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 |
|---|---|---|---|---|---|---|
54,701 | Flawed multiple linear regression in academia? Heteroscedasticity's effect on p-value? | This isn't heteroscedasticity you are looking at, but truncation.
You can see this very clearly in the first plot: No combination of the fitted + residual exceeds a certain number, causing this sudden imaginary diagonal line, past which no observations exist. In the scale-location plot, this strange shape reveals that ... | Flawed multiple linear regression in academia? Heteroscedasticity's effect on p-value? | This isn't heteroscedasticity you are looking at, but truncation.
You can see this very clearly in the first plot: No combination of the fitted + residual exceeds a certain number, causing this sudden | Flawed multiple linear regression in academia? Heteroscedasticity's effect on p-value?
This isn't heteroscedasticity you are looking at, but truncation.
You can see this very clearly in the first plot: No combination of the fitted + residual exceeds a certain number, causing this sudden imaginary diagonal line, past wh... | Flawed multiple linear regression in academia? Heteroscedasticity's effect on p-value?
This isn't heteroscedasticity you are looking at, but truncation.
You can see this very clearly in the first plot: No combination of the fitted + residual exceeds a certain number, causing this sudden |
54,702 | Is classification using linear regression called logistic regression or linear disriminant analysis? | They are both close, but in different ways
If you run ordinary least-squares regression with a binary class variable as the outcome (label) variable, you get exactly the 2-class case of linear discriminant analysis. So LDA (in the 2-class case) is linear regression run on a classification problem. It's conceptually di... | Is classification using linear regression called logistic regression or linear disriminant analysis? | They are both close, but in different ways
If you run ordinary least-squares regression with a binary class variable as the outcome (label) variable, you get exactly the 2-class case of linear discri | Is classification using linear regression called logistic regression or linear disriminant analysis?
They are both close, but in different ways
If you run ordinary least-squares regression with a binary class variable as the outcome (label) variable, you get exactly the 2-class case of linear discriminant analysis. So... | Is classification using linear regression called logistic regression or linear disriminant analysis?
They are both close, but in different ways
If you run ordinary least-squares regression with a binary class variable as the outcome (label) variable, you get exactly the 2-class case of linear discri |
54,703 | Bernouilli ML parameter estimation from indirect observations | The model is a mixture of Bernoullis, with likelihood
$$L(p)=\prod_{t=1}^n \{pa_t+(1-p)b_t\}$$
a polynomial of degree $n$ in $p$.
Since this distribution is not an exponential family, there is no sufficient statistic of fixed dimension and hence no way to update the maximum likelihood estimator in the way you describe... | Bernouilli ML parameter estimation from indirect observations | The model is a mixture of Bernoullis, with likelihood
$$L(p)=\prod_{t=1}^n \{pa_t+(1-p)b_t\}$$
a polynomial of degree $n$ in $p$.
Since this distribution is not an exponential family, there is no suf | Bernouilli ML parameter estimation from indirect observations
The model is a mixture of Bernoullis, with likelihood
$$L(p)=\prod_{t=1}^n \{pa_t+(1-p)b_t\}$$
a polynomial of degree $n$ in $p$.
Since this distribution is not an exponential family, there is no sufficient statistic of fixed dimension and hence no way to u... | Bernouilli ML parameter estimation from indirect observations
The model is a mixture of Bernoullis, with likelihood
$$L(p)=\prod_{t=1}^n \{pa_t+(1-p)b_t\}$$
a polynomial of degree $n$ in $p$.
Since this distribution is not an exponential family, there is no suf |
54,704 | KL divergence for joint probability distributions? | KL divergence is defined between two distributions, period. If this is marginal or joint distributions is immaterial. You want them to have the same support. So you do the same as in single dimension. Asker in a comment says
Thanks for your useful answer. In the KL divergence, we must calculate
$\log p/q$ for probabil... | KL divergence for joint probability distributions? | KL divergence is defined between two distributions, period. If this is marginal or joint distributions is immaterial. You want them to have the same support. So you do the same as in single dimension. | KL divergence for joint probability distributions?
KL divergence is defined between two distributions, period. If this is marginal or joint distributions is immaterial. You want them to have the same support. So you do the same as in single dimension. Asker in a comment says
Thanks for your useful answer. In the KL di... | KL divergence for joint probability distributions?
KL divergence is defined between two distributions, period. If this is marginal or joint distributions is immaterial. You want them to have the same support. So you do the same as in single dimension. |
54,705 | What does it mean for OLS residuals to be independent from the fitted values? | First consider what the definition of independence of two vector-valued random variables $x$ and $y$ comes down to: the probability that $x$ is in some event $\mathcal A$ and $y$ is in some event $\mathcal B$ is the product of the chances of these events.
It helps to recast this in terms of conditional probabilities: i... | What does it mean for OLS residuals to be independent from the fitted values? | First consider what the definition of independence of two vector-valued random variables $x$ and $y$ comes down to: the probability that $x$ is in some event $\mathcal A$ and $y$ is in some event $\ma | What does it mean for OLS residuals to be independent from the fitted values?
First consider what the definition of independence of two vector-valued random variables $x$ and $y$ comes down to: the probability that $x$ is in some event $\mathcal A$ and $y$ is in some event $\mathcal B$ is the product of the chances of ... | What does it mean for OLS residuals to be independent from the fitted values?
First consider what the definition of independence of two vector-valued random variables $x$ and $y$ comes down to: the probability that $x$ is in some event $\mathcal A$ and $y$ is in some event $\ma |
54,706 | What does it mean for OLS residuals to be independent from the fitted values? | Geometrically, it means the error is orthogonal to the prediction.
Roughly, OLS finds a point in the column space of $X$ which is closest to $y$ (assuming $y$ is not already in the column space). In the included picture, the vector in the grey plane is the prediction, and the vector outside the plane (connected to the... | What does it mean for OLS residuals to be independent from the fitted values? | Geometrically, it means the error is orthogonal to the prediction.
Roughly, OLS finds a point in the column space of $X$ which is closest to $y$ (assuming $y$ is not already in the column space). In | What does it mean for OLS residuals to be independent from the fitted values?
Geometrically, it means the error is orthogonal to the prediction.
Roughly, OLS finds a point in the column space of $X$ which is closest to $y$ (assuming $y$ is not already in the column space). In the included picture, the vector in the gr... | What does it mean for OLS residuals to be independent from the fitted values?
Geometrically, it means the error is orthogonal to the prediction.
Roughly, OLS finds a point in the column space of $X$ which is closest to $y$ (assuming $y$ is not already in the column space). In |
54,707 | Is there no such thing as a multivariate generalized linear mixed model? | Yes, there is such a thing as a Multivariate (multi-response) Generalized Linear Mixed Model (MGLMM)
Many popular software packages for fitting GLMMs are unable to handle multiple responses, especially those that work utilise the frequentist paradigm. However if you adopt a Bayesian approach then there are a number of ... | Is there no such thing as a multivariate generalized linear mixed model? | Yes, there is such a thing as a Multivariate (multi-response) Generalized Linear Mixed Model (MGLMM)
Many popular software packages for fitting GLMMs are unable to handle multiple responses, especiall | Is there no such thing as a multivariate generalized linear mixed model?
Yes, there is such a thing as a Multivariate (multi-response) Generalized Linear Mixed Model (MGLMM)
Many popular software packages for fitting GLMMs are unable to handle multiple responses, especially those that work utilise the frequentist parad... | Is there no such thing as a multivariate generalized linear mixed model?
Yes, there is such a thing as a Multivariate (multi-response) Generalized Linear Mixed Model (MGLMM)
Many popular software packages for fitting GLMMs are unable to handle multiple responses, especiall |
54,708 | Understand a statement about P value | If your data is $\mathcal{D}$, and your hypothesis $H_0$ then the p-value is $ p = \mathbb{P}(\mathcal{D}\mid H_0)$.
The $p$ value tells you the following:
If $H_0$ is true, how likely is the data I'm currently observing ?
So if $p$ is very low, it only means the data cannot easily happen in a world in which $H_0$ is... | Understand a statement about P value | If your data is $\mathcal{D}$, and your hypothesis $H_0$ then the p-value is $ p = \mathbb{P}(\mathcal{D}\mid H_0)$.
The $p$ value tells you the following:
If $H_0$ is true, how likely is the data I' | Understand a statement about P value
If your data is $\mathcal{D}$, and your hypothesis $H_0$ then the p-value is $ p = \mathbb{P}(\mathcal{D}\mid H_0)$.
The $p$ value tells you the following:
If $H_0$ is true, how likely is the data I'm currently observing ?
So if $p$ is very low, it only means the data cannot easil... | Understand a statement about P value
If your data is $\mathcal{D}$, and your hypothesis $H_0$ then the p-value is $ p = \mathbb{P}(\mathcal{D}\mid H_0)$.
The $p$ value tells you the following:
If $H_0$ is true, how likely is the data I' |
54,709 | Understand a statement about P value | Answering without equations:
p value is a measure of surprise. Given that the null hypothesis is true (Design of the experiment) what is the chances that you stumble upon a value at least this extreme in your data.
You compute p on the test data. You never can include the actual data. If you can get the actual data the... | Understand a statement about P value | Answering without equations:
p value is a measure of surprise. Given that the null hypothesis is true (Design of the experiment) what is the chances that you stumble upon a value at least this extreme | Understand a statement about P value
Answering without equations:
p value is a measure of surprise. Given that the null hypothesis is true (Design of the experiment) what is the chances that you stumble upon a value at least this extreme in your data.
You compute p on the test data. You never can include the actual dat... | Understand a statement about P value
Answering without equations:
p value is a measure of surprise. Given that the null hypothesis is true (Design of the experiment) what is the chances that you stumble upon a value at least this extreme |
54,710 | How is this connection between Beta and Binomial possible? | The function you have plotted is the kernel of a beta density function (i.e., it is a positive multiple of the beta density). Since you have really just plotted the binomial likelihood function for a particular observed outcome, from a Bayesian perspective your plotted function is proportionate to the posterior densit... | How is this connection between Beta and Binomial possible? | The function you have plotted is the kernel of a beta density function (i.e., it is a positive multiple of the beta density). Since you have really just plotted the binomial likelihood function for a | How is this connection between Beta and Binomial possible?
The function you have plotted is the kernel of a beta density function (i.e., it is a positive multiple of the beta density). Since you have really just plotted the binomial likelihood function for a particular observed outcome, from a Bayesian perspective you... | How is this connection between Beta and Binomial possible?
The function you have plotted is the kernel of a beta density function (i.e., it is a positive multiple of the beta density). Since you have really just plotted the binomial likelihood function for a |
54,711 | How to use a for loop in this Bernoulli exercise in R? [closed] | The instruction in the exercise to use a loop is bad advice. The rbinom function is already capable of simulating vectors of values, so there is no need for a loop. The simplest thing to do here is to create an $r \times N$ matrix of simulated Bernoulli random variables taking $N=100$ so that you have enough sample s... | How to use a for loop in this Bernoulli exercise in R? [closed] | The instruction in the exercise to use a loop is bad advice. The rbinom function is already capable of simulating vectors of values, so there is no need for a loop. The simplest thing to do here is | How to use a for loop in this Bernoulli exercise in R? [closed]
The instruction in the exercise to use a loop is bad advice. The rbinom function is already capable of simulating vectors of values, so there is no need for a loop. The simplest thing to do here is to create an $r \times N$ matrix of simulated Bernoulli ... | How to use a for loop in this Bernoulli exercise in R? [closed]
The instruction in the exercise to use a loop is bad advice. The rbinom function is already capable of simulating vectors of values, so there is no need for a loop. The simplest thing to do here is |
54,712 | How to use a for loop in this Bernoulli exercise in R? [closed] | To begin, I agree with @Ben's(+1) statements about avoiding explicit
loops when possible. I have used for loops because they seem
to be required for your exercise.
Standardization is done outside the for loop, using means and standard deviations from the 10,000 averages a.
Here is a simulation in R for the case $n = 25... | How to use a for loop in this Bernoulli exercise in R? [closed] | To begin, I agree with @Ben's(+1) statements about avoiding explicit
loops when possible. I have used for loops because they seem
to be required for your exercise.
Standardization is done outside the | How to use a for loop in this Bernoulli exercise in R? [closed]
To begin, I agree with @Ben's(+1) statements about avoiding explicit
loops when possible. I have used for loops because they seem
to be required for your exercise.
Standardization is done outside the for loop, using means and standard deviations from the 1... | How to use a for loop in this Bernoulli exercise in R? [closed]
To begin, I agree with @Ben's(+1) statements about avoiding explicit
loops when possible. I have used for loops because they seem
to be required for your exercise.
Standardization is done outside the |
54,713 | Is the following textbook definition of $p$-value correct? | The issue I have with this is that as it stands it is not a definition, as long as there is no formal definition what "in favour of $H_1$" actually means. Furthermore, as you probably know, Fisher and others have defined tests and p-values without specifying a $H_1$.
Here's an attempt to make the "definition" correct. ... | Is the following textbook definition of $p$-value correct? | The issue I have with this is that as it stands it is not a definition, as long as there is no formal definition what "in favour of $H_1$" actually means. Furthermore, as you probably know, Fisher and | Is the following textbook definition of $p$-value correct?
The issue I have with this is that as it stands it is not a definition, as long as there is no formal definition what "in favour of $H_1$" actually means. Furthermore, as you probably know, Fisher and others have defined tests and p-values without specifying a ... | Is the following textbook definition of $p$-value correct?
The issue I have with this is that as it stands it is not a definition, as long as there is no formal definition what "in favour of $H_1$" actually means. Furthermore, as you probably know, Fisher and |
54,714 | Is the following textbook definition of $p$-value correct? | That is the correct definition for a test with a simple null hypothesis. For a test with a composite null hypothesis (i.e., more than one possible parameter value in the null space) things are complicated a bit by the fact that the p-value is the supremum over the conditional probabilities over the parameters in the n... | Is the following textbook definition of $p$-value correct? | That is the correct definition for a test with a simple null hypothesis. For a test with a composite null hypothesis (i.e., more than one possible parameter value in the null space) things are compli | Is the following textbook definition of $p$-value correct?
That is the correct definition for a test with a simple null hypothesis. For a test with a composite null hypothesis (i.e., more than one possible parameter value in the null space) things are complicated a bit by the fact that the p-value is the supremum over... | Is the following textbook definition of $p$-value correct?
That is the correct definition for a test with a simple null hypothesis. For a test with a composite null hypothesis (i.e., more than one possible parameter value in the null space) things are compli |
54,715 | Is the following textbook definition of $p$-value correct? | The more general definition of a p-value is
the p-value is the probability of getting a result that is at least as extreme as the observed result, provided that $H_0$ is correct.
The definition is not clear about what 'extreme' means. One example of a p-value is a p-value that defines the degree of extremeness as val... | Is the following textbook definition of $p$-value correct? | The more general definition of a p-value is
the p-value is the probability of getting a result that is at least as extreme as the observed result, provided that $H_0$ is correct.
The definition is n | Is the following textbook definition of $p$-value correct?
The more general definition of a p-value is
the p-value is the probability of getting a result that is at least as extreme as the observed result, provided that $H_0$ is correct.
The definition is not clear about what 'extreme' means. One example of a p-value... | Is the following textbook definition of $p$-value correct?
The more general definition of a p-value is
the p-value is the probability of getting a result that is at least as extreme as the observed result, provided that $H_0$ is correct.
The definition is n |
54,716 | Is the following textbook definition of $p$-value correct? | It's obviously a translation, but logically correct. P-value is probability, under $H_0$, of a more extreme result of the test statistic in the direction(s) of the alternative hypothesis than the observed value of the test statistic. [For a two-sided alternative, two probabilities are added to get the P-value.]
Conside... | Is the following textbook definition of $p$-value correct? | It's obviously a translation, but logically correct. P-value is probability, under $H_0$, of a more extreme result of the test statistic in the direction(s) of the alternative hypothesis than the obse | Is the following textbook definition of $p$-value correct?
It's obviously a translation, but logically correct. P-value is probability, under $H_0$, of a more extreme result of the test statistic in the direction(s) of the alternative hypothesis than the observed value of the test statistic. [For a two-sided alternativ... | Is the following textbook definition of $p$-value correct?
It's obviously a translation, but logically correct. P-value is probability, under $H_0$, of a more extreme result of the test statistic in the direction(s) of the alternative hypothesis than the obse |
54,717 | If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional distribution? | A simple counterexample:
$$
P(X = 1, Y = 2) = \frac{1}{3}\\
P(X = 2, Y = 3) = \frac{1}{3}\\
P(X = 3, Y = 1) = \frac{1}{3}
$$
Then $P(X \le 1 | Y = 2) = 1$, but $P(Y \le 1 | X = 2) = 0$. | If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional d | A simple counterexample:
$$
P(X = 1, Y = 2) = \frac{1}{3}\\
P(X = 2, Y = 3) = \frac{1}{3}\\
P(X = 3, Y = 1) = \frac{1}{3}
$$
Then $P(X \le 1 | Y = 2) = 1$, but $P(Y \le 1 | X = 2) = 0$. | If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional distribution?
A simple counterexample:
$$
P(X = 1, Y = 2) = \frac{1}{3}\\
P(X = 2, Y = 3) = \frac{1}{3}\\
P(X = 3, Y = 1) = \frac{1}{3}
$$
Then $P(X \le 1 | Y = 2) = 1$, but $P(Y \le 1 | X = 2) = 0$. | If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional d
A simple counterexample:
$$
P(X = 1, Y = 2) = \frac{1}{3}\\
P(X = 2, Y = 3) = \frac{1}{3}\\
P(X = 3, Y = 1) = \frac{1}{3}
$$
Then $P(X \le 1 | Y = 2) = 1$, but $P(Y \le 1 | X = 2) = 0$. |
54,718 | If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional distribution? | How about $X=-Y = \begin{cases} 0 \\ 1 \end{cases}\quad$ each with probability $1/2.$ | If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional d | How about $X=-Y = \begin{cases} 0 \\ 1 \end{cases}\quad$ each with probability $1/2.$ | If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional distribution?
How about $X=-Y = \begin{cases} 0 \\ 1 \end{cases}\quad$ each with probability $1/2.$ | If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional d
How about $X=-Y = \begin{cases} 0 \\ 1 \end{cases}\quad$ each with probability $1/2.$ |
54,719 | If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional distribution? | There are already answers with simple examples, so why one more? Because it is interesting to look at the general pattern. The wanted property is a kind of symmetry, so we should look for some asymmetrical joint distribution for $X,Y)$. Si if $(X,Y)$ has a permutable distribution in the sense that $(X,Y)$ and $(Y,X)$ h... | If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional d | There are already answers with simple examples, so why one more? Because it is interesting to look at the general pattern. The wanted property is a kind of symmetry, so we should look for some asymmet | If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional distribution?
There are already answers with simple examples, so why one more? Because it is interesting to look at the general pattern. The wanted property is a kind of symmetry, so we should look for some asymmetrical j... | If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional d
There are already answers with simple examples, so why one more? Because it is interesting to look at the general pattern. The wanted property is a kind of symmetry, so we should look for some asymmet |
54,720 | If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional distribution? | Not necessarily true. Let $X$ and $Y$ be discrete random variables that take the values in {1, 2, 3} each with probability $\frac{1}3$; i.e. they have a discrete uniform distribution. Consider the joint probability mass function represented by the matrix below where the element in row $i$ and column $j$ is $P[X=i, Y=j... | If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional d | Not necessarily true. Let $X$ and $Y$ be discrete random variables that take the values in {1, 2, 3} each with probability $\frac{1}3$; i.e. they have a discrete uniform distribution. Consider the jo | If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional distribution?
Not necessarily true. Let $X$ and $Y$ be discrete random variables that take the values in {1, 2, 3} each with probability $\frac{1}3$; i.e. they have a discrete uniform distribution. Consider the joint pro... | If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional d
Not necessarily true. Let $X$ and $Y$ be discrete random variables that take the values in {1, 2, 3} each with probability $\frac{1}3$; i.e. they have a discrete uniform distribution. Consider the jo |
54,721 | Generating text from language model | "Weighted choice sampling" means that you sample each category with some predefined probability, so you basically sample from a categorical distribution. If each category has fixed probability, there is not much else that you can do about. There are some methods for special cases like Gumbel-max trick when the probabil... | Generating text from language model | "Weighted choice sampling" means that you sample each category with some predefined probability, so you basically sample from a categorical distribution. If each category has fixed probability, there | Generating text from language model
"Weighted choice sampling" means that you sample each category with some predefined probability, so you basically sample from a categorical distribution. If each category has fixed probability, there is not much else that you can do about. There are some methods for special cases lik... | Generating text from language model
"Weighted choice sampling" means that you sample each category with some predefined probability, so you basically sample from a categorical distribution. If each category has fixed probability, there |
54,722 | Generating text from language model | There are multiple methods for sampling utterances from a trained language model (LM). What you're doing is certainly a valid approach, and fairly modern. Having, I'll just outline a few more approaches here that people have found empirically useful. These are commonly used in large LM such as those found in GPT-2 or R... | Generating text from language model | There are multiple methods for sampling utterances from a trained language model (LM). What you're doing is certainly a valid approach, and fairly modern. Having, I'll just outline a few more approach | Generating text from language model
There are multiple methods for sampling utterances from a trained language model (LM). What you're doing is certainly a valid approach, and fairly modern. Having, I'll just outline a few more approaches here that people have found empirically useful. These are commonly used in large ... | Generating text from language model
There are multiple methods for sampling utterances from a trained language model (LM). What you're doing is certainly a valid approach, and fairly modern. Having, I'll just outline a few more approach |
54,723 | Separating $X$ from $Y$ in $E[(X^T Y))^p]$ for $p = 3$ and $4$? | The answer is that for nonnegative integer values $P$, it is in principle possible, but gets progressively more complicated as $P$ increases. To this end, one can write
\begin{align}
\mathbf{E} \left[ \left( X^T Y \right)^P\right] &= \mathbf{E} \left[ \left( \sum_{i = 1}^D X_i Y_i \right)^P \right] \\
&= \mathbf{E} \le... | Separating $X$ from $Y$ in $E[(X^T Y))^p]$ for $p = 3$ and $4$? | The answer is that for nonnegative integer values $P$, it is in principle possible, but gets progressively more complicated as $P$ increases. To this end, one can write
\begin{align}
\mathbf{E} \left[ | Separating $X$ from $Y$ in $E[(X^T Y))^p]$ for $p = 3$ and $4$?
The answer is that for nonnegative integer values $P$, it is in principle possible, but gets progressively more complicated as $P$ increases. To this end, one can write
\begin{align}
\mathbf{E} \left[ \left( X^T Y \right)^P\right] &= \mathbf{E} \left[ \lef... | Separating $X$ from $Y$ in $E[(X^T Y))^p]$ for $p = 3$ and $4$?
The answer is that for nonnegative integer values $P$, it is in principle possible, but gets progressively more complicated as $P$ increases. To this end, one can write
\begin{align}
\mathbf{E} \left[ |
54,724 | Deriving the marginal multivariate Dirichlet distribution | Since$$p(\theta_1,\ldots,\theta_{k-1})\propto (1-\theta_1-\cdots-\theta_{k-1})^{\alpha_{k}-1}\prod_{i=1}^{k-1} \theta_i^{\alpha_i-1}$$over the $\mathbb R^k$-simplex,
$$\mathfrak S = \left\{(\theta_1,\ldots,\theta_{k-1})\in\mathbb R^{k-1}_+\,;\,\sum_{i=1}^{k-1} \theta_i\le 1\right\}$$
integrating out $\theta_{k-1}$ prod... | Deriving the marginal multivariate Dirichlet distribution | Since$$p(\theta_1,\ldots,\theta_{k-1})\propto (1-\theta_1-\cdots-\theta_{k-1})^{\alpha_{k}-1}\prod_{i=1}^{k-1} \theta_i^{\alpha_i-1}$$over the $\mathbb R^k$-simplex,
$$\mathfrak S = \left\{(\theta_1,\ | Deriving the marginal multivariate Dirichlet distribution
Since$$p(\theta_1,\ldots,\theta_{k-1})\propto (1-\theta_1-\cdots-\theta_{k-1})^{\alpha_{k}-1}\prod_{i=1}^{k-1} \theta_i^{\alpha_i-1}$$over the $\mathbb R^k$-simplex,
$$\mathfrak S = \left\{(\theta_1,\ldots,\theta_{k-1})\in\mathbb R^{k-1}_+\,;\,\sum_{i=1}^{k-1} \... | Deriving the marginal multivariate Dirichlet distribution
Since$$p(\theta_1,\ldots,\theta_{k-1})\propto (1-\theta_1-\cdots-\theta_{k-1})^{\alpha_{k}-1}\prod_{i=1}^{k-1} \theta_i^{\alpha_i-1}$$over the $\mathbb R^k$-simplex,
$$\mathfrak S = \left\{(\theta_1,\ |
54,725 | causal impact estimation | Thank you for including a causal diagram!
Answer: Simply regress $Y$ on $T$ like this:
$$Y=aT+b.$$
There is no backdoor path from $T$ to $Y,$ so you don't need to condition on anything. In fact, if you want the full causal effect of $T$ on $Y,$ you need to NOT condition on $x_2.$
You have a mediation situation, so ther... | causal impact estimation | Thank you for including a causal diagram!
Answer: Simply regress $Y$ on $T$ like this:
$$Y=aT+b.$$
There is no backdoor path from $T$ to $Y,$ so you don't need to condition on anything. In fact, if yo | causal impact estimation
Thank you for including a causal diagram!
Answer: Simply regress $Y$ on $T$ like this:
$$Y=aT+b.$$
There is no backdoor path from $T$ to $Y,$ so you don't need to condition on anything. In fact, if you want the full causal effect of $T$ on $Y,$ you need to NOT condition on $x_2.$
You have a med... | causal impact estimation
Thank you for including a causal diagram!
Answer: Simply regress $Y$ on $T$ like this:
$$Y=aT+b.$$
There is no backdoor path from $T$ to $Y,$ so you don't need to condition on anything. In fact, if yo |
54,726 | causal impact estimation | To simplify, I am going to make the problem linear in parameters. You have a structural-form equation for the outcome $y$, the intermediate outcome equation for $x$, and an independence assumption:
$$ \begin{align*} y_i &=\beta_1+\beta_t \cdot t_i + \beta_x \cdot x_i + \varepsilon_i \\ x_i &= \alpha_1+\alpha_t \cdot t... | causal impact estimation | To simplify, I am going to make the problem linear in parameters. You have a structural-form equation for the outcome $y$, the intermediate outcome equation for $x$, and an independence assumption:
$$ | causal impact estimation
To simplify, I am going to make the problem linear in parameters. You have a structural-form equation for the outcome $y$, the intermediate outcome equation for $x$, and an independence assumption:
$$ \begin{align*} y_i &=\beta_1+\beta_t \cdot t_i + \beta_x \cdot x_i + \varepsilon_i \\ x_i &= ... | causal impact estimation
To simplify, I am going to make the problem linear in parameters. You have a structural-form equation for the outcome $y$, the intermediate outcome equation for $x$, and an independence assumption:
$$ |
54,727 | Confidence interval for the difference of two fitted values from a linear regression model | Taking the difference of the two predicted values gives:
$$
(\hat{\beta_0} + \hat{\beta_1} 90 + \hat{\beta_2} 5 + \hat{\beta_3} 5^2) - (\hat{\beta_0} + \hat{\beta_1} 90 + \hat{\beta_2} 2 + \hat{\beta_3} 2^2) = (5 - 2)\beta_2 + (5^2 - 2^2)\beta_3 = 3\beta_2 + 21\beta_3.
$$
This is a linear combination of the coefficient... | Confidence interval for the difference of two fitted values from a linear regression model | Taking the difference of the two predicted values gives:
$$
(\hat{\beta_0} + \hat{\beta_1} 90 + \hat{\beta_2} 5 + \hat{\beta_3} 5^2) - (\hat{\beta_0} + \hat{\beta_1} 90 + \hat{\beta_2} 2 + \hat{\beta_ | Confidence interval for the difference of two fitted values from a linear regression model
Taking the difference of the two predicted values gives:
$$
(\hat{\beta_0} + \hat{\beta_1} 90 + \hat{\beta_2} 5 + \hat{\beta_3} 5^2) - (\hat{\beta_0} + \hat{\beta_1} 90 + \hat{\beta_2} 2 + \hat{\beta_3} 2^2) = (5 - 2)\beta_2 + (5... | Confidence interval for the difference of two fitted values from a linear regression model
Taking the difference of the two predicted values gives:
$$
(\hat{\beta_0} + \hat{\beta_1} 90 + \hat{\beta_2} 5 + \hat{\beta_3} 5^2) - (\hat{\beta_0} + \hat{\beta_1} 90 + \hat{\beta_2} 2 + \hat{\beta_ |
54,728 | Confidence interval for the difference of two fitted values from a linear regression model | An alternative approach (I agree it is devious, bit it is also interesting) is to transform your function
$$y=\beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3x_2^2 + \epsilon$$
into
$$y=\beta_0 + \beta_1 x_1 + \beta_2 \frac{x_2}{3} + \beta_3(x_2-2)(x_2-5) + \epsilon$$
This is the same quadratic polynomial but now you have... | Confidence interval for the difference of two fitted values from a linear regression model | An alternative approach (I agree it is devious, bit it is also interesting) is to transform your function
$$y=\beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3x_2^2 + \epsilon$$
into
$$y=\beta_0 + \beta_1 | Confidence interval for the difference of two fitted values from a linear regression model
An alternative approach (I agree it is devious, bit it is also interesting) is to transform your function
$$y=\beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3x_2^2 + \epsilon$$
into
$$y=\beta_0 + \beta_1 x_1 + \beta_2 \frac{x_2}{3} ... | Confidence interval for the difference of two fitted values from a linear regression model
An alternative approach (I agree it is devious, bit it is also interesting) is to transform your function
$$y=\beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3x_2^2 + \epsilon$$
into
$$y=\beta_0 + \beta_1 |
54,729 | Given two normal populations,, classifying a given data point | Consider the two possible normal populations as models, $p(\mathcal{M_1}), p(\mathcal{M_2})$, that is, two hypothesis that compete to explain your datum $X$.
The prior information tells which is the prior odds,
$$\frac{p(\mathcal{M_1})}{p(\mathcal{M_2})} = \frac{\pi_1}{\pi_2}$$
You can apply the standard formula
$$\un... | Given two normal populations,, classifying a given data point | Consider the two possible normal populations as models, $p(\mathcal{M_1}), p(\mathcal{M_2})$, that is, two hypothesis that compete to explain your datum $X$.
The prior information tells which is the | Given two normal populations,, classifying a given data point
Consider the two possible normal populations as models, $p(\mathcal{M_1}), p(\mathcal{M_2})$, that is, two hypothesis that compete to explain your datum $X$.
The prior information tells which is the prior odds,
$$\frac{p(\mathcal{M_1})}{p(\mathcal{M_2})} = ... | Given two normal populations,, classifying a given data point
Consider the two possible normal populations as models, $p(\mathcal{M_1}), p(\mathcal{M_2})$, that is, two hypothesis that compete to explain your datum $X$.
The prior information tells which is the |
54,730 | Interpreting coefficients from ordinal regression R `polr` function | Nicely laid out question, Dylan. I'll take a stab at answering it but will keep my answer practical (i.e., without using mathematical equations).
Will you change the sign of the hxcopd coefficient for reporting purposes?
The first thing you need to determine when looking at the Coefficients output produced by polr is w... | Interpreting coefficients from ordinal regression R `polr` function | Nicely laid out question, Dylan. I'll take a stab at answering it but will keep my answer practical (i.e., without using mathematical equations).
Will you change the sign of the hxcopd coefficient for | Interpreting coefficients from ordinal regression R `polr` function
Nicely laid out question, Dylan. I'll take a stab at answering it but will keep my answer practical (i.e., without using mathematical equations).
Will you change the sign of the hxcopd coefficient for reporting purposes?
The first thing you need to det... | Interpreting coefficients from ordinal regression R `polr` function
Nicely laid out question, Dylan. I'll take a stab at answering it but will keep my answer practical (i.e., without using mathematical equations).
Will you change the sign of the hxcopd coefficient for |
54,731 | Is it true that we can always increase statistical power/estimator precision by increasing sample size? | Fisher information increases
The Fisher information (which is not the property of an estimator) scales with the size of the sample. See for instance here: https://en.m.wikipedia.org/wiki/Fisher_information#Discrepancy_in_definition
, if the data are i.i.d. the difference between two versions is simply a factor of $n$,... | Is it true that we can always increase statistical power/estimator precision by increasing sample si | Fisher information increases
The Fisher information (which is not the property of an estimator) scales with the size of the sample. See for instance here: https://en.m.wikipedia.org/wiki/Fisher_inform | Is it true that we can always increase statistical power/estimator precision by increasing sample size?
Fisher information increases
The Fisher information (which is not the property of an estimator) scales with the size of the sample. See for instance here: https://en.m.wikipedia.org/wiki/Fisher_information#Discrepanc... | Is it true that we can always increase statistical power/estimator precision by increasing sample si
Fisher information increases
The Fisher information (which is not the property of an estimator) scales with the size of the sample. See for instance here: https://en.m.wikipedia.org/wiki/Fisher_inform |
54,732 | Is it true that we can always increase statistical power/estimator precision by increasing sample size? | An example of a statistic that does not increase in Fisher information as sample size increases is the matching statistic. The matching statistic $m$ (Vernon, 1936) is computed between a pair of vectors of ranked scores, as the number of paired ranks that match. Gordon Rae (1987, 1991) showed that, when the population ... | Is it true that we can always increase statistical power/estimator precision by increasing sample si | An example of a statistic that does not increase in Fisher information as sample size increases is the matching statistic. The matching statistic $m$ (Vernon, 1936) is computed between a pair of vecto | Is it true that we can always increase statistical power/estimator precision by increasing sample size?
An example of a statistic that does not increase in Fisher information as sample size increases is the matching statistic. The matching statistic $m$ (Vernon, 1936) is computed between a pair of vectors of ranked sco... | Is it true that we can always increase statistical power/estimator precision by increasing sample si
An example of a statistic that does not increase in Fisher information as sample size increases is the matching statistic. The matching statistic $m$ (Vernon, 1936) is computed between a pair of vecto |
54,733 | Finding a confidence interval of a MAPE | Interesting question. I have been active in both academic and applied forecasting for quite a while, and I can't recall anyone discussing CIs for MAPEs ever.
I don't think your calculation is very helpful. As an example, assume that the true holdout actuals are lognormally distributed with log-mean $\mu=1$ and log-SD $... | Finding a confidence interval of a MAPE | Interesting question. I have been active in both academic and applied forecasting for quite a while, and I can't recall anyone discussing CIs for MAPEs ever.
I don't think your calculation is very hel | Finding a confidence interval of a MAPE
Interesting question. I have been active in both academic and applied forecasting for quite a while, and I can't recall anyone discussing CIs for MAPEs ever.
I don't think your calculation is very helpful. As an example, assume that the true holdout actuals are lognormally distri... | Finding a confidence interval of a MAPE
Interesting question. I have been active in both academic and applied forecasting for quite a while, and I can't recall anyone discussing CIs for MAPEs ever.
I don't think your calculation is very hel |
54,734 | What is the defintion of variation in a box plot? | A boxplot invites you to characterize variation in many different ways, by comparing the quantities shown on the plot: extremes, extremes of the whiskers, quartiles, and median. That gives 21 different measures of variation in each one! On that basis I can identify (with some difficulty, because many of the boxplots ... | What is the defintion of variation in a box plot? | A boxplot invites you to characterize variation in many different ways, by comparing the quantities shown on the plot: extremes, extremes of the whiskers, quartiles, and median. That gives 21 differe | What is the defintion of variation in a box plot?
A boxplot invites you to characterize variation in many different ways, by comparing the quantities shown on the plot: extremes, extremes of the whiskers, quartiles, and median. That gives 21 different measures of variation in each one! On that basis I can identify (w... | What is the defintion of variation in a box plot?
A boxplot invites you to characterize variation in many different ways, by comparing the quantities shown on the plot: extremes, extremes of the whiskers, quartiles, and median. That gives 21 differe |
54,735 | What is the difference between Non-Negative Matrix Factorization (NMF) and Factor Analysis (FA)? | NMF/PMF are typically used to make low-rank decompositions. They can be used like a truncated SVD, just for dimension reduction. They can also be used like factor analysis, to attempt to identify latent variables that theory says underly the data.
A truncated rank-$k$ SVD asks for the best decomposition of the data mat... | What is the difference between Non-Negative Matrix Factorization (NMF) and Factor Analysis (FA)? | NMF/PMF are typically used to make low-rank decompositions. They can be used like a truncated SVD, just for dimension reduction. They can also be used like factor analysis, to attempt to identify late | What is the difference between Non-Negative Matrix Factorization (NMF) and Factor Analysis (FA)?
NMF/PMF are typically used to make low-rank decompositions. They can be used like a truncated SVD, just for dimension reduction. They can also be used like factor analysis, to attempt to identify latent variables that theor... | What is the difference between Non-Negative Matrix Factorization (NMF) and Factor Analysis (FA)?
NMF/PMF are typically used to make low-rank decompositions. They can be used like a truncated SVD, just for dimension reduction. They can also be used like factor analysis, to attempt to identify late |
54,736 | Mathematical/Statistical Assumptions Underlying Machine and Deep Learning Methods | There's no such thing as universal statistical or machine learning assumptions. There are lots of different statistical/ML methods, with different assumptions among them. You might ask about what assumptions underlie a specific method, or what goes wrong if you violate an assumption of a certain method, but there's no ... | Mathematical/Statistical Assumptions Underlying Machine and Deep Learning Methods | There's no such thing as universal statistical or machine learning assumptions. There are lots of different statistical/ML methods, with different assumptions among them. You might ask about what assu | Mathematical/Statistical Assumptions Underlying Machine and Deep Learning Methods
There's no such thing as universal statistical or machine learning assumptions. There are lots of different statistical/ML methods, with different assumptions among them. You might ask about what assumptions underlie a specific method, or... | Mathematical/Statistical Assumptions Underlying Machine and Deep Learning Methods
There's no such thing as universal statistical or machine learning assumptions. There are lots of different statistical/ML methods, with different assumptions among them. You might ask about what assu |
54,737 | Mathematical/Statistical Assumptions Underlying Machine and Deep Learning Methods | I would disagree slightly with the opening statement of Sycorax's excellent and detailed answer "There's no such thing as universal statistical or machine learning assumptions" - in supervised machine learning, in general, it is assumed that your data is drawn IID from a probability distribution, and that any test/new ... | Mathematical/Statistical Assumptions Underlying Machine and Deep Learning Methods | I would disagree slightly with the opening statement of Sycorax's excellent and detailed answer "There's no such thing as universal statistical or machine learning assumptions" - in supervised machine | Mathematical/Statistical Assumptions Underlying Machine and Deep Learning Methods
I would disagree slightly with the opening statement of Sycorax's excellent and detailed answer "There's no such thing as universal statistical or machine learning assumptions" - in supervised machine learning, in general, it is assumed t... | Mathematical/Statistical Assumptions Underlying Machine and Deep Learning Methods
I would disagree slightly with the opening statement of Sycorax's excellent and detailed answer "There's no such thing as universal statistical or machine learning assumptions" - in supervised machine |
54,738 | Mathematical/Statistical Assumptions Underlying Machine and Deep Learning Methods | Assumptions essentially add information. This added information is more useful if you have less data. For example, contrast two OLS regression relationships
$Y \sim X + Z$
$Y \sim X + X^2 + X^3 + Z + Z^2 + Z^3 + X*Z + (X*Z)^2 + (X*Z)^3$
The first has more assumptions because it is a special case of the second. It's a... | Mathematical/Statistical Assumptions Underlying Machine and Deep Learning Methods | Assumptions essentially add information. This added information is more useful if you have less data. For example, contrast two OLS regression relationships
$Y \sim X + Z$
$Y \sim X + X^2 + X^3 + Z + | Mathematical/Statistical Assumptions Underlying Machine and Deep Learning Methods
Assumptions essentially add information. This added information is more useful if you have less data. For example, contrast two OLS regression relationships
$Y \sim X + Z$
$Y \sim X + X^2 + X^3 + Z + Z^2 + Z^3 + X*Z + (X*Z)^2 + (X*Z)^3$
... | Mathematical/Statistical Assumptions Underlying Machine and Deep Learning Methods
Assumptions essentially add information. This added information is more useful if you have less data. For example, contrast two OLS regression relationships
$Y \sim X + Z$
$Y \sim X + X^2 + X^3 + Z + |
54,739 | Is the MA($\infty$) process with i.i.d. noise strictly stationary? | This process is always strictly stationary by definition.
Recall that the process is (strictly) stationary when all $n$-variate distributions formed by selecting any pattern $(s_1,s_2,\ldots,s_{n-1})$ of (integral) indexes are identical: that is, for all $n\ge 1$ and integral $s$ and $t,$
$$(X_s, X_{s+s_1}, \ldots, X_{... | Is the MA($\infty$) process with i.i.d. noise strictly stationary? | This process is always strictly stationary by definition.
Recall that the process is (strictly) stationary when all $n$-variate distributions formed by selecting any pattern $(s_1,s_2,\ldots,s_{n-1})$ | Is the MA($\infty$) process with i.i.d. noise strictly stationary?
This process is always strictly stationary by definition.
Recall that the process is (strictly) stationary when all $n$-variate distributions formed by selecting any pattern $(s_1,s_2,\ldots,s_{n-1})$ of (integral) indexes are identical: that is, for al... | Is the MA($\infty$) process with i.i.d. noise strictly stationary?
This process is always strictly stationary by definition.
Recall that the process is (strictly) stationary when all $n$-variate distributions formed by selecting any pattern $(s_1,s_2,\ldots,s_{n-1})$ |
54,740 | Is the MA($\infty$) process with i.i.d. noise strictly stationary? | Strict stationarity do not imply weak essentially because is possible that the first two moments are not finite. If we add that
$V[\epsilon_t]=\sigma^2< \infty$
strictly stationarity imply weak also.
Seems me useful to note that stationarity is strongly related to ergodicity and, then, memory (this discussion can help:... | Is the MA($\infty$) process with i.i.d. noise strictly stationary? | Strict stationarity do not imply weak essentially because is possible that the first two moments are not finite. If we add that
$V[\epsilon_t]=\sigma^2< \infty$
strictly stationarity imply weak also.
| Is the MA($\infty$) process with i.i.d. noise strictly stationary?
Strict stationarity do not imply weak essentially because is possible that the first two moments are not finite. If we add that
$V[\epsilon_t]=\sigma^2< \infty$
strictly stationarity imply weak also.
Seems me useful to note that stationarity is strongly... | Is the MA($\infty$) process with i.i.d. noise strictly stationary?
Strict stationarity do not imply weak essentially because is possible that the first two moments are not finite. If we add that
$V[\epsilon_t]=\sigma^2< \infty$
strictly stationarity imply weak also.
|
54,741 | Visualizing the hinge loss and 0-1 loss | The x-axis is the score output from a classifier, often interpreted as the estimated/predicted log-odds. The y-axis is the loss for a single datapoint with true label $y = 1$.
In notation, if we denote the score output from the classifier as $\hat s$, the plots are the graphs of the functions:
$$ f(\hat s) = \text{Zero... | Visualizing the hinge loss and 0-1 loss | The x-axis is the score output from a classifier, often interpreted as the estimated/predicted log-odds. The y-axis is the loss for a single datapoint with true label $y = 1$.
In notation, if we denot | Visualizing the hinge loss and 0-1 loss
The x-axis is the score output from a classifier, often interpreted as the estimated/predicted log-odds. The y-axis is the loss for a single datapoint with true label $y = 1$.
In notation, if we denote the score output from the classifier as $\hat s$, the plots are the graphs of ... | Visualizing the hinge loss and 0-1 loss
The x-axis is the score output from a classifier, often interpreted as the estimated/predicted log-odds. The y-axis is the loss for a single datapoint with true label $y = 1$.
In notation, if we denot |
54,742 | How to include an interaction with sex | Algebra lights the way.
The purpose of an "interaction" between a binary variable like gender and another variable (let's just call it "$X$") is to model the possibility that how a response (call it "$Y$") is associated with $X$ may depend on the binary variable. Specifically, it allows for the slope (aka coefficient)... | How to include an interaction with sex | Algebra lights the way.
The purpose of an "interaction" between a binary variable like gender and another variable (let's just call it "$X$") is to model the possibility that how a response (call it " | How to include an interaction with sex
Algebra lights the way.
The purpose of an "interaction" between a binary variable like gender and another variable (let's just call it "$X$") is to model the possibility that how a response (call it "$Y$") is associated with $X$ may depend on the binary variable. Specifically, it... | How to include an interaction with sex
Algebra lights the way.
The purpose of an "interaction" between a binary variable like gender and another variable (let's just call it "$X$") is to model the possibility that how a response (call it " |
54,743 | How to include an interaction with sex | If you have an interaction with sex, then this means that you create a new variable that did not exist before.
For instance:
let the outcome (dependent variable) be the probability of a baby
let sex be a variable which is either 0 or 1
and let's say we interact it with condom use which is either 0 or 1 as well.
Then ... | How to include an interaction with sex | If you have an interaction with sex, then this means that you create a new variable that did not exist before.
For instance:
let the outcome (dependent variable) be the probability of a baby
let sex | How to include an interaction with sex
If you have an interaction with sex, then this means that you create a new variable that did not exist before.
For instance:
let the outcome (dependent variable) be the probability of a baby
let sex be a variable which is either 0 or 1
and let's say we interact it with condom use... | How to include an interaction with sex
If you have an interaction with sex, then this means that you create a new variable that did not exist before.
For instance:
let the outcome (dependent variable) be the probability of a baby
let sex |
54,744 | How to linearize a non linear function | The following argument indicates how to address such questions generally.
Let's suppose there is a vector parameter $\theta\in\Theta\subset\mathbb{R}^p$ and a one-to-one differentiable reparameterization $\alpha = h(\theta)$ (where $h:\mathbb{R}^p\to\mathbb{R}$) and, if necessary, a re-expression of the variable $x$ in... | How to linearize a non linear function | The following argument indicates how to address such questions generally.
Let's suppose there is a vector parameter $\theta\in\Theta\subset\mathbb{R}^p$ and a one-to-one differentiable reparameterizat | How to linearize a non linear function
The following argument indicates how to address such questions generally.
Let's suppose there is a vector parameter $\theta\in\Theta\subset\mathbb{R}^p$ and a one-to-one differentiable reparameterization $\alpha = h(\theta)$ (where $h:\mathbb{R}^p\to\mathbb{R}$) and, if necessary,... | How to linearize a non linear function
The following argument indicates how to address such questions generally.
Let's suppose there is a vector parameter $\theta\in\Theta\subset\mathbb{R}^p$ and a one-to-one differentiable reparameterizat |
54,745 | How to linearize a non linear function | Your first example is a model with two effective parameters:$$y=\beta_0+\beta_x x+\varepsilon$$
You have two degrees of freedom $\alpha,\beta$ so you were able to linearize the model. Having the same degrees of freedom is not a sufficient condition but it's necessary. I show thesufficient conditions further in answer.
... | How to linearize a non linear function | Your first example is a model with two effective parameters:$$y=\beta_0+\beta_x x+\varepsilon$$
You have two degrees of freedom $\alpha,\beta$ so you were able to linearize the model. Having the same | How to linearize a non linear function
Your first example is a model with two effective parameters:$$y=\beta_0+\beta_x x+\varepsilon$$
You have two degrees of freedom $\alpha,\beta$ so you were able to linearize the model. Having the same degrees of freedom is not a sufficient condition but it's necessary. I show thesu... | How to linearize a non linear function
Your first example is a model with two effective parameters:$$y=\beta_0+\beta_x x+\varepsilon$$
You have two degrees of freedom $\alpha,\beta$ so you were able to linearize the model. Having the same |
54,746 | What is the PDF of a Normal convolved with a Laplace | Let's work it out from first principles, beginning with the hard work of computing a convolution.
As an auxiliary calculation, consider the distribution of $W=X+Y$ where $Y$ has an Exponential distribution with pdf $$f_Y(y) = e^{-y}\,\mathcal{I}(y\gt 0)$$ and $X$ has a Normal$(\mu,\sigma^2)$ distribution with pdf $f_X(... | What is the PDF of a Normal convolved with a Laplace | Let's work it out from first principles, beginning with the hard work of computing a convolution.
As an auxiliary calculation, consider the distribution of $W=X+Y$ where $Y$ has an Exponential distrib | What is the PDF of a Normal convolved with a Laplace
Let's work it out from first principles, beginning with the hard work of computing a convolution.
As an auxiliary calculation, consider the distribution of $W=X+Y$ where $Y$ has an Exponential distribution with pdf $$f_Y(y) = e^{-y}\,\mathcal{I}(y\gt 0)$$ and $X$ has... | What is the PDF of a Normal convolved with a Laplace
Let's work it out from first principles, beginning with the hard work of computing a convolution.
As an auxiliary calculation, consider the distribution of $W=X+Y$ where $Y$ has an Exponential distrib |
54,747 | Difference Between Scipy.optimize.least_squares and Scipy.optimize.curve_fit | There is no fundamental difference between curve_fit and least_squares. Moreover, if you don't use method = 'lm'they do exactly the same thing. You can check it in a source code of curve_fit fucntion on a Github:
if method == 'lm':
...
res = leastsq(func, p0, Dfun=jac, full_output=1, **kwargs)
...
else:
... | Difference Between Scipy.optimize.least_squares and Scipy.optimize.curve_fit | There is no fundamental difference between curve_fit and least_squares. Moreover, if you don't use method = 'lm'they do exactly the same thing. You can check it in a source code of curve_fit fucntion | Difference Between Scipy.optimize.least_squares and Scipy.optimize.curve_fit
There is no fundamental difference between curve_fit and least_squares. Moreover, if you don't use method = 'lm'they do exactly the same thing. You can check it in a source code of curve_fit fucntion on a Github:
if method == 'lm':
...
... | Difference Between Scipy.optimize.least_squares and Scipy.optimize.curve_fit
There is no fundamental difference between curve_fit and least_squares. Moreover, if you don't use method = 'lm'they do exactly the same thing. You can check it in a source code of curve_fit fucntion |
54,748 | Why is Kullback Leibler Divergence always positive? | Intuitive understanding is somewhat subjective, but I can at least offer my perspective:
Kullback-Leibler divergence is a concept from Information Theory. It tells you how much longer --- how many bits --- on average are your messages going to be if you use a suboptimal coding scheme.
For every probability distribution... | Why is Kullback Leibler Divergence always positive? | Intuitive understanding is somewhat subjective, but I can at least offer my perspective:
Kullback-Leibler divergence is a concept from Information Theory. It tells you how much longer --- how many bit | Why is Kullback Leibler Divergence always positive?
Intuitive understanding is somewhat subjective, but I can at least offer my perspective:
Kullback-Leibler divergence is a concept from Information Theory. It tells you how much longer --- how many bits --- on average are your messages going to be if you use a suboptim... | Why is Kullback Leibler Divergence always positive?
Intuitive understanding is somewhat subjective, but I can at least offer my perspective:
Kullback-Leibler divergence is a concept from Information Theory. It tells you how much longer --- how many bit |
54,749 | Does paired sample `t test` need pre test? | The general sentiment on Cross Validated is that formal testing of normality is not helpful: either you have too few observations to reject, or you have so many that the tests become sensitive to deviations from normality that are not practically significant because your data are “normal enough”. Graphical examination ... | Does paired sample `t test` need pre test? | The general sentiment on Cross Validated is that formal testing of normality is not helpful: either you have too few observations to reject, or you have so many that the tests become sensitive to devi | Does paired sample `t test` need pre test?
The general sentiment on Cross Validated is that formal testing of normality is not helpful: either you have too few observations to reject, or you have so many that the tests become sensitive to deviations from normality that are not practically significant because your data ... | Does paired sample `t test` need pre test?
The general sentiment on Cross Validated is that formal testing of normality is not helpful: either you have too few observations to reject, or you have so many that the tests become sensitive to devi |
54,750 | An unbiased estimate for population variance | Presumably $Q_s(X) = 1 - P_s(X) = (n-X)/n.$
Writing $q=1-p$, let's work out the expectation of $n^2P_s(X)Q_s(X)$ using the definition of expectation, the formula for Binomial probabilities, and the Binomial Theorem:
$$\eqalign{
E\left[n^2P_s(X)Q_s(X)\right] &= E\left[X(n-X)\right] \\
&= \sum_x \Pr(X=x)\, x(n-x) & \text... | An unbiased estimate for population variance | Presumably $Q_s(X) = 1 - P_s(X) = (n-X)/n.$
Writing $q=1-p$, let's work out the expectation of $n^2P_s(X)Q_s(X)$ using the definition of expectation, the formula for Binomial probabilities, and the Bi | An unbiased estimate for population variance
Presumably $Q_s(X) = 1 - P_s(X) = (n-X)/n.$
Writing $q=1-p$, let's work out the expectation of $n^2P_s(X)Q_s(X)$ using the definition of expectation, the formula for Binomial probabilities, and the Binomial Theorem:
$$\eqalign{
E\left[n^2P_s(X)Q_s(X)\right] &= E\left[X(n-X)\... | An unbiased estimate for population variance
Presumably $Q_s(X) = 1 - P_s(X) = (n-X)/n.$
Writing $q=1-p$, let's work out the expectation of $n^2P_s(X)Q_s(X)$ using the definition of expectation, the formula for Binomial probabilities, and the Bi |
54,751 | An unbiased estimate for population variance | Here's an approach using the following variance formula and rule
$Var(\hat{p})=\frac{p(1-p)}{n}=E[\hat{p}^2]-E[\hat{p}]^2$
where $\hat{p}$ is the sample proportion of times an indicator variable is 1 in a simple random sample of size $n$, i.e. the mean of an indicator variable, and $p$ is the corresponding population p... | An unbiased estimate for population variance | Here's an approach using the following variance formula and rule
$Var(\hat{p})=\frac{p(1-p)}{n}=E[\hat{p}^2]-E[\hat{p}]^2$
where $\hat{p}$ is the sample proportion of times an indicator variable is 1 | An unbiased estimate for population variance
Here's an approach using the following variance formula and rule
$Var(\hat{p})=\frac{p(1-p)}{n}=E[\hat{p}^2]-E[\hat{p}]^2$
where $\hat{p}$ is the sample proportion of times an indicator variable is 1 in a simple random sample of size $n$, i.e. the mean of an indicator variab... | An unbiased estimate for population variance
Here's an approach using the following variance formula and rule
$Var(\hat{p})=\frac{p(1-p)}{n}=E[\hat{p}^2]-E[\hat{p}]^2$
where $\hat{p}$ is the sample proportion of times an indicator variable is 1 |
54,752 | Comparing Coefficients of Two Time Series Models | Let us define $\delta_i = \beta_{1,i} - \gamma_{1,i}$, with i indexing the samples.
You would ideally regress $\delta_i$ over 1: $\delta_i = b.1 +\eta_i $.
If I make no mistake, your question can indeed be rephrased:: What is the value of $b$? Is $b$ significantly different from 0?
However, as noticed by @F.Tusel, ther... | Comparing Coefficients of Two Time Series Models | Let us define $\delta_i = \beta_{1,i} - \gamma_{1,i}$, with i indexing the samples.
You would ideally regress $\delta_i$ over 1: $\delta_i = b.1 +\eta_i $.
If I make no mistake, your question can inde | Comparing Coefficients of Two Time Series Models
Let us define $\delta_i = \beta_{1,i} - \gamma_{1,i}$, with i indexing the samples.
You would ideally regress $\delta_i$ over 1: $\delta_i = b.1 +\eta_i $.
If I make no mistake, your question can indeed be rephrased:: What is the value of $b$? Is $b$ significantly differ... | Comparing Coefficients of Two Time Series Models
Let us define $\delta_i = \beta_{1,i} - \gamma_{1,i}$, with i indexing the samples.
You would ideally regress $\delta_i$ over 1: $\delta_i = b.1 +\eta_i $.
If I make no mistake, your question can inde |
54,753 | Comparing Coefficients of Two Time Series Models | I fully concur with the last paragraph of @AlexC-L's answer which is in essence a paired comparisons method. I have a feeling, though, that you do not want to look at the raw differences $\delta_i = \beta_i - \gamma_i$. The $\beta_i$ and $\gamma_i$ are presumably estimated by regression and are affected by uncertainty:... | Comparing Coefficients of Two Time Series Models | I fully concur with the last paragraph of @AlexC-L's answer which is in essence a paired comparisons method. I have a feeling, though, that you do not want to look at the raw differences $\delta_i = \ | Comparing Coefficients of Two Time Series Models
I fully concur with the last paragraph of @AlexC-L's answer which is in essence a paired comparisons method. I have a feeling, though, that you do not want to look at the raw differences $\delta_i = \beta_i - \gamma_i$. The $\beta_i$ and $\gamma_i$ are presumably estimat... | Comparing Coefficients of Two Time Series Models
I fully concur with the last paragraph of @AlexC-L's answer which is in essence a paired comparisons method. I have a feeling, though, that you do not want to look at the raw differences $\delta_i = \ |
54,754 | Comparing Coefficients of Two Time Series Models | I assume when saying "test wether the independent variable has a significantly larger effect on the dependent variable in the adjusted panel than in the unadjusted panel", you are actually trying to find out which model can better describe the uncertainties and relations among the observations.
So instead of comparing ... | Comparing Coefficients of Two Time Series Models | I assume when saying "test wether the independent variable has a significantly larger effect on the dependent variable in the adjusted panel than in the unadjusted panel", you are actually trying to f | Comparing Coefficients of Two Time Series Models
I assume when saying "test wether the independent variable has a significantly larger effect on the dependent variable in the adjusted panel than in the unadjusted panel", you are actually trying to find out which model can better describe the uncertainties and relations... | Comparing Coefficients of Two Time Series Models
I assume when saying "test wether the independent variable has a significantly larger effect on the dependent variable in the adjusted panel than in the unadjusted panel", you are actually trying to f |
54,755 | Comparing Coefficients of Two Time Series Models | Note: This answer does not take into consideration that $\beta_i$ and $\gamma_i$ are themselves estimated (thank @Turell for pointing that out). I make another try in another answer.
You have n $\beta_i$ and n $\gamma_i$ that you want to compare. If n is large enough, you might turn this problem into the comparison be... | Comparing Coefficients of Two Time Series Models | Note: This answer does not take into consideration that $\beta_i$ and $\gamma_i$ are themselves estimated (thank @Turell for pointing that out). I make another try in another answer.
You have n $\bet | Comparing Coefficients of Two Time Series Models
Note: This answer does not take into consideration that $\beta_i$ and $\gamma_i$ are themselves estimated (thank @Turell for pointing that out). I make another try in another answer.
You have n $\beta_i$ and n $\gamma_i$ that you want to compare. If n is large enough, y... | Comparing Coefficients of Two Time Series Models
Note: This answer does not take into consideration that $\beta_i$ and $\gamma_i$ are themselves estimated (thank @Turell for pointing that out). I make another try in another answer.
You have n $\bet |
54,756 | Law of total probability and conditioning on multiple events | \begin{align}
P(Y=y|X) &=E(1_{Y=y}|X) \\ &\overset{Tower\ property}{=}E\color{green}{\bigg(}E\color{red}{(}1_{Y=y}|X \color{red}{)}|(X,Z)\color{green}{\bigg)}
\\
&\overset{Tower\ property}{=}E\color{green}{\bigg(}E\color{red}{(}1_{Y=y}|(X,Z) \color{red}{)}|X\color{green}{\bigg)}
\\
&= E\color{green}{\bigg(}g(X,Z) |X\c... | Law of total probability and conditioning on multiple events | \begin{align}
P(Y=y|X) &=E(1_{Y=y}|X) \\ &\overset{Tower\ property}{=}E\color{green}{\bigg(}E\color{red}{(}1_{Y=y}|X \color{red}{)}|(X,Z)\color{green}{\bigg)}
\\
&\overset{Tower\ property}{=}E\color{ | Law of total probability and conditioning on multiple events
\begin{align}
P(Y=y|X) &=E(1_{Y=y}|X) \\ &\overset{Tower\ property}{=}E\color{green}{\bigg(}E\color{red}{(}1_{Y=y}|X \color{red}{)}|(X,Z)\color{green}{\bigg)}
\\
&\overset{Tower\ property}{=}E\color{green}{\bigg(}E\color{red}{(}1_{Y=y}|(X,Z) \color{red}{)}|X... | Law of total probability and conditioning on multiple events
\begin{align}
P(Y=y|X) &=E(1_{Y=y}|X) \\ &\overset{Tower\ property}{=}E\color{green}{\bigg(}E\color{red}{(}1_{Y=y}|X \color{red}{)}|(X,Z)\color{green}{\bigg)}
\\
&\overset{Tower\ property}{=}E\color{ |
54,757 | Law of total probability and conditioning on multiple events | Your purported proof of $(3.4)$, without using independent of $Z$ and $X$, is not correct. It is not valid to form an event that includes a condition, because that condition then escapes the other probability operator in the law of total probability. In fact, the equation is not true in general (i.e., without the ind... | Law of total probability and conditioning on multiple events | Your purported proof of $(3.4)$, without using independent of $Z$ and $X$, is not correct. It is not valid to form an event that includes a condition, because that condition then escapes the other pr | Law of total probability and conditioning on multiple events
Your purported proof of $(3.4)$, without using independent of $Z$ and $X$, is not correct. It is not valid to form an event that includes a condition, because that condition then escapes the other probability operator in the law of total probability. In fac... | Law of total probability and conditioning on multiple events
Your purported proof of $(3.4)$, without using independent of $Z$ and $X$, is not correct. It is not valid to form an event that includes a condition, because that condition then escapes the other pr |
54,758 | Law of total probability and conditioning on multiple events | I think there is a mistake in your proof when you define the event $A$ as $Y=y|X=x$, this definition does not make sense. You cannot include conditionality in an event (what would be a realization of such an event?), you can just talk about probability of an event conditionally to some other event. Conditioning on an e... | Law of total probability and conditioning on multiple events | I think there is a mistake in your proof when you define the event $A$ as $Y=y|X=x$, this definition does not make sense. You cannot include conditionality in an event (what would be a realization of | Law of total probability and conditioning on multiple events
I think there is a mistake in your proof when you define the event $A$ as $Y=y|X=x$, this definition does not make sense. You cannot include conditionality in an event (what would be a realization of such an event?), you can just talk about probability of an ... | Law of total probability and conditioning on multiple events
I think there is a mistake in your proof when you define the event $A$ as $Y=y|X=x$, this definition does not make sense. You cannot include conditionality in an event (what would be a realization of |
54,759 | Concept of a z-score for a gamma distribution | The $z$ score expressed how many standard deviations a given observation from a symmetric distribution is away from the mean. Negative $z$ scores indicate an observation below, positive $z$ scores above the mean.
The logic doesn't make much sense for asymmetric distributions, even when standard deviations exist. In a s... | Concept of a z-score for a gamma distribution | The $z$ score expressed how many standard deviations a given observation from a symmetric distribution is away from the mean. Negative $z$ scores indicate an observation below, positive $z$ scores abo | Concept of a z-score for a gamma distribution
The $z$ score expressed how many standard deviations a given observation from a symmetric distribution is away from the mean. Negative $z$ scores indicate an observation below, positive $z$ scores above the mean.
The logic doesn't make much sense for asymmetric distribution... | Concept of a z-score for a gamma distribution
The $z$ score expressed how many standard deviations a given observation from a symmetric distribution is away from the mean. Negative $z$ scores indicate an observation below, positive $z$ scores abo |
54,760 | Convergence of Poisson Random Variable | Since $X$ is discrete, you can simplify a little:
$$\lim_{n\to\infty}p(X_n=0) = \lim_{n\to\infty}\text{e}^{-{1 \over n}} = \text{e}^{\lim_{n\to\infty}{-{1\over n}}} = \text{e}^0=1$$
where we can go from the second to the third term by the continuity of the exponentiation function.
The second statement follows from the ... | Convergence of Poisson Random Variable | Since $X$ is discrete, you can simplify a little:
$$\lim_{n\to\infty}p(X_n=0) = \lim_{n\to\infty}\text{e}^{-{1 \over n}} = \text{e}^{\lim_{n\to\infty}{-{1\over n}}} = \text{e}^0=1$$
where we can go fr | Convergence of Poisson Random Variable
Since $X$ is discrete, you can simplify a little:
$$\lim_{n\to\infty}p(X_n=0) = \lim_{n\to\infty}\text{e}^{-{1 \over n}} = \text{e}^{\lim_{n\to\infty}{-{1\over n}}} = \text{e}^0=1$$
where we can go from the second to the third term by the continuity of the exponentiation function.... | Convergence of Poisson Random Variable
Since $X$ is discrete, you can simplify a little:
$$\lim_{n\to\infty}p(X_n=0) = \lim_{n\to\infty}\text{e}^{-{1 \over n}} = \text{e}^{\lim_{n\to\infty}{-{1\over n}}} = \text{e}^0=1$$
where we can go fr |
54,761 | Convergence of Poisson Random Variable | Alternate answer for part-2:
Let $X = \lim_{n\rightarrow\infty}X_n$, then we have
$$
\begin{align*}
P(X=k) & = \lim_{n\rightarrow\infty}\,P(X_n=k) \\
& = \lim_{n\rightarrow\infty}\frac{1}{e^{\frac{1}{n}}n^{k}k!}
& = \begin{cases}
& 1;\qquad X_n = 0 \\
& 0;\qquad \text{otherwise}
\end{cases}
\end{align*}
$$
Now, using ... | Convergence of Poisson Random Variable | Alternate answer for part-2:
Let $X = \lim_{n\rightarrow\infty}X_n$, then we have
$$
\begin{align*}
P(X=k) & = \lim_{n\rightarrow\infty}\,P(X_n=k) \\
& = \lim_{n\rightarrow\infty}\frac{1}{e^{\frac{1}{ | Convergence of Poisson Random Variable
Alternate answer for part-2:
Let $X = \lim_{n\rightarrow\infty}X_n$, then we have
$$
\begin{align*}
P(X=k) & = \lim_{n\rightarrow\infty}\,P(X_n=k) \\
& = \lim_{n\rightarrow\infty}\frac{1}{e^{\frac{1}{n}}n^{k}k!}
& = \begin{cases}
& 1;\qquad X_n = 0 \\
& 0;\qquad \text{otherwise}
... | Convergence of Poisson Random Variable
Alternate answer for part-2:
Let $X = \lim_{n\rightarrow\infty}X_n$, then we have
$$
\begin{align*}
P(X=k) & = \lim_{n\rightarrow\infty}\,P(X_n=k) \\
& = \lim_{n\rightarrow\infty}\frac{1}{e^{\frac{1}{ |
54,762 | Why does the proportions_ztest function in statsmodels produce different values than the formula for a 1-proportion Z test? | proportions_ztest seems to work exactly as documented.
Unfortunately what the documentation says it does is just not what you're expecting it to do.
By default this function uses the sample proportion in calculating the standard error of $p-p_0$. There's an option (via a boolean function argument) to change that defau... | Why does the proportions_ztest function in statsmodels produce different values than the formula for | proportions_ztest seems to work exactly as documented.
Unfortunately what the documentation says it does is just not what you're expecting it to do.
By default this function uses the sample proportio | Why does the proportions_ztest function in statsmodels produce different values than the formula for a 1-proportion Z test?
proportions_ztest seems to work exactly as documented.
Unfortunately what the documentation says it does is just not what you're expecting it to do.
By default this function uses the sample propo... | Why does the proportions_ztest function in statsmodels produce different values than the formula for
proportions_ztest seems to work exactly as documented.
Unfortunately what the documentation says it does is just not what you're expecting it to do.
By default this function uses the sample proportio |
54,763 | Why does the proportions_ztest function in statsmodels produce different values than the formula for a 1-proportion Z test? | Even though @Glen_b answer is completely right, it doesn't mention the parameter they are referring to.
For others who may fall into this same problem, the right way of using proportions_ztest the way the OP tried to use it is as follows:
from statsmodels.stats.proportion import proportions_ztest
proportions_ztest(10, ... | Why does the proportions_ztest function in statsmodels produce different values than the formula for | Even though @Glen_b answer is completely right, it doesn't mention the parameter they are referring to.
For others who may fall into this same problem, the right way of using proportions_ztest the way | Why does the proportions_ztest function in statsmodels produce different values than the formula for a 1-proportion Z test?
Even though @Glen_b answer is completely right, it doesn't mention the parameter they are referring to.
For others who may fall into this same problem, the right way of using proportions_ztest the... | Why does the proportions_ztest function in statsmodels produce different values than the formula for
Even though @Glen_b answer is completely right, it doesn't mention the parameter they are referring to.
For others who may fall into this same problem, the right way of using proportions_ztest the way |
54,764 | time series model with additional, time-independent regressors? | For time dependent regressors, it is pretty straightforward. Many classes of time series models can handle them, including from the ARIMA family (ex: ARIMAX and regression with ARIMA errors), BSTS, Facebook Prophet, and others.
The tricky part is time independent regressors: Most people don't realize that time indepen... | time series model with additional, time-independent regressors? | For time dependent regressors, it is pretty straightforward. Many classes of time series models can handle them, including from the ARIMA family (ex: ARIMAX and regression with ARIMA errors), BSTS, Fa | time series model with additional, time-independent regressors?
For time dependent regressors, it is pretty straightforward. Many classes of time series models can handle them, including from the ARIMA family (ex: ARIMAX and regression with ARIMA errors), BSTS, Facebook Prophet, and others.
The tricky part is time ind... | time series model with additional, time-independent regressors?
For time dependent regressors, it is pretty straightforward. Many classes of time series models can handle them, including from the ARIMA family (ex: ARIMAX and regression with ARIMA errors), BSTS, Fa |
54,765 | Why high correlation coefficient doesn't guarantee high VIF? | What do you consider to be a high correlation coefficient? What do you consider to be a low VIF?
The VIF is calculated by regressing predictor $i$ on all the other predictors, and then calculating $VIF = \frac{1}{1 - R_i^2}$. If you consider a VIF of 5 to be high, you'd only get a high VIF if $R_i^2$ was greater than ... | Why high correlation coefficient doesn't guarantee high VIF? | What do you consider to be a high correlation coefficient? What do you consider to be a low VIF?
The VIF is calculated by regressing predictor $i$ on all the other predictors, and then calculating $VI | Why high correlation coefficient doesn't guarantee high VIF?
What do you consider to be a high correlation coefficient? What do you consider to be a low VIF?
The VIF is calculated by regressing predictor $i$ on all the other predictors, and then calculating $VIF = \frac{1}{1 - R_i^2}$. If you consider a VIF of 5 to be... | Why high correlation coefficient doesn't guarantee high VIF?
What do you consider to be a high correlation coefficient? What do you consider to be a low VIF?
The VIF is calculated by regressing predictor $i$ on all the other predictors, and then calculating $VI |
54,766 | What does "weighted logistic regression" mean? | Let's begin with a weighted average, which slightly modifies the formula for an ordinary average:
$$\bar{x}^w=\frac{\sum_i w_i x_i}{\sum w_i}$$
An unweighted average would correspond to using $w_i=1$ (though any other constant would do as well).
Why would we do that?
Imagine, for example, that each value occurred mult... | What does "weighted logistic regression" mean? | Let's begin with a weighted average, which slightly modifies the formula for an ordinary average:
$$\bar{x}^w=\frac{\sum_i w_i x_i}{\sum w_i}$$
An unweighted average would correspond to using $w_i=1$ | What does "weighted logistic regression" mean?
Let's begin with a weighted average, which slightly modifies the formula for an ordinary average:
$$\bar{x}^w=\frac{\sum_i w_i x_i}{\sum w_i}$$
An unweighted average would correspond to using $w_i=1$ (though any other constant would do as well).
Why would we do that?
Imag... | What does "weighted logistic regression" mean?
Let's begin with a weighted average, which slightly modifies the formula for an ordinary average:
$$\bar{x}^w=\frac{\sum_i w_i x_i}{\sum w_i}$$
An unweighted average would correspond to using $w_i=1$ |
54,767 | What does "weighted logistic regression" mean? | Weighted logistic regression is used when you have an imbalanced dataset. Let's understand with an example.
Let's assume you have a dataset with patient details and you need to predict whether patient has cancer or not. Such datasets are generally imbalanced. If you have 10,000 data points who is having cancer and 1,00... | What does "weighted logistic regression" mean? | Weighted logistic regression is used when you have an imbalanced dataset. Let's understand with an example.
Let's assume you have a dataset with patient details and you need to predict whether patient | What does "weighted logistic regression" mean?
Weighted logistic regression is used when you have an imbalanced dataset. Let's understand with an example.
Let's assume you have a dataset with patient details and you need to predict whether patient has cancer or not. Such datasets are generally imbalanced. If you have 1... | What does "weighted logistic regression" mean?
Weighted logistic regression is used when you have an imbalanced dataset. Let's understand with an example.
Let's assume you have a dataset with patient details and you need to predict whether patient |
54,768 | Decision tree that fits a regression at leaf nodes | Although regression trees with constant fits in the terminal nodes are still much more widely used in practice, there is a long history of literature on regression trees that fit regression models (or other kinds of statistical models) in the nodes of the tree. RECPAM by Ciampi et al. (1988) is pioneering work in the s... | Decision tree that fits a regression at leaf nodes | Although regression trees with constant fits in the terminal nodes are still much more widely used in practice, there is a long history of literature on regression trees that fit regression models (or | Decision tree that fits a regression at leaf nodes
Although regression trees with constant fits in the terminal nodes are still much more widely used in practice, there is a long history of literature on regression trees that fit regression models (or other kinds of statistical models) in the nodes of the tree. RECPAM ... | Decision tree that fits a regression at leaf nodes
Although regression trees with constant fits in the terminal nodes are still much more widely used in practice, there is a long history of literature on regression trees that fit regression models (or |
54,769 | How to evaluate whether model is overfitting or underfitting when using cross_val_score and GridSearchCV? | You need to check the accuracy difference between train and test set for each fold result. If your model gives you high training accuracy but low test accuracy so your model is overfitting. If your model does not give good training accuracy you can say your model is underfitting.
GridSearchCV is trying to find the best... | How to evaluate whether model is overfitting or underfitting when using cross_val_score and GridSear | You need to check the accuracy difference between train and test set for each fold result. If your model gives you high training accuracy but low test accuracy so your model is overfitting. If your mo | How to evaluate whether model is overfitting or underfitting when using cross_val_score and GridSearchCV?
You need to check the accuracy difference between train and test set for each fold result. If your model gives you high training accuracy but low test accuracy so your model is overfitting. If your model does not g... | How to evaluate whether model is overfitting or underfitting when using cross_val_score and GridSear
You need to check the accuracy difference between train and test set for each fold result. If your model gives you high training accuracy but low test accuracy so your model is overfitting. If your mo |
54,770 | How to evaluate whether model is overfitting or underfitting when using cross_val_score and GridSearchCV? | I am trying to answer each of the questions, you asked.
....... However, averaging scores you get from cross validation returns just a single score. Should this be interpreted as the train or the test score from the previous case? or neither?
I am not sure which library or package, you are using for cross-validation. I... | How to evaluate whether model is overfitting or underfitting when using cross_val_score and GridSear | I am trying to answer each of the questions, you asked.
....... However, averaging scores you get from cross validation returns just a single score. Should this be interpreted as the train or the test | How to evaluate whether model is overfitting or underfitting when using cross_val_score and GridSearchCV?
I am trying to answer each of the questions, you asked.
....... However, averaging scores you get from cross validation returns just a single score. Should this be interpreted as the train or the test score from th... | How to evaluate whether model is overfitting or underfitting when using cross_val_score and GridSear
I am trying to answer each of the questions, you asked.
....... However, averaging scores you get from cross validation returns just a single score. Should this be interpreted as the train or the test |
54,771 | Average number of random permutations of a sequence, before seeing a sorted sequence | Let us start by assuming $N$ unique entries in our vector.
The action of randomly shuffling a vector and checking whether it is sorted in a particular order afterwards is equivalent to picking a permutation at random and checking whether it is one very specific one, namely the one that orders the vector in the order we... | Average number of random permutations of a sequence, before seeing a sorted sequence | Let us start by assuming $N$ unique entries in our vector.
The action of randomly shuffling a vector and checking whether it is sorted in a particular order afterwards is equivalent to picking a permu | Average number of random permutations of a sequence, before seeing a sorted sequence
Let us start by assuming $N$ unique entries in our vector.
The action of randomly shuffling a vector and checking whether it is sorted in a particular order afterwards is equivalent to picking a permutation at random and checking wheth... | Average number of random permutations of a sequence, before seeing a sorted sequence
Let us start by assuming $N$ unique entries in our vector.
The action of randomly shuffling a vector and checking whether it is sorted in a particular order afterwards is equivalent to picking a permu |
54,772 | What is the difference between "controlling for a variable" and interaction? | It makes more sense to say that someone becomes heavier if one is taller and/or consumes more soda than that someone becomes taller if (s)he is heavier and consumes more soda. So I assume you mean that the dependent/explained/left-hand-side/y-variable is weight and the independent/explanatory/right-hand-side/x-variable... | What is the difference between "controlling for a variable" and interaction? | It makes more sense to say that someone becomes heavier if one is taller and/or consumes more soda than that someone becomes taller if (s)he is heavier and consumes more soda. So I assume you mean tha | What is the difference between "controlling for a variable" and interaction?
It makes more sense to say that someone becomes heavier if one is taller and/or consumes more soda than that someone becomes taller if (s)he is heavier and consumes more soda. So I assume you mean that the dependent/explained/left-hand-side/y-... | What is the difference between "controlling for a variable" and interaction?
It makes more sense to say that someone becomes heavier if one is taller and/or consumes more soda than that someone becomes taller if (s)he is heavier and consumes more soda. So I assume you mean tha |
54,773 | Is it possible of overfit using Propensity score matching with the MatchIt R package? | First, I would caution anyone without a background in applied statistics from performing advanced analyses like propensity score matching. The ease of the software makes it seem like the procedure itself is easy, when in fact there are many considerations required to make a valid inference. I'm also pretty skeptical of... | Is it possible of overfit using Propensity score matching with the MatchIt R package? | First, I would caution anyone without a background in applied statistics from performing advanced analyses like propensity score matching. The ease of the software makes it seem like the procedure its | Is it possible of overfit using Propensity score matching with the MatchIt R package?
First, I would caution anyone without a background in applied statistics from performing advanced analyses like propensity score matching. The ease of the software makes it seem like the procedure itself is easy, when in fact there ar... | Is it possible of overfit using Propensity score matching with the MatchIt R package?
First, I would caution anyone without a background in applied statistics from performing advanced analyses like propensity score matching. The ease of the software makes it seem like the procedure its |
54,774 | Should I remove duplicates from my dataset for my machine learning problem? | You should probably remove them. Duplicates are an extreme case of nonrandom sampling, and they bias your fitted model. Including them will essentially lead to the model overfitting this subset of points.
I say probably because you should (1) be sure they are not real data that coincidentally have values that are iden... | Should I remove duplicates from my dataset for my machine learning problem? | You should probably remove them. Duplicates are an extreme case of nonrandom sampling, and they bias your fitted model. Including them will essentially lead to the model overfitting this subset of poi | Should I remove duplicates from my dataset for my machine learning problem?
You should probably remove them. Duplicates are an extreme case of nonrandom sampling, and they bias your fitted model. Including them will essentially lead to the model overfitting this subset of points.
I say probably because you should (1) ... | Should I remove duplicates from my dataset for my machine learning problem?
You should probably remove them. Duplicates are an extreme case of nonrandom sampling, and they bias your fitted model. Including them will essentially lead to the model overfitting this subset of poi |
54,775 | Selecting ARIMA orders based on ACF-PACF vs. auto.arima | First, it is very hard to use (P)ACF plots to identify ARIMA(p,d,q) models if both p and q are nonzero. See Hyndman & Athanasopoulos:
If p and q are both positive, then the plots do not help in finding suitable values of p and q.
Second, your peaks at lag 4 only barely exceed the confidence bands.
I would always pref... | Selecting ARIMA orders based on ACF-PACF vs. auto.arima | First, it is very hard to use (P)ACF plots to identify ARIMA(p,d,q) models if both p and q are nonzero. See Hyndman & Athanasopoulos:
If p and q are both positive, then the plots do not help in findi | Selecting ARIMA orders based on ACF-PACF vs. auto.arima
First, it is very hard to use (P)ACF plots to identify ARIMA(p,d,q) models if both p and q are nonzero. See Hyndman & Athanasopoulos:
If p and q are both positive, then the plots do not help in finding suitable values of p and q.
Second, your peaks at lag 4 only... | Selecting ARIMA orders based on ACF-PACF vs. auto.arima
First, it is very hard to use (P)ACF plots to identify ARIMA(p,d,q) models if both p and q are nonzero. See Hyndman & Athanasopoulos:
If p and q are both positive, then the plots do not help in findi |
54,776 | Why are emmeans package means different than regular means? | The fundamental difference between estimated marginal means (EMMs) and ordinary marginal means of data (OMMs) is that OMMs summarize the data, while EMMs summarize a model. Thus, if you fit a different model to the data, the EMMs are potentially different. EMMs are not just one thing.
To be a bit more precise, EMMs inv... | Why are emmeans package means different than regular means? | The fundamental difference between estimated marginal means (EMMs) and ordinary marginal means of data (OMMs) is that OMMs summarize the data, while EMMs summarize a model. Thus, if you fit a differen | Why are emmeans package means different than regular means?
The fundamental difference between estimated marginal means (EMMs) and ordinary marginal means of data (OMMs) is that OMMs summarize the data, while EMMs summarize a model. Thus, if you fit a different model to the data, the EMMs are potentially different. EMM... | Why are emmeans package means different than regular means?
The fundamental difference between estimated marginal means (EMMs) and ordinary marginal means of data (OMMs) is that OMMs summarize the data, while EMMs summarize a model. Thus, if you fit a differen |
54,777 | Why are emmeans package means different than regular means? | You are indeed right that this difference can be explained from the missing data you have. In particular, when you have missing data that are of the missing at random type, then the observed data are not a representative sample of your target population. In this case, the simple sample means will be biased and should n... | Why are emmeans package means different than regular means? | You are indeed right that this difference can be explained from the missing data you have. In particular, when you have missing data that are of the missing at random type, then the observed data are | Why are emmeans package means different than regular means?
You are indeed right that this difference can be explained from the missing data you have. In particular, when you have missing data that are of the missing at random type, then the observed data are not a representative sample of your target population. In th... | Why are emmeans package means different than regular means?
You are indeed right that this difference can be explained from the missing data you have. In particular, when you have missing data that are of the missing at random type, then the observed data are |
54,778 | Simulating random walk with "known" prediction | I started by shortening your series to 20 realizations, so we could actually see something.
set.seed(420)
x=rnorm(20)
y=rep(NA,length(x))
y[1]=x[1]
for (i in 2:length(x)) y[i]=y[i-1]+x[i]*0.7
Then I simulated five trajectories (each of length six). First, I draw 5 normal random variables with mean 'y[15]. Then I draw ... | Simulating random walk with "known" prediction | I started by shortening your series to 20 realizations, so we could actually see something.
set.seed(420)
x=rnorm(20)
y=rep(NA,length(x))
y[1]=x[1]
for (i in 2:length(x)) y[i]=y[i-1]+x[i]*0.7
Then I | Simulating random walk with "known" prediction
I started by shortening your series to 20 realizations, so we could actually see something.
set.seed(420)
x=rnorm(20)
y=rep(NA,length(x))
y[1]=x[1]
for (i in 2:length(x)) y[i]=y[i-1]+x[i]*0.7
Then I simulated five trajectories (each of length six). First, I draw 5 normal ... | Simulating random walk with "known" prediction
I started by shortening your series to 20 realizations, so we could actually see something.
set.seed(420)
x=rnorm(20)
y=rep(NA,length(x))
y[1]=x[1]
for (i in 2:length(x)) y[i]=y[i-1]+x[i]*0.7
Then I |
54,779 | Simulating random walk with "known" prediction | The specification of the sort of data that you want to generate is very broad, and there are many ways to simulate sort of random walks with paths that have a tendency to 'return to the mean'.
For instance, you can change your code like:
set.seed(420)
a = 0.9
b = 0.7
n = 10^4
x=rnorm(n)
y=rep(NA,n)
y[1]=x[1]
for (i in ... | Simulating random walk with "known" prediction | The specification of the sort of data that you want to generate is very broad, and there are many ways to simulate sort of random walks with paths that have a tendency to 'return to the mean'.
For ins | Simulating random walk with "known" prediction
The specification of the sort of data that you want to generate is very broad, and there are many ways to simulate sort of random walks with paths that have a tendency to 'return to the mean'.
For instance, you can change your code like:
set.seed(420)
a = 0.9
b = 0.7
n = 1... | Simulating random walk with "known" prediction
The specification of the sort of data that you want to generate is very broad, and there are many ways to simulate sort of random walks with paths that have a tendency to 'return to the mean'.
For ins |
54,780 | Simulating random walk with "known" prediction | A random walk process that you're using has a constant (or zero) drift and variance:
$$dW_t=\xi_t,\\\xi_t\sim \mathcal N(0,\sigma^2)$$
You want "arbitrary variablility", i.e. $\sigma^2$ is not only changing with time, but also in some arbitrary way, whatever you meant by this word. If you meant that it's stochastic, u... | Simulating random walk with "known" prediction | A random walk process that you're using has a constant (or zero) drift and variance:
$$dW_t=\xi_t,\\\xi_t\sim \mathcal N(0,\sigma^2)$$
You want "arbitrary variablility", i.e. $\sigma^2$ is not only c | Simulating random walk with "known" prediction
A random walk process that you're using has a constant (or zero) drift and variance:
$$dW_t=\xi_t,\\\xi_t\sim \mathcal N(0,\sigma^2)$$
You want "arbitrary variablility", i.e. $\sigma^2$ is not only changing with time, but also in some arbitrary way, whatever you meant by ... | Simulating random walk with "known" prediction
A random walk process that you're using has a constant (or zero) drift and variance:
$$dW_t=\xi_t,\\\xi_t\sim \mathcal N(0,\sigma^2)$$
You want "arbitrary variablility", i.e. $\sigma^2$ is not only c |
54,781 | What is the probability that Person A will require more tosses of a particular coin than Person B to obtain the first head? | Your first way and result are correct. For the second way, we can exchange the summation order, and $i$ should start from $2$:
$$\begin{align}p&=\pi^2\sum_{i=2}^\infty (1-\pi)^{i-1}\sum_{j=1}^{i-1}(1-\pi)^{j-1}= \pi^2\sum_{i=2}^\infty (1-\pi)^{i-1} \left(\frac{1-(1-\pi)^{i-1}}{\pi}\right)\\&=\pi\sum_{i=2}^\infty(1-\pi)... | What is the probability that Person A will require more tosses of a particular coin than Person B to | Your first way and result are correct. For the second way, we can exchange the summation order, and $i$ should start from $2$:
$$\begin{align}p&=\pi^2\sum_{i=2}^\infty (1-\pi)^{i-1}\sum_{j=1}^{i-1}(1- | What is the probability that Person A will require more tosses of a particular coin than Person B to obtain the first head?
Your first way and result are correct. For the second way, we can exchange the summation order, and $i$ should start from $2$:
$$\begin{align}p&=\pi^2\sum_{i=2}^\infty (1-\pi)^{i-1}\sum_{j=1}^{i-1... | What is the probability that Person A will require more tosses of a particular coin than Person B to
Your first way and result are correct. For the second way, we can exchange the summation order, and $i$ should start from $2$:
$$\begin{align}p&=\pi^2\sum_{i=2}^\infty (1-\pi)^{i-1}\sum_{j=1}^{i-1}(1- |
54,782 | How to predict by hand in R using splines regression? [closed] | As per my comment, once you fit your model, you can extract the values of the predictors included in the model using the model.matrix() function:
require(stats)
require(splines)
require(graphics)
fm1 <- lm(weight ~ bs(height, df = 5), data = women)
summary(fm1)
model.matrix(fm1)
The R output produced by model.matr... | How to predict by hand in R using splines regression? [closed] | As per my comment, once you fit your model, you can extract the values of the predictors included in the model using the model.matrix() function:
require(stats)
require(splines)
require(graphics)
fm | How to predict by hand in R using splines regression? [closed]
As per my comment, once you fit your model, you can extract the values of the predictors included in the model using the model.matrix() function:
require(stats)
require(splines)
require(graphics)
fm1 <- lm(weight ~ bs(height, df = 5), data = women)
summa... | How to predict by hand in R using splines regression? [closed]
As per my comment, once you fit your model, you can extract the values of the predictors included in the model using the model.matrix() function:
require(stats)
require(splines)
require(graphics)
fm |
54,783 | Should I cross-validate a logistic regression model that will not be used to make predictions? | You want to identify "variables that are most strongly related to the outcomes of complaints against practitioners in a profession," but not to predict future outcomes of complaints. Presumably, the idea is to generate hypotheses about factors that might be manipulated in future work to reduce undesirable outcomes. Cro... | Should I cross-validate a logistic regression model that will not be used to make predictions? | You want to identify "variables that are most strongly related to the outcomes of complaints against practitioners in a profession," but not to predict future outcomes of complaints. Presumably, the i | Should I cross-validate a logistic regression model that will not be used to make predictions?
You want to identify "variables that are most strongly related to the outcomes of complaints against practitioners in a profession," but not to predict future outcomes of complaints. Presumably, the idea is to generate hypoth... | Should I cross-validate a logistic regression model that will not be used to make predictions?
You want to identify "variables that are most strongly related to the outcomes of complaints against practitioners in a profession," but not to predict future outcomes of complaints. Presumably, the i |
54,784 | Should I cross-validate a logistic regression model that will not be used to make predictions? | Your question is, IMHO, slightly off the point. In statistics book often a distinction is made between inference and prediction (e.g. in Harrell 2001 Regression Modeling Strategies, or in Shmueli 2010's paper To explain or to predict?). In your case, I would argue you are actually interested in using the data to form a... | Should I cross-validate a logistic regression model that will not be used to make predictions? | Your question is, IMHO, slightly off the point. In statistics book often a distinction is made between inference and prediction (e.g. in Harrell 2001 Regression Modeling Strategies, or in Shmueli 2010 | Should I cross-validate a logistic regression model that will not be used to make predictions?
Your question is, IMHO, slightly off the point. In statistics book often a distinction is made between inference and prediction (e.g. in Harrell 2001 Regression Modeling Strategies, or in Shmueli 2010's paper To explain or to... | Should I cross-validate a logistic regression model that will not be used to make predictions?
Your question is, IMHO, slightly off the point. In statistics book often a distinction is made between inference and prediction (e.g. in Harrell 2001 Regression Modeling Strategies, or in Shmueli 2010 |
54,785 | Should I cross-validate a logistic regression model that will not be used to make predictions? | The model will not be used to make predictions on future outcomes, but to make inferences about decisions during the time period.
Having lots of hope, that I am not mistaken, I understand, that your goal is to make causal inferernce. This means you want to say "such and such decision caused different probability of ou... | Should I cross-validate a logistic regression model that will not be used to make predictions? | The model will not be used to make predictions on future outcomes, but to make inferences about decisions during the time period.
Having lots of hope, that I am not mistaken, I understand, that your | Should I cross-validate a logistic regression model that will not be used to make predictions?
The model will not be used to make predictions on future outcomes, but to make inferences about decisions during the time period.
Having lots of hope, that I am not mistaken, I understand, that your goal is to make causal in... | Should I cross-validate a logistic regression model that will not be used to make predictions?
The model will not be used to make predictions on future outcomes, but to make inferences about decisions during the time period.
Having lots of hope, that I am not mistaken, I understand, that your |
54,786 | What does beta distribution "has support" mean? | I think it is better read this way:
the beta prior has and only has support over all valid probabilities
However, I think it is a bit redundant in terms of its meaning? | What does beta distribution "has support" mean? | I think it is better read this way:
the beta prior has and only has support over all valid probabilities
However, I think it is a bit redundant in terms of its meaning? | What does beta distribution "has support" mean?
I think it is better read this way:
the beta prior has and only has support over all valid probabilities
However, I think it is a bit redundant in terms of its meaning? | What does beta distribution "has support" mean?
I think it is better read this way:
the beta prior has and only has support over all valid probabilities
However, I think it is a bit redundant in terms of its meaning? |
54,787 | What does beta distribution "has support" mean? | For a single probability parameter, the interval $\mathscr{P} \equiv [0,1]$ is the set of "all valid probabilities and only valid probabilities". Thus, when they say that the Beta distribution "has support over" $\mathscr{P}$, what they mean is that for any random variable $p \sim \text{Beta}$ we have:
$$\begin{equati... | What does beta distribution "has support" mean? | For a single probability parameter, the interval $\mathscr{P} \equiv [0,1]$ is the set of "all valid probabilities and only valid probabilities". Thus, when they say that the Beta distribution "has s | What does beta distribution "has support" mean?
For a single probability parameter, the interval $\mathscr{P} \equiv [0,1]$ is the set of "all valid probabilities and only valid probabilities". Thus, when they say that the Beta distribution "has support over" $\mathscr{P}$, what they mean is that for any random variab... | What does beta distribution "has support" mean?
For a single probability parameter, the interval $\mathscr{P} \equiv [0,1]$ is the set of "all valid probabilities and only valid probabilities". Thus, when they say that the Beta distribution "has s |
54,788 | What does beta distribution "has support" mean? | I wanted to provide a non-technical answer to your question.
The beta distribution is commonly used in the context of modeling proportions.
As an example, let’s say you select (at random) 100 geographic sites for a study and you keep track for each site what proportion of the site’s area can be classified as “forest... | What does beta distribution "has support" mean? | I wanted to provide a non-technical answer to your question.
The beta distribution is commonly used in the context of modeling proportions.
As an example, let’s say you select (at random) 100 geogr | What does beta distribution "has support" mean?
I wanted to provide a non-technical answer to your question.
The beta distribution is commonly used in the context of modeling proportions.
As an example, let’s say you select (at random) 100 geographic sites for a study and you keep track for each site what proportion... | What does beta distribution "has support" mean?
I wanted to provide a non-technical answer to your question.
The beta distribution is commonly used in the context of modeling proportions.
As an example, let’s say you select (at random) 100 geogr |
54,789 | Can anyone give a concrete example to illustrate what is an uniform prior? | Let's say you don't know the probability of head, $p$, of a coin. You decide to conduct an experiment to estimate what it is, via Bayesian analysis. It requires you to choose a prior, and in general you're free to choose one of the feasible ones. If you don't know or don't want to assume anything about this $p$, you ca... | Can anyone give a concrete example to illustrate what is an uniform prior? | Let's say you don't know the probability of head, $p$, of a coin. You decide to conduct an experiment to estimate what it is, via Bayesian analysis. It requires you to choose a prior, and in general y | Can anyone give a concrete example to illustrate what is an uniform prior?
Let's say you don't know the probability of head, $p$, of a coin. You decide to conduct an experiment to estimate what it is, via Bayesian analysis. It requires you to choose a prior, and in general you're free to choose one of the feasible ones... | Can anyone give a concrete example to illustrate what is an uniform prior?
Let's say you don't know the probability of head, $p$, of a coin. You decide to conduct an experiment to estimate what it is, via Bayesian analysis. It requires you to choose a prior, and in general y |
54,790 | Can anyone give a concrete example to illustrate what is an uniform prior? | The notion of uniform prior understood as a prior with a constant density $\pi(\theta)=c$ is not well-defined (or even meaningful) as it depends on both
the dominating measure that determines the density function of the prior (i.e., one measures volume);
the parameterisation $\theta$ of the sampling model $f(x|\theta)... | Can anyone give a concrete example to illustrate what is an uniform prior? | The notion of uniform prior understood as a prior with a constant density $\pi(\theta)=c$ is not well-defined (or even meaningful) as it depends on both
the dominating measure that determines the den | Can anyone give a concrete example to illustrate what is an uniform prior?
The notion of uniform prior understood as a prior with a constant density $\pi(\theta)=c$ is not well-defined (or even meaningful) as it depends on both
the dominating measure that determines the density function of the prior (i.e., one measure... | Can anyone give a concrete example to illustrate what is an uniform prior?
The notion of uniform prior understood as a prior with a constant density $\pi(\theta)=c$ is not well-defined (or even meaningful) as it depends on both
the dominating measure that determines the den |
54,791 | Can anyone give a concrete example to illustrate what is an uniform prior? | When the prior distribution $\pi$, of the parameter $\theta$ to be estimated is the Uniform distribution, i.e. $\pi(\theta)\sim U(a,b)$, we refer to prior $\pi$ as a uniform or uninformative prior. I'm not sure what's not to understand here except the basics of Bayesian inference and the Uniform distribution.
The best ... | Can anyone give a concrete example to illustrate what is an uniform prior? | When the prior distribution $\pi$, of the parameter $\theta$ to be estimated is the Uniform distribution, i.e. $\pi(\theta)\sim U(a,b)$, we refer to prior $\pi$ as a uniform or uninformative prior. I' | Can anyone give a concrete example to illustrate what is an uniform prior?
When the prior distribution $\pi$, of the parameter $\theta$ to be estimated is the Uniform distribution, i.e. $\pi(\theta)\sim U(a,b)$, we refer to prior $\pi$ as a uniform or uninformative prior. I'm not sure what's not to understand here exce... | Can anyone give a concrete example to illustrate what is an uniform prior?
When the prior distribution $\pi$, of the parameter $\theta$ to be estimated is the Uniform distribution, i.e. $\pi(\theta)\sim U(a,b)$, we refer to prior $\pi$ as a uniform or uninformative prior. I' |
54,792 | Big Sample size, Small coefficients, significant results. What should I do? | I think it's been asked before. It's useful to realize that, without a prespecified sample size and alpha level, the $p$-value is just a measure of the sample size you ultimately wind up with. Not appealing. An approach I use is this: at what sample size would a 0.05 level be appropriate? Scale accordingly. For instanc... | Big Sample size, Small coefficients, significant results. What should I do? | I think it's been asked before. It's useful to realize that, without a prespecified sample size and alpha level, the $p$-value is just a measure of the sample size you ultimately wind up with. Not app | Big Sample size, Small coefficients, significant results. What should I do?
I think it's been asked before. It's useful to realize that, without a prespecified sample size and alpha level, the $p$-value is just a measure of the sample size you ultimately wind up with. Not appealing. An approach I use is this: at what s... | Big Sample size, Small coefficients, significant results. What should I do?
I think it's been asked before. It's useful to realize that, without a prespecified sample size and alpha level, the $p$-value is just a measure of the sample size you ultimately wind up with. Not app |
54,793 | Big Sample size, Small coefficients, significant results. What should I do? | You have encountered the gulf between "statistically significant" and "meaningful". As you point out, with sufficient sample size, you can assign statistical significance to arbitrarily small differences - there is no difference too small that can't be called "significant" with large enough N. You need to use domain kn... | Big Sample size, Small coefficients, significant results. What should I do? | You have encountered the gulf between "statistically significant" and "meaningful". As you point out, with sufficient sample size, you can assign statistical significance to arbitrarily small differen | Big Sample size, Small coefficients, significant results. What should I do?
You have encountered the gulf between "statistically significant" and "meaningful". As you point out, with sufficient sample size, you can assign statistical significance to arbitrarily small differences - there is no difference too small that ... | Big Sample size, Small coefficients, significant results. What should I do?
You have encountered the gulf between "statistically significant" and "meaningful". As you point out, with sufficient sample size, you can assign statistical significance to arbitrarily small differen |
54,794 | Big Sample size, Small coefficients, significant results. What should I do? | When you have many samples and the observed effect is very small (small for the specified application), you can safely conclude that the independent variables do not have an important effect on the dependent variable. The effect size can be “statistically significant” and unimportant at the same time.
Using small sampl... | Big Sample size, Small coefficients, significant results. What should I do? | When you have many samples and the observed effect is very small (small for the specified application), you can safely conclude that the independent variables do not have an important effect on the de | Big Sample size, Small coefficients, significant results. What should I do?
When you have many samples and the observed effect is very small (small for the specified application), you can safely conclude that the independent variables do not have an important effect on the dependent variable. The effect size can be “st... | Big Sample size, Small coefficients, significant results. What should I do?
When you have many samples and the observed effect is very small (small for the specified application), you can safely conclude that the independent variables do not have an important effect on the de |
54,795 | Big Sample size, Small coefficients, significant results. What should I do? | I think you should decide on an "expected minimal effect size", i.e. the minimal coefficients you care to include in your model. Say, do you care about coefficients less than 0.0001, or 1, or 100? To clarify, the effect size is the degree to which the null hypothesis is false, or how large the coefficient actually is. ... | Big Sample size, Small coefficients, significant results. What should I do? | I think you should decide on an "expected minimal effect size", i.e. the minimal coefficients you care to include in your model. Say, do you care about coefficients less than 0.0001, or 1, or 100? To | Big Sample size, Small coefficients, significant results. What should I do?
I think you should decide on an "expected minimal effect size", i.e. the minimal coefficients you care to include in your model. Say, do you care about coefficients less than 0.0001, or 1, or 100? To clarify, the effect size is the degree to wh... | Big Sample size, Small coefficients, significant results. What should I do?
I think you should decide on an "expected minimal effect size", i.e. the minimal coefficients you care to include in your model. Say, do you care about coefficients less than 0.0001, or 1, or 100? To |
54,796 | Combining together principal components from PCA performed on different subsets of a large dataset | Note: though the old answer (below the line) was accepted, the comment below alerted me to the fact that I had misinterpreted the question. My old answer pertains to comparing PCs on different batches of observations (i.e. different rows). But the question is actually about doing PCs on different batches of variables (... | Combining together principal components from PCA performed on different subsets of a large dataset | Note: though the old answer (below the line) was accepted, the comment below alerted me to the fact that I had misinterpreted the question. My old answer pertains to comparing PCs on different batches | Combining together principal components from PCA performed on different subsets of a large dataset
Note: though the old answer (below the line) was accepted, the comment below alerted me to the fact that I had misinterpreted the question. My old answer pertains to comparing PCs on different batches of observations (i.e... | Combining together principal components from PCA performed on different subsets of a large dataset
Note: though the old answer (below the line) was accepted, the comment below alerted me to the fact that I had misinterpreted the question. My old answer pertains to comparing PCs on different batches |
54,797 | How to form groups before randomizing the treatment assignment? | A reasonable approach here is to use block randomisation, where you create non-randomised blocks of subjects, grouping together like subjects (e.g., by prior skill variables), and then you create random groups by randomly allocating people from the blocks into the different treatment groups. This gives you randomised ... | How to form groups before randomizing the treatment assignment? | A reasonable approach here is to use block randomisation, where you create non-randomised blocks of subjects, grouping together like subjects (e.g., by prior skill variables), and then you create rand | How to form groups before randomizing the treatment assignment?
A reasonable approach here is to use block randomisation, where you create non-randomised blocks of subjects, grouping together like subjects (e.g., by prior skill variables), and then you create random groups by randomly allocating people from the blocks ... | How to form groups before randomizing the treatment assignment?
A reasonable approach here is to use block randomisation, where you create non-randomised blocks of subjects, grouping together like subjects (e.g., by prior skill variables), and then you create rand |
54,798 | How to form groups before randomizing the treatment assignment? | You want to balance prior skills across classes, because they may influence the outcome and confound the results. This is analogous to clinical trials where covariates that are known to influence the prognosis need to be controlled for. This is done at the level of assignment through stratified randomization (https://w... | How to form groups before randomizing the treatment assignment? | You want to balance prior skills across classes, because they may influence the outcome and confound the results. This is analogous to clinical trials where covariates that are known to influence the | How to form groups before randomizing the treatment assignment?
You want to balance prior skills across classes, because they may influence the outcome and confound the results. This is analogous to clinical trials where covariates that are known to influence the prognosis need to be controlled for. This is done at the... | How to form groups before randomizing the treatment assignment?
You want to balance prior skills across classes, because they may influence the outcome and confound the results. This is analogous to clinical trials where covariates that are known to influence the |
54,799 | How to form groups before randomizing the treatment assignment? | What is nice is that regardless of how you allocate the students to the classes, as long as you randomly assign the classes to treatment, your effect estimate will be free of confounding. Ideally, though, you want the class composition to be the same on average between the treatment groups (i.e., for the distribution o... | How to form groups before randomizing the treatment assignment? | What is nice is that regardless of how you allocate the students to the classes, as long as you randomly assign the classes to treatment, your effect estimate will be free of confounding. Ideally, tho | How to form groups before randomizing the treatment assignment?
What is nice is that regardless of how you allocate the students to the classes, as long as you randomly assign the classes to treatment, your effect estimate will be free of confounding. Ideally, though, you want the class composition to be the same on av... | How to form groups before randomizing the treatment assignment?
What is nice is that regardless of how you allocate the students to the classes, as long as you randomly assign the classes to treatment, your effect estimate will be free of confounding. Ideally, tho |
54,800 | How to form groups before randomizing the treatment assignment? | You can do better by randomizing treatment at the student level and then forming the classes deterministically. Consider the following procedure.
1) Match the N students based on baseline skill and any other important covariates you want to include, e.g. gender, race, etc. This will form N/2 student pairs. References ... | How to form groups before randomizing the treatment assignment? | You can do better by randomizing treatment at the student level and then forming the classes deterministically. Consider the following procedure.
1) Match the N students based on baseline skill and a | How to form groups before randomizing the treatment assignment?
You can do better by randomizing treatment at the student level and then forming the classes deterministically. Consider the following procedure.
1) Match the N students based on baseline skill and any other important covariates you want to include, e.g. ... | How to form groups before randomizing the treatment assignment?
You can do better by randomizing treatment at the student level and then forming the classes deterministically. Consider the following procedure.
1) Match the N students based on baseline skill and a |
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