are the associated Bayesian pr | As Maurits M mentions in another answer whether independence makes sense is really problem specific. The OP question asked:
"I am wondering why most set-ups assume that the priors above are independe | For creating the joint posterior distribution for multiple variables, are the associated Bayesian priors usually assume independent of each other?
As Maurits M mentions in another answer whether independence makes sense is really problem specific. The OP question asked:
"I am wondering why most set-ups assume that the... | For creating the joint posterior distribution for multiple variables, are the associated Bayesian pr
As Maurits M mentions in another answer whether independence makes sense is really problem specific. The OP question asked:
"I am wondering why most set-ups assume that the priors above are independe |
40,595 | Derivative of the Joint Distribution Interpretation | The first-order partial derivatives of a multivariate joint distribution function can be considered as giving the density of the differentiated variable, jointly with the cumulative probability of the other variable(s). One simple way to see this interpretation is to convert the partial derivative to a density integra... | Derivative of the Joint Distribution Interpretation | The first-order partial derivatives of a multivariate joint distribution function can be considered as giving the density of the differentiated variable, jointly with the cumulative probability of the | Derivative of the Joint Distribution Interpretation
The first-order partial derivatives of a multivariate joint distribution function can be considered as giving the density of the differentiated variable, jointly with the cumulative probability of the other variable(s). One simple way to see this interpretation is to... | Derivative of the Joint Distribution Interpretation
The first-order partial derivatives of a multivariate joint distribution function can be considered as giving the density of the differentiated variable, jointly with the cumulative probability of the |
40,596 | Derivative of the Joint Distribution Interpretation | If you take the joint CDF over xy and derive it over just one of the variables - you're left with marginal PDF for that same variable.
Let's prove using a simple joint distribution of two i.i.d. RVs X and Y ~Expo(1)
On one hand, we can get to the marginal PDF through the joint PDF:
$$
F_{XY}(x,y) = \iint_{XY} f_{XY}(... | Derivative of the Joint Distribution Interpretation | If you take the joint CDF over xy and derive it over just one of the variables - you're left with marginal PDF for that same variable.
Let's prove using a simple joint distribution of two i.i.d. RVs X | Derivative of the Joint Distribution Interpretation
If you take the joint CDF over xy and derive it over just one of the variables - you're left with marginal PDF for that same variable.
Let's prove using a simple joint distribution of two i.i.d. RVs X and Y ~Expo(1)
On one hand, we can get to the marginal PDF through ... | Derivative of the Joint Distribution Interpretation
If you take the joint CDF over xy and derive it over just one of the variables - you're left with marginal PDF for that same variable.
Let's prove using a simple joint distribution of two i.i.d. RVs X |
40,597 | Study path to Bayesian thinking? | I've started down my own path towards understanding the bayesian way of thinking and I'll share my perspective. I started off reading classic papers on the various samplers and going through derivations for the conjugate cases, and I don't think that got me very far. True, the elites are going to write their own sample... | Study path to Bayesian thinking? | I've started down my own path towards understanding the bayesian way of thinking and I'll share my perspective. I started off reading classic papers on the various samplers and going through derivatio | Study path to Bayesian thinking?
I've started down my own path towards understanding the bayesian way of thinking and I'll share my perspective. I started off reading classic papers on the various samplers and going through derivations for the conjugate cases, and I don't think that got me very far. True, the elites ar... | Study path to Bayesian thinking?
I've started down my own path towards understanding the bayesian way of thinking and I'll share my perspective. I started off reading classic papers on the various samplers and going through derivatio |
40,598 | Study path to Bayesian thinking? | From your business point of view you might be motivated by Bayesian decision theory, which is a way to apply Bayesian inference to make decisions under uncertainty.
If so, you'd find the topics that introductions to Bayesian analysis often focus on (such as specifying various prior and likelihood distributions and perf... | Study path to Bayesian thinking? | From your business point of view you might be motivated by Bayesian decision theory, which is a way to apply Bayesian inference to make decisions under uncertainty.
If so, you'd find the topics that i | Study path to Bayesian thinking?
From your business point of view you might be motivated by Bayesian decision theory, which is a way to apply Bayesian inference to make decisions under uncertainty.
If so, you'd find the topics that introductions to Bayesian analysis often focus on (such as specifying various prior and ... | Study path to Bayesian thinking?
From your business point of view you might be motivated by Bayesian decision theory, which is a way to apply Bayesian inference to make decisions under uncertainty.
If so, you'd find the topics that i |
40,599 | Study path to Bayesian thinking? | I had a course on Bayesian Data Analysis in the last semester. It assumes no previous background. Here's the course homepage where the instructor has put all of the materials: https://michael-franke.github.io/BDACM_2017/
We used the Kruschke textbook for the course. It worked out fine. I don't think there's much proble... | Study path to Bayesian thinking? | I had a course on Bayesian Data Analysis in the last semester. It assumes no previous background. Here's the course homepage where the instructor has put all of the materials: https://michael-franke.g | Study path to Bayesian thinking?
I had a course on Bayesian Data Analysis in the last semester. It assumes no previous background. Here's the course homepage where the instructor has put all of the materials: https://michael-franke.github.io/BDACM_2017/
We used the Kruschke textbook for the course. It worked out fine. ... | Study path to Bayesian thinking?
I had a course on Bayesian Data Analysis in the last semester. It assumes no previous background. Here's the course homepage where the instructor has put all of the materials: https://michael-franke.g |
40,600 | Is this a valid way to construct a confidence interval? | Hanley and Lippman-Hand (1983) give something like the following argument which provides motivation for the rule. Taking $n$ as fixed, $P(X=0|p) =(1-p)^n$.
Solving $(1-p)^n \geq \alpha\,$ for $p\,$ we get $p\geq 1-\alpha^{\frac{1}{n}}$. The smallest $p$ that keeps the probability of $0$ no less than $\alpha$ is $1-\al... | Is this a valid way to construct a confidence interval? | Hanley and Lippman-Hand (1983) give something like the following argument which provides motivation for the rule. Taking $n$ as fixed, $P(X=0|p) =(1-p)^n$.
Solving $(1-p)^n \geq \alpha\,$ for $p\,$ w | Is this a valid way to construct a confidence interval?
Hanley and Lippman-Hand (1983) give something like the following argument which provides motivation for the rule. Taking $n$ as fixed, $P(X=0|p) =(1-p)^n$.
Solving $(1-p)^n \geq \alpha\,$ for $p\,$ we get $p\geq 1-\alpha^{\frac{1}{n}}$. The smallest $p$ that keep... | Is this a valid way to construct a confidence interval?
Hanley and Lippman-Hand (1983) give something like the following argument which provides motivation for the rule. Taking $n$ as fixed, $P(X=0|p) =(1-p)^n$.
Solving $(1-p)^n \geq \alpha\,$ for $p\,$ w |
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