idx int64 1 56k | question stringlengths 15 155 | answer stringlengths 2 29.2k ⌀ | question_cut stringlengths 15 100 | answer_cut stringlengths 2 200 ⌀ | conversation stringlengths 47 29.3k | conversation_cut stringlengths 47 301 |
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53,501 | Continuous and differentiable bell-shaped distribution on $[a, b]$ | The Truncated normal distribution obeys all prerequisites:
It's bell shaped
It's continuous
Its support is $x \in [a,b]$
It's differentiable, i.e. $\nabla_x p(x)$ exists for all $x \in [a,b]$ | Continuous and differentiable bell-shaped distribution on $[a, b]$ | The Truncated normal distribution obeys all prerequisites:
It's bell shaped
It's continuous
Its support is $x \in [a,b]$
It's differentiable, i.e. $\nabla_x p(x)$ exists for all $x \in [a,b]$ | Continuous and differentiable bell-shaped distribution on $[a, b]$
The Truncated normal distribution obeys all prerequisites:
It's bell shaped
It's continuous
Its support is $x \in [a,b]$
It's differentiable, i.e. $\nabla_x p(x)$ exists for all $x \in [a,b]$ | Continuous and differentiable bell-shaped distribution on $[a, b]$
The Truncated normal distribution obeys all prerequisites:
It's bell shaped
It's continuous
Its support is $x \in [a,b]$
It's differentiable, i.e. $\nabla_x p(x)$ exists for all $x \in [a,b]$ |
53,502 | Continuous and differentiable bell-shaped distribution on $[a, b]$ | One option is to transform a beta distribution.
$Beta(3,3)$ has your desired properties on $[0,1]$.
Now subtract $1/2$ to center the distribution.
Next, multiply to stretch or compress the distribution.
Finally, add your desired mean. | Continuous and differentiable bell-shaped distribution on $[a, b]$ | One option is to transform a beta distribution.
$Beta(3,3)$ has your desired properties on $[0,1]$.
Now subtract $1/2$ to center the distribution.
Next, multiply to stretch or compress the distributio | Continuous and differentiable bell-shaped distribution on $[a, b]$
One option is to transform a beta distribution.
$Beta(3,3)$ has your desired properties on $[0,1]$.
Now subtract $1/2$ to center the distribution.
Next, multiply to stretch or compress the distribution.
Finally, add your desired mean. | Continuous and differentiable bell-shaped distribution on $[a, b]$
One option is to transform a beta distribution.
$Beta(3,3)$ has your desired properties on $[0,1]$.
Now subtract $1/2$ to center the distribution.
Next, multiply to stretch or compress the distributio |
53,503 | Can we construct a pair of random variables having any given covariance? | Covariances cannot have arbitrary values in comparison to variances; $|\operatorname{Cov}(X,Y)| \leq \sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}$. So, Yes, it is not possible to find random variables that have the alleged covariance matrix, which is, as you have discovered, not a positive semidefinite matrix. For... | Can we construct a pair of random variables having any given covariance? | Covariances cannot have arbitrary values in comparison to variances; $|\operatorname{Cov}(X,Y)| \leq \sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}$. So, Yes, it is not possible to find random vari | Can we construct a pair of random variables having any given covariance?
Covariances cannot have arbitrary values in comparison to variances; $|\operatorname{Cov}(X,Y)| \leq \sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}$. So, Yes, it is not possible to find random variables that have the alleged covariance matrix, ... | Can we construct a pair of random variables having any given covariance?
Covariances cannot have arbitrary values in comparison to variances; $|\operatorname{Cov}(X,Y)| \leq \sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}$. So, Yes, it is not possible to find random vari |
53,504 | What do you do in regression if $(X^\text{T} X)^{-1}$ does not exist? | The matrix $X^\text{T} X$ is the Gramian matrix of the design matrix (assuming here that the design matrix has elements that are all real numbers). If the Gram-determinant is zero (i.e., if $\text{det} (X^\text{T} X) = 0$) then the Gramian matrix is not invertible, which means that the design matrix has at least one c... | What do you do in regression if $(X^\text{T} X)^{-1}$ does not exist? | The matrix $X^\text{T} X$ is the Gramian matrix of the design matrix (assuming here that the design matrix has elements that are all real numbers). If the Gram-determinant is zero (i.e., if $\text{de | What do you do in regression if $(X^\text{T} X)^{-1}$ does not exist?
The matrix $X^\text{T} X$ is the Gramian matrix of the design matrix (assuming here that the design matrix has elements that are all real numbers). If the Gram-determinant is zero (i.e., if $\text{det} (X^\text{T} X) = 0$) then the Gramian matrix is... | What do you do in regression if $(X^\text{T} X)^{-1}$ does not exist?
The matrix $X^\text{T} X$ is the Gramian matrix of the design matrix (assuming here that the design matrix has elements that are all real numbers). If the Gram-determinant is zero (i.e., if $\text{de |
53,505 | What do you do in regression if $(X^\text{T} X)^{-1}$ does not exist? | There are two common settings where the problem occurs in regression, and these are treated differently.
The first is when the number of columns $p$ of $X$ is greater than the number of rows $n$. In that case you can't do the regression, and you need some sort of dimension-reduction approach. You might do subset selec... | What do you do in regression if $(X^\text{T} X)^{-1}$ does not exist? | There are two common settings where the problem occurs in regression, and these are treated differently.
The first is when the number of columns $p$ of $X$ is greater than the number of rows $n$. In t | What do you do in regression if $(X^\text{T} X)^{-1}$ does not exist?
There are two common settings where the problem occurs in regression, and these are treated differently.
The first is when the number of columns $p$ of $X$ is greater than the number of rows $n$. In that case you can't do the regression, and you need... | What do you do in regression if $(X^\text{T} X)^{-1}$ does not exist?
There are two common settings where the problem occurs in regression, and these are treated differently.
The first is when the number of columns $p$ of $X$ is greater than the number of rows $n$. In t |
53,506 | Post hoc contrasts when only certain contrasts make sense | In what follows I assume that you want to control the FWER (in the strong sense). In general, if you want to test a fixed number of arbitrary planned contrasts (in your case: treatment A vs. control A, treatment B vs. control B), the Holm method can be used to control the FWER strongly within the family of hypothesis t... | Post hoc contrasts when only certain contrasts make sense | In what follows I assume that you want to control the FWER (in the strong sense). In general, if you want to test a fixed number of arbitrary planned contrasts (in your case: treatment A vs. control A | Post hoc contrasts when only certain contrasts make sense
In what follows I assume that you want to control the FWER (in the strong sense). In general, if you want to test a fixed number of arbitrary planned contrasts (in your case: treatment A vs. control A, treatment B vs. control B), the Holm method can be used to c... | Post hoc contrasts when only certain contrasts make sense
In what follows I assume that you want to control the FWER (in the strong sense). In general, if you want to test a fixed number of arbitrary planned contrasts (in your case: treatment A vs. control A |
53,507 | Post hoc contrasts when only certain contrasts make sense | This is a good question.
In your situation, it's common and perfectly ok to just run two t tests for the contrasts of interest, and do the Boneferroni correction manually by multiplying the p values by 2. With only two contrasts, there will be very little difference between this and other, harder to justify corrections... | Post hoc contrasts when only certain contrasts make sense | This is a good question.
In your situation, it's common and perfectly ok to just run two t tests for the contrasts of interest, and do the Boneferroni correction manually by multiplying the p values b | Post hoc contrasts when only certain contrasts make sense
This is a good question.
In your situation, it's common and perfectly ok to just run two t tests for the contrasts of interest, and do the Boneferroni correction manually by multiplying the p values by 2. With only two contrasts, there will be very little differ... | Post hoc contrasts when only certain contrasts make sense
This is a good question.
In your situation, it's common and perfectly ok to just run two t tests for the contrasts of interest, and do the Boneferroni correction manually by multiplying the p values b |
53,508 | heavier tails means that it is less sensitive to outlying data for logistic and probit | The statement is not about generating extreme values from either a logistic or a normal, it's about trying to fit a logistic or a Normal to pre-existing data that may have extreme values.
You have data, and you are trying to fit a model to it. However, a data point is not an outlier if it is "expected", in some sense, ... | heavier tails means that it is less sensitive to outlying data for logistic and probit | The statement is not about generating extreme values from either a logistic or a normal, it's about trying to fit a logistic or a Normal to pre-existing data that may have extreme values.
You have dat | heavier tails means that it is less sensitive to outlying data for logistic and probit
The statement is not about generating extreme values from either a logistic or a normal, it's about trying to fit a logistic or a Normal to pre-existing data that may have extreme values.
You have data, and you are trying to fit a mo... | heavier tails means that it is less sensitive to outlying data for logistic and probit
The statement is not about generating extreme values from either a logistic or a normal, it's about trying to fit a logistic or a Normal to pre-existing data that may have extreme values.
You have dat |
53,509 | Acceptance-Rejection Technique Theorem Proof | The probability a proposal is accepted is the sum over $j$'s that the value $j$ is (1) generated and then (2) accepted:
\begin{align}\mathbb P(\text{proposal accepted})&=\sum_{j=1}^\infty \mathbb P(\text{proposal accepted and }Y=j)\\
&=\sum_{j=1}^\infty \mathbb P(\text{proposal accepted }|Y=j)\mathbb P(Y=j)\\
&=\sum_{j... | Acceptance-Rejection Technique Theorem Proof | The probability a proposal is accepted is the sum over $j$'s that the value $j$ is (1) generated and then (2) accepted:
\begin{align}\mathbb P(\text{proposal accepted})&=\sum_{j=1}^\infty \mathbb P(\t | Acceptance-Rejection Technique Theorem Proof
The probability a proposal is accepted is the sum over $j$'s that the value $j$ is (1) generated and then (2) accepted:
\begin{align}\mathbb P(\text{proposal accepted})&=\sum_{j=1}^\infty \mathbb P(\text{proposal accepted and }Y=j)\\
&=\sum_{j=1}^\infty \mathbb P(\text{propo... | Acceptance-Rejection Technique Theorem Proof
The probability a proposal is accepted is the sum over $j$'s that the value $j$ is (1) generated and then (2) accepted:
\begin{align}\mathbb P(\text{proposal accepted})&=\sum_{j=1}^\infty \mathbb P(\t |
53,510 | Meaning of "$\stackrel{p}\longrightarrow$" in math notation (arrow with a p over it) | It means convergence in probability. In your case, it's about random processes rather than random variables. It says that the series of random processes will converge towards a single random process. | Meaning of "$\stackrel{p}\longrightarrow$" in math notation (arrow with a p over it) | It means convergence in probability. In your case, it's about random processes rather than random variables. It says that the series of random processes will converge towards a single random process. | Meaning of "$\stackrel{p}\longrightarrow$" in math notation (arrow with a p over it)
It means convergence in probability. In your case, it's about random processes rather than random variables. It says that the series of random processes will converge towards a single random process. | Meaning of "$\stackrel{p}\longrightarrow$" in math notation (arrow with a p over it)
It means convergence in probability. In your case, it's about random processes rather than random variables. It says that the series of random processes will converge towards a single random process. |
53,511 | Why use supervised binning on train data if it leaks data? | As already noticed in the comments and another answer, you need to train the binning algorithm using training data only, in such a case it has no chance to leak the test data, as it hasn't seen it.
But you seem to be concerned with the fact that the binning algorithm uses the labels, so it "leaks" the labels to the fea... | Why use supervised binning on train data if it leaks data? | As already noticed in the comments and another answer, you need to train the binning algorithm using training data only, in such a case it has no chance to leak the test data, as it hasn't seen it.
Bu | Why use supervised binning on train data if it leaks data?
As already noticed in the comments and another answer, you need to train the binning algorithm using training data only, in such a case it has no chance to leak the test data, as it hasn't seen it.
But you seem to be concerned with the fact that the binning alg... | Why use supervised binning on train data if it leaks data?
As already noticed in the comments and another answer, you need to train the binning algorithm using training data only, in such a case it has no chance to leak the test data, as it hasn't seen it.
Bu |
53,512 | Why use supervised binning on train data if it leaks data? | If you only use training data for supervised binning, you cannot leak information from the test dataset, simply because you are not using it. So, no, when done right, there is no leakage. | Why use supervised binning on train data if it leaks data? | If you only use training data for supervised binning, you cannot leak information from the test dataset, simply because you are not using it. So, no, when done right, there is no leakage. | Why use supervised binning on train data if it leaks data?
If you only use training data for supervised binning, you cannot leak information from the test dataset, simply because you are not using it. So, no, when done right, there is no leakage. | Why use supervised binning on train data if it leaks data?
If you only use training data for supervised binning, you cannot leak information from the test dataset, simply because you are not using it. So, no, when done right, there is no leakage. |
53,513 | If my data doesn't completely follow the Zipf's law, how do I justify it mathematically? | The literature on the mathematical theory underlying Zipf's law is quite vast, and includes a large number of underlying theoretical models in which the law emerges. Zipf's law is related to power laws through the fact that it asserts a power-law relationship for the rank versus frequency of the objects under analysis... | If my data doesn't completely follow the Zipf's law, how do I justify it mathematically? | The literature on the mathematical theory underlying Zipf's law is quite vast, and includes a large number of underlying theoretical models in which the law emerges. Zipf's law is related to power la | If my data doesn't completely follow the Zipf's law, how do I justify it mathematically?
The literature on the mathematical theory underlying Zipf's law is quite vast, and includes a large number of underlying theoretical models in which the law emerges. Zipf's law is related to power laws through the fact that it ass... | If my data doesn't completely follow the Zipf's law, how do I justify it mathematically?
The literature on the mathematical theory underlying Zipf's law is quite vast, and includes a large number of underlying theoretical models in which the law emerges. Zipf's law is related to power la |
53,514 | Does autocorrelation imply temporal dependence? | Removing non-stationarity just makes statistical structure of of your time series independent of absolute time-steps. It will typically reduce the auto-correlation but will not remove them. In some cases, it might happen that the time series consists of a deterministic seasonal component with some white-noise superimpo... | Does autocorrelation imply temporal dependence? | Removing non-stationarity just makes statistical structure of of your time series independent of absolute time-steps. It will typically reduce the auto-correlation but will not remove them. In some ca | Does autocorrelation imply temporal dependence?
Removing non-stationarity just makes statistical structure of of your time series independent of absolute time-steps. It will typically reduce the auto-correlation but will not remove them. In some cases, it might happen that the time series consists of a deterministic se... | Does autocorrelation imply temporal dependence?
Removing non-stationarity just makes statistical structure of of your time series independent of absolute time-steps. It will typically reduce the auto-correlation but will not remove them. In some ca |
53,515 | Does autocorrelation imply temporal dependence? | Correlation does not imply causation, neither the other way around, nor when it regards time. The same applies to autocorrelation. Correlation(s) measure particular kinds of relationships between variables, while there may be many other non-linear relationships that are possible, so correlation and causation or depende... | Does autocorrelation imply temporal dependence? | Correlation does not imply causation, neither the other way around, nor when it regards time. The same applies to autocorrelation. Correlation(s) measure particular kinds of relationships between vari | Does autocorrelation imply temporal dependence?
Correlation does not imply causation, neither the other way around, nor when it regards time. The same applies to autocorrelation. Correlation(s) measure particular kinds of relationships between variables, while there may be many other non-linear relationships that are p... | Does autocorrelation imply temporal dependence?
Correlation does not imply causation, neither the other way around, nor when it regards time. The same applies to autocorrelation. Correlation(s) measure particular kinds of relationships between vari |
53,516 | Does autocorrelation imply temporal dependence? | No, a stationary TS can still have a ACF showing a temporal dependency. Autocorrelation is the dependency of on point on the previous ones. This temporal dependency can be a drift or an oscillation and those parts will indeed be removed by making it stationary. But you can still have a dependency on the previous point,... | Does autocorrelation imply temporal dependence? | No, a stationary TS can still have a ACF showing a temporal dependency. Autocorrelation is the dependency of on point on the previous ones. This temporal dependency can be a drift or an oscillation an | Does autocorrelation imply temporal dependence?
No, a stationary TS can still have a ACF showing a temporal dependency. Autocorrelation is the dependency of on point on the previous ones. This temporal dependency can be a drift or an oscillation and those parts will indeed be removed by making it stationary. But you ca... | Does autocorrelation imply temporal dependence?
No, a stationary TS can still have a ACF showing a temporal dependency. Autocorrelation is the dependency of on point on the previous ones. This temporal dependency can be a drift or an oscillation an |
53,517 | Do you need large amounts of data to estimate parameters in extreme value distributions? | It's good to have more data , always :) However, consider why we have EVT: to work with less data! Why would you need EVT if you could collect infinite amount of data? You'd simply fit the underlying distribution and calculate any metrics on it. Because only a fraction of data goes to tails, we'd need to collect enormo... | Do you need large amounts of data to estimate parameters in extreme value distributions? | It's good to have more data , always :) However, consider why we have EVT: to work with less data! Why would you need EVT if you could collect infinite amount of data? You'd simply fit the underlying | Do you need large amounts of data to estimate parameters in extreme value distributions?
It's good to have more data , always :) However, consider why we have EVT: to work with less data! Why would you need EVT if you could collect infinite amount of data? You'd simply fit the underlying distribution and calculate any ... | Do you need large amounts of data to estimate parameters in extreme value distributions?
It's good to have more data , always :) However, consider why we have EVT: to work with less data! Why would you need EVT if you could collect infinite amount of data? You'd simply fit the underlying |
53,518 | Do you need large amounts of data to estimate parameters in extreme value distributions? | The Fisher information matrix tells you how much information there is in each observed value about your parameters. If your observations are independent, then the information in $n$ samples is $n$ times the Fisher information matrix. The inverse of the Fisher information matrix is a lower bound on the covariance of you... | Do you need large amounts of data to estimate parameters in extreme value distributions? | The Fisher information matrix tells you how much information there is in each observed value about your parameters. If your observations are independent, then the information in $n$ samples is $n$ tim | Do you need large amounts of data to estimate parameters in extreme value distributions?
The Fisher information matrix tells you how much information there is in each observed value about your parameters. If your observations are independent, then the information in $n$ samples is $n$ times the Fisher information matri... | Do you need large amounts of data to estimate parameters in extreme value distributions?
The Fisher information matrix tells you how much information there is in each observed value about your parameters. If your observations are independent, then the information in $n$ samples is $n$ tim |
53,519 | Trying to figure out which statistical method to use | First, I hope that this score is not the only criterion used for ranking residency applicants. When I was on a residency admissions committee, scores from interviews with faculty were only one part of the process. We would then meet together to go down the list carefully, reviewing all the candidates and frequently reo... | Trying to figure out which statistical method to use | First, I hope that this score is not the only criterion used for ranking residency applicants. When I was on a residency admissions committee, scores from interviews with faculty were only one part of | Trying to figure out which statistical method to use
First, I hope that this score is not the only criterion used for ranking residency applicants. When I was on a residency admissions committee, scores from interviews with faculty were only one part of the process. We would then meet together to go down the list caref... | Trying to figure out which statistical method to use
First, I hope that this score is not the only criterion used for ranking residency applicants. When I was on a residency admissions committee, scores from interviews with faculty were only one part of |
53,520 | GLM negative binomial - what to do when one category has only zeros? | Adding to the other answers with some experimental calculations. The large standard error for managementD is caused by small sample size. The standard error you've got is based on an approximation, based on the loglikelihood function being approximately quadratic, which it is not. We can try to get a confidence interva... | GLM negative binomial - what to do when one category has only zeros? | Adding to the other answers with some experimental calculations. The large standard error for managementD is caused by small sample size. The standard error you've got is based on an approximation, ba | GLM negative binomial - what to do when one category has only zeros?
Adding to the other answers with some experimental calculations. The large standard error for managementD is caused by small sample size. The standard error you've got is based on an approximation, based on the loglikelihood function being approximate... | GLM negative binomial - what to do when one category has only zeros?
Adding to the other answers with some experimental calculations. The large standard error for managementD is caused by small sample size. The standard error you've got is based on an approximation, ba |
53,521 | GLM negative binomial - what to do when one category has only zeros? | I dissent somewhat from the first answer by @dariober.
Adding 1 is a fudge.
There is no substantive reason for disbelieving zeros as recorded in the sample.
Most important, model fits are reasonable, the only oddity being the rather wide confidence intervals in one case. There is some robustness, as Poisson and neg... | GLM negative binomial - what to do when one category has only zeros? | I dissent somewhat from the first answer by @dariober.
Adding 1 is a fudge.
There is no substantive reason for disbelieving zeros as recorded in the sample.
Most important, model fits are reasonabl | GLM negative binomial - what to do when one category has only zeros?
I dissent somewhat from the first answer by @dariober.
Adding 1 is a fudge.
There is no substantive reason for disbelieving zeros as recorded in the sample.
Most important, model fits are reasonable, the only oddity being the rather wide confidence... | GLM negative binomial - what to do when one category has only zeros?
I dissent somewhat from the first answer by @dariober.
Adding 1 is a fudge.
There is no substantive reason for disbelieving zeros as recorded in the sample.
Most important, model fits are reasonabl |
53,522 | GLM negative binomial - what to do when one category has only zeros? | As requested by the OP in comments, I'm going to give an example in R of applying likelihood ratio test (LRT) to test differences between management groups as suggested by @GordonSmyth. I'm not sure I'm getting this right so please check it - credit goes to Gordon, faults are mine.
With LRT we check for significant di... | GLM negative binomial - what to do when one category has only zeros? | As requested by the OP in comments, I'm going to give an example in R of applying likelihood ratio test (LRT) to test differences between management groups as suggested by @GordonSmyth. I'm not sure | GLM negative binomial - what to do when one category has only zeros?
As requested by the OP in comments, I'm going to give an example in R of applying likelihood ratio test (LRT) to test differences between management groups as suggested by @GordonSmyth. I'm not sure I'm getting this right so please check it - credit ... | GLM negative binomial - what to do when one category has only zeros?
As requested by the OP in comments, I'm going to give an example in R of applying likelihood ratio test (LRT) to test differences between management groups as suggested by @GordonSmyth. I'm not sure |
53,523 | GLM negative binomial - what to do when one category has only zeros? | For an explanation of why this is happening see GLM for count data with all zeroes in one category.
I know I could do a bayesian approach but before I go into that, I would like to know if there's a work-around that allows me to do this analysis using a frequentist
I think you could just add 1 to all observations and... | GLM negative binomial - what to do when one category has only zeros? | For an explanation of why this is happening see GLM for count data with all zeroes in one category.
I know I could do a bayesian approach but before I go into that, I would like to know if there's a | GLM negative binomial - what to do when one category has only zeros?
For an explanation of why this is happening see GLM for count data with all zeroes in one category.
I know I could do a bayesian approach but before I go into that, I would like to know if there's a work-around that allows me to do this analysis usin... | GLM negative binomial - what to do when one category has only zeros?
For an explanation of why this is happening see GLM for count data with all zeroes in one category.
I know I could do a bayesian approach but before I go into that, I would like to know if there's a |
53,524 | Can probability distributions be used as an alternative for regression models? | You can, but not without consequences.
Linear regression is a pretty flexible model in terms of your ability to define the functional relationship between the features and the dependent variable. If you use multivariate distribution, you are limited by what kind of relationships between variables are possible under th... | Can probability distributions be used as an alternative for regression models? | You can, but not without consequences.
Linear regression is a pretty flexible model in terms of your ability to define the functional relationship between the features and the dependent variable. If | Can probability distributions be used as an alternative for regression models?
You can, but not without consequences.
Linear regression is a pretty flexible model in terms of your ability to define the functional relationship between the features and the dependent variable. If you use multivariate distribution, you ar... | Can probability distributions be used as an alternative for regression models?
You can, but not without consequences.
Linear regression is a pretty flexible model in terms of your ability to define the functional relationship between the features and the dependent variable. If |
53,525 | Can probability distributions be used as an alternative for regression models? | I am fairly sure it's valid to treat regression model as a joint probability distribution. Let $s,h,w$ be salary, height, weight, then a linear regression
$$
s = \beta_0 + \beta_1h + \beta_2w + \epsilon \\
\epsilon \sim N(0, \sigma^2)
$$
where $\beta_0, \beta_1, \beta_2$ are regression coefficients posits the distr... | Can probability distributions be used as an alternative for regression models? | I am fairly sure it's valid to treat regression model as a joint probability distribution. Let $s,h,w$ be salary, height, weight, then a linear regression
$$
s = \beta_0 + \beta_1h + \beta_2w + \eps | Can probability distributions be used as an alternative for regression models?
I am fairly sure it's valid to treat regression model as a joint probability distribution. Let $s,h,w$ be salary, height, weight, then a linear regression
$$
s = \beta_0 + \beta_1h + \beta_2w + \epsilon \\
\epsilon \sim N(0, \sigma^2)
$$... | Can probability distributions be used as an alternative for regression models?
I am fairly sure it's valid to treat regression model as a joint probability distribution. Let $s,h,w$ be salary, height, weight, then a linear regression
$$
s = \beta_0 + \beta_1h + \beta_2w + \eps |
53,526 | Can probability distributions be used as an alternative for regression models? | Yes, you definitely can. Moreover, it does not have to be any linear kind of regression. In the most general case, the following method will allow you to effectively turn any parametric or even non-parametric probability density estimation into a regressor for the desired output (i.e., the 3rd variable in your case, wh... | Can probability distributions be used as an alternative for regression models? | Yes, you definitely can. Moreover, it does not have to be any linear kind of regression. In the most general case, the following method will allow you to effectively turn any parametric or even non-pa | Can probability distributions be used as an alternative for regression models?
Yes, you definitely can. Moreover, it does not have to be any linear kind of regression. In the most general case, the following method will allow you to effectively turn any parametric or even non-parametric probability density estimation i... | Can probability distributions be used as an alternative for regression models?
Yes, you definitely can. Moreover, it does not have to be any linear kind of regression. In the most general case, the following method will allow you to effectively turn any parametric or even non-pa |
53,527 | Delta method for Poisson ratio | Use a vector version of the delta method. You have convergence of
$$\sqrt{n}(\bar X-\lambda,\, \bar Y-\theta)$$ to a bivariate Normal, and
the function
$$f(\bar X, \bar Y)=\frac{\bar X}{\bar X+\bar Y}$$
is differentiable (away from $\lambda=\theta=0$), so the delta method applies.
That isn't how I'd actually work out ... | Delta method for Poisson ratio | Use a vector version of the delta method. You have convergence of
$$\sqrt{n}(\bar X-\lambda,\, \bar Y-\theta)$$ to a bivariate Normal, and
the function
$$f(\bar X, \bar Y)=\frac{\bar X}{\bar X+\bar Y | Delta method for Poisson ratio
Use a vector version of the delta method. You have convergence of
$$\sqrt{n}(\bar X-\lambda,\, \bar Y-\theta)$$ to a bivariate Normal, and
the function
$$f(\bar X, \bar Y)=\frac{\bar X}{\bar X+\bar Y}$$
is differentiable (away from $\lambda=\theta=0$), so the delta method applies.
That i... | Delta method for Poisson ratio
Use a vector version of the delta method. You have convergence of
$$\sqrt{n}(\bar X-\lambda,\, \bar Y-\theta)$$ to a bivariate Normal, and
the function
$$f(\bar X, \bar Y)=\frac{\bar X}{\bar X+\bar Y |
53,528 | Delta method for Poisson ratio | This is just a visual comment on Thomas Lumley's answer (+1), illustrating it by simulation (blue) against his approximating normal distribution (red) using R
set.seed(2021)
lambda <- 2
theta <- 5
n <- 1000
cases <- 10^5
Xbar <- rpois(cases, n * lambda) / n
Ybar <- rpois(cases, n * theta ) / n
ratio <- Xbar... | Delta method for Poisson ratio | This is just a visual comment on Thomas Lumley's answer (+1), illustrating it by simulation (blue) against his approximating normal distribution (red) using R
set.seed(2021)
lambda <- 2
theta <- 5
n | Delta method for Poisson ratio
This is just a visual comment on Thomas Lumley's answer (+1), illustrating it by simulation (blue) against his approximating normal distribution (red) using R
set.seed(2021)
lambda <- 2
theta <- 5
n <- 1000
cases <- 10^5
Xbar <- rpois(cases, n * lambda) / n
Ybar <- rpois(cases,... | Delta method for Poisson ratio
This is just a visual comment on Thomas Lumley's answer (+1), illustrating it by simulation (blue) against his approximating normal distribution (red) using R
set.seed(2021)
lambda <- 2
theta <- 5
n |
53,529 | Central Limit Theorem with Bounded Sum of Variances? | Heuristically, if the sum of the variances isn't infinite, there is some residual shape information in the sum about some of the individual variables. For example, if
$$\mathrm{var}[X_1]=\epsilon\sum_i \mathrm{var}[X_i]$$
then $X_1$ makes up $\epsilon>0$ of the limiting random variable and the shape of $X_i$ (tails, mo... | Central Limit Theorem with Bounded Sum of Variances? | Heuristically, if the sum of the variances isn't infinite, there is some residual shape information in the sum about some of the individual variables. For example, if
$$\mathrm{var}[X_1]=\epsilon\sum_ | Central Limit Theorem with Bounded Sum of Variances?
Heuristically, if the sum of the variances isn't infinite, there is some residual shape information in the sum about some of the individual variables. For example, if
$$\mathrm{var}[X_1]=\epsilon\sum_i \mathrm{var}[X_i]$$
then $X_1$ makes up $\epsilon>0$ of the limit... | Central Limit Theorem with Bounded Sum of Variances?
Heuristically, if the sum of the variances isn't infinite, there is some residual shape information in the sum about some of the individual variables. For example, if
$$\mathrm{var}[X_1]=\epsilon\sum_ |
53,530 | Definition of a one-sided test | $H_0: \theta = \theta_0$ versus $H_1: \theta \gt \theta_0$ is OK, but some authors might write
$H_0: \theta \le \theta_0$ versus $H_1: \theta \gt \theta_0.$
$H_0: \theta = \theta_0$ versus $H_1: \theta \lt \theta_0$ is OK, but
some authors might write
$H_0: \theta \ge \theta_0$ versus $H_1: \theta \lt \theta_0.$
In eac... | Definition of a one-sided test | $H_0: \theta = \theta_0$ versus $H_1: \theta \gt \theta_0$ is OK, but some authors might write
$H_0: \theta \le \theta_0$ versus $H_1: \theta \gt \theta_0.$
$H_0: \theta = \theta_0$ versus $H_1: \thet | Definition of a one-sided test
$H_0: \theta = \theta_0$ versus $H_1: \theta \gt \theta_0$ is OK, but some authors might write
$H_0: \theta \le \theta_0$ versus $H_1: \theta \gt \theta_0.$
$H_0: \theta = \theta_0$ versus $H_1: \theta \lt \theta_0$ is OK, but
some authors might write
$H_0: \theta \ge \theta_0$ versus $H_... | Definition of a one-sided test
$H_0: \theta = \theta_0$ versus $H_1: \theta \gt \theta_0$ is OK, but some authors might write
$H_0: \theta \le \theta_0$ versus $H_1: \theta \gt \theta_0.$
$H_0: \theta = \theta_0$ versus $H_1: \thet |
53,531 | Definition of a one-sided test | It would be more appropriate to write them this way.
$H_0: \theta = \theta_0~~$ versus $~~H_1: \theta \ne \theta_0$ (two-sided)
$H_0: \theta \le \theta_0~~$ versus $~~H_1: \theta > \theta_0$ (one-sided [upper-tailed])
$H_0: \theta \ge \theta_0~~$ versus $~~H_1: \theta < \theta_0$ (one-sided [lower-tailed]) | Definition of a one-sided test | It would be more appropriate to write them this way.
$H_0: \theta = \theta_0~~$ versus $~~H_1: \theta \ne \theta_0$ (two-sided)
$H_0: \theta \le \theta_0~~$ versus $~~H_1: \theta > \theta_0$ (one-side | Definition of a one-sided test
It would be more appropriate to write them this way.
$H_0: \theta = \theta_0~~$ versus $~~H_1: \theta \ne \theta_0$ (two-sided)
$H_0: \theta \le \theta_0~~$ versus $~~H_1: \theta > \theta_0$ (one-sided [upper-tailed])
$H_0: \theta \ge \theta_0~~$ versus $~~H_1: \theta < \theta_0$ (one-sid... | Definition of a one-sided test
It would be more appropriate to write them this way.
$H_0: \theta = \theta_0~~$ versus $~~H_1: \theta \ne \theta_0$ (two-sided)
$H_0: \theta \le \theta_0~~$ versus $~~H_1: \theta > \theta_0$ (one-side |
53,532 | Test to determine whether coin is fair or not | Ten tosses of a coin. Test $H_0: p = 1/2$ against $H_0: p \ne 1/2.$ Comment at the start: there is not a lot of information in only ten tosses of a coin, so in order to reject $H_0$ we will have to observe very few heads (0 or 1) or very many (9 or 10).
Normal approximation: Under $H_0,$ Number $X$ of heads see in $n ... | Test to determine whether coin is fair or not | Ten tosses of a coin. Test $H_0: p = 1/2$ against $H_0: p \ne 1/2.$ Comment at the start: there is not a lot of information in only ten tosses of a coin, so in order to reject $H_0$ we will have to o | Test to determine whether coin is fair or not
Ten tosses of a coin. Test $H_0: p = 1/2$ against $H_0: p \ne 1/2.$ Comment at the start: there is not a lot of information in only ten tosses of a coin, so in order to reject $H_0$ we will have to observe very few heads (0 or 1) or very many (9 or 10).
Normal approximatio... | Test to determine whether coin is fair or not
Ten tosses of a coin. Test $H_0: p = 1/2$ against $H_0: p \ne 1/2.$ Comment at the start: there is not a lot of information in only ten tosses of a coin, so in order to reject $H_0$ we will have to o |
53,533 | Test to determine whether coin is fair or not | Another way to extract probabilities is to simulate :
Imagine a fair coin.
Toss it 10 times and write down the number of heads.
According to the central limit theorem, the number of heads in 10 tosses will follow a Normal-like distribution with mean
Then do it all over again. And again. And again. N times (maybe N = 10... | Test to determine whether coin is fair or not | Another way to extract probabilities is to simulate :
Imagine a fair coin.
Toss it 10 times and write down the number of heads.
According to the central limit theorem, the number of heads in 10 tosses | Test to determine whether coin is fair or not
Another way to extract probabilities is to simulate :
Imagine a fair coin.
Toss it 10 times and write down the number of heads.
According to the central limit theorem, the number of heads in 10 tosses will follow a Normal-like distribution with mean
Then do it all over agai... | Test to determine whether coin is fair or not
Another way to extract probabilities is to simulate :
Imagine a fair coin.
Toss it 10 times and write down the number of heads.
According to the central limit theorem, the number of heads in 10 tosses |
53,534 | Test to determine whether coin is fair or not | The formula to calculate the approximate confidence limits for a binomial test is:
$z_{alpha/2}*\sqrt{p*q/n}$
In your case for a fair coin p = q = 0.5 and using $z_{alpha/2}=1.96$ for a 95% confidence limit.
The range of heads for 10 flips is expected to be between
$ 10*(0.5 \pm 1.96*\sqrt{0.025})$ or 1.9 to 8.1 heads
... | Test to determine whether coin is fair or not | The formula to calculate the approximate confidence limits for a binomial test is:
$z_{alpha/2}*\sqrt{p*q/n}$
In your case for a fair coin p = q = 0.5 and using $z_{alpha/2}=1.96$ for a 95% confidence | Test to determine whether coin is fair or not
The formula to calculate the approximate confidence limits for a binomial test is:
$z_{alpha/2}*\sqrt{p*q/n}$
In your case for a fair coin p = q = 0.5 and using $z_{alpha/2}=1.96$ for a 95% confidence limit.
The range of heads for 10 flips is expected to be between
$ 10*(0.... | Test to determine whether coin is fair or not
The formula to calculate the approximate confidence limits for a binomial test is:
$z_{alpha/2}*\sqrt{p*q/n}$
In your case for a fair coin p = q = 0.5 and using $z_{alpha/2}=1.96$ for a 95% confidence |
53,535 | Test to determine whether coin is fair or not | The Red Bead experiment and Deming is where I start.
An unfair coin is a special or assignable cause of variation. When should we look for a special cause? When we see something beyond 3 standard deviations.
10 flips might not be enough.
.5 +/- 3 times sqrt of (.5*.5/N), where N is number of flips.
More than ~ 9.743 H ... | Test to determine whether coin is fair or not | The Red Bead experiment and Deming is where I start.
An unfair coin is a special or assignable cause of variation. When should we look for a special cause? When we see something beyond 3 standard devi | Test to determine whether coin is fair or not
The Red Bead experiment and Deming is where I start.
An unfair coin is a special or assignable cause of variation. When should we look for a special cause? When we see something beyond 3 standard deviations.
10 flips might not be enough.
.5 +/- 3 times sqrt of (.5*.5/N), wh... | Test to determine whether coin is fair or not
The Red Bead experiment and Deming is where I start.
An unfair coin is a special or assignable cause of variation. When should we look for a special cause? When we see something beyond 3 standard devi |
53,536 | Iteratively Reweighted Least Squares - Weights Confusion | The IWLS algorithm for generalised linear models is different from that for a heteroscedastic linear model because it accounts for two things:
the non-linear link function
the variance-mean relationship
The likelihood score equations look like
$$\frac{d\mu}{d\beta}\frac{1}{V(\mu)}(Y-\mu)=0$$
so the variance is in the... | Iteratively Reweighted Least Squares - Weights Confusion | The IWLS algorithm for generalised linear models is different from that for a heteroscedastic linear model because it accounts for two things:
the non-linear link function
the variance-mean relations | Iteratively Reweighted Least Squares - Weights Confusion
The IWLS algorithm for generalised linear models is different from that for a heteroscedastic linear model because it accounts for two things:
the non-linear link function
the variance-mean relationship
The likelihood score equations look like
$$\frac{d\mu}{d\b... | Iteratively Reweighted Least Squares - Weights Confusion
The IWLS algorithm for generalised linear models is different from that for a heteroscedastic linear model because it accounts for two things:
the non-linear link function
the variance-mean relations |
53,537 | Does it make sense to do PCA before a Tree-Boosting model? | In general, good features will improve the performance of any model, and should require fewer steps / result in faster convergence. One nice example of this is whether you want to use the distance from the hole for modeling the golf putting probability of success, or whether you design a new feature based on the geomet... | Does it make sense to do PCA before a Tree-Boosting model? | In general, good features will improve the performance of any model, and should require fewer steps / result in faster convergence. One nice example of this is whether you want to use the distance fro | Does it make sense to do PCA before a Tree-Boosting model?
In general, good features will improve the performance of any model, and should require fewer steps / result in faster convergence. One nice example of this is whether you want to use the distance from the hole for modeling the golf putting probability of succe... | Does it make sense to do PCA before a Tree-Boosting model?
In general, good features will improve the performance of any model, and should require fewer steps / result in faster convergence. One nice example of this is whether you want to use the distance fro |
53,538 | Is multivariate normal the only distribution with this property? | No, the bivariate normal is not the only distribution with the property that $E[X\mid Y=y]$ is a linear function of $y$ and also that $E[Y\mid X=x]$ is a linear function of $x$; many other distributions enjoy the same property.
For example, suppose that $(X,Y)$ is uniformly distributed on the triangle with vertices $(0... | Is multivariate normal the only distribution with this property? | No, the bivariate normal is not the only distribution with the property that $E[X\mid Y=y]$ is a linear function of $y$ and also that $E[Y\mid X=x]$ is a linear function of $x$; many other distributio | Is multivariate normal the only distribution with this property?
No, the bivariate normal is not the only distribution with the property that $E[X\mid Y=y]$ is a linear function of $y$ and also that $E[Y\mid X=x]$ is a linear function of $x$; many other distributions enjoy the same property.
For example, suppose that $... | Is multivariate normal the only distribution with this property?
No, the bivariate normal is not the only distribution with the property that $E[X\mid Y=y]$ is a linear function of $y$ and also that $E[Y\mid X=x]$ is a linear function of $x$; many other distributio |
53,539 | Is multivariate normal the only distribution with this property? | No - it is not just a property of bivariate normals. For example
Let $A,B,C$ be i.i.d. with finite mean $\mu$. Then let $X=A+B$ and $Y=A+C$.
$E[A \mid X=x] =E[B \mid X=x] = \frac12 E[A+B \mid X=x]=\frac12 E[X \mid X=x]= \frac 12x$.
So $E[Y \mid X=x]=E[A \mid X=x] +E[C \mid X=x] = \frac 12x+\mu$ which is linear in $x... | Is multivariate normal the only distribution with this property? | No - it is not just a property of bivariate normals. For example
Let $A,B,C$ be i.i.d. with finite mean $\mu$. Then let $X=A+B$ and $Y=A+C$.
$E[A \mid X=x] =E[B \mid X=x] = \frac12 E[A+B \mid X=x]=\ | Is multivariate normal the only distribution with this property?
No - it is not just a property of bivariate normals. For example
Let $A,B,C$ be i.i.d. with finite mean $\mu$. Then let $X=A+B$ and $Y=A+C$.
$E[A \mid X=x] =E[B \mid X=x] = \frac12 E[A+B \mid X=x]=\frac12 E[X \mid X=x]= \frac 12x$.
So $E[Y \mid X=x]=E[... | Is multivariate normal the only distribution with this property?
No - it is not just a property of bivariate normals. For example
Let $A,B,C$ be i.i.d. with finite mean $\mu$. Then let $X=A+B$ and $Y=A+C$.
$E[A \mid X=x] =E[B \mid X=x] = \frac12 E[A+B \mid X=x]=\ |
53,540 | Calculate single absolute standardized difference across levels of a categorical treatment variable cobalt::bal.tab | Author of cobalt here. What the reviewer is requesting doesn't really make a lot of sense. The bias in an effect estimate is a function of the mean difference of each level of the categorical variable. You could create a one-dimensional summary of balance for that categorical variable, e.g., as the maximum SMD for that... | Calculate single absolute standardized difference across levels of a categorical treatment variable | Author of cobalt here. What the reviewer is requesting doesn't really make a lot of sense. The bias in an effect estimate is a function of the mean difference of each level of the categorical variable | Calculate single absolute standardized difference across levels of a categorical treatment variable cobalt::bal.tab
Author of cobalt here. What the reviewer is requesting doesn't really make a lot of sense. The bias in an effect estimate is a function of the mean difference of each level of the categorical variable. Yo... | Calculate single absolute standardized difference across levels of a categorical treatment variable
Author of cobalt here. What the reviewer is requesting doesn't really make a lot of sense. The bias in an effect estimate is a function of the mean difference of each level of the categorical variable |
53,541 | Loss function in for gamma objective function in regression in XGBoost? | I'm not actually sure whether "gamma regression" is officially defined (it doesn't appear to have a wikipedia page for example), but if I were to define it (and some googling around suggests I'm not alone here), I would define it as setting up my regression problem so that for a given input vector $\underline{x}$, I pr... | Loss function in for gamma objective function in regression in XGBoost? | I'm not actually sure whether "gamma regression" is officially defined (it doesn't appear to have a wikipedia page for example), but if I were to define it (and some googling around suggests I'm not a | Loss function in for gamma objective function in regression in XGBoost?
I'm not actually sure whether "gamma regression" is officially defined (it doesn't appear to have a wikipedia page for example), but if I were to define it (and some googling around suggests I'm not alone here), I would define it as setting up my r... | Loss function in for gamma objective function in regression in XGBoost?
I'm not actually sure whether "gamma regression" is officially defined (it doesn't appear to have a wikipedia page for example), but if I were to define it (and some googling around suggests I'm not a |
53,542 | Loss function in for gamma objective function in regression in XGBoost? | Thanks everybody for the contributions! I am late to the party, but I wanted to add one more point regarding the log-link function, which was to me still unclear.
I take the formula for the Gamma distribution from the bottom of gazza89's answer:
$$
\frac{1}{\Gamma(k)(\frac{\mu}{k})^k}x^{k-1}e^{-\frac{xk}{\mu}}
$$
Using... | Loss function in for gamma objective function in regression in XGBoost? | Thanks everybody for the contributions! I am late to the party, but I wanted to add one more point regarding the log-link function, which was to me still unclear.
I take the formula for the Gamma dist | Loss function in for gamma objective function in regression in XGBoost?
Thanks everybody for the contributions! I am late to the party, but I wanted to add one more point regarding the log-link function, which was to me still unclear.
I take the formula for the Gamma distribution from the bottom of gazza89's answer:
$$... | Loss function in for gamma objective function in regression in XGBoost?
Thanks everybody for the contributions! I am late to the party, but I wanted to add one more point regarding the log-link function, which was to me still unclear.
I take the formula for the Gamma dist |
53,543 | Testing Hypothesis with different alternatives | Doing it right the first time is best.
First, in practice, this should be an unlikely situation.
Maybe you have re-engineered a pharmaceutical process hoping
that the new process has a higher yield than the current one
with $\mu_0 = 100,$ so you'd take data from runs of the new process,
average them and test $H_0: \mu... | Testing Hypothesis with different alternatives | Doing it right the first time is best.
First, in practice, this should be an unlikely situation.
Maybe you have re-engineered a pharmaceutical process hoping
that the new process has a higher yield t | Testing Hypothesis with different alternatives
Doing it right the first time is best.
First, in practice, this should be an unlikely situation.
Maybe you have re-engineered a pharmaceutical process hoping
that the new process has a higher yield than the current one
with $\mu_0 = 100,$ so you'd take data from runs of t... | Testing Hypothesis with different alternatives
Doing it right the first time is best.
First, in practice, this should be an unlikely situation.
Maybe you have re-engineered a pharmaceutical process hoping
that the new process has a higher yield t |
53,544 | Testing Hypothesis with different alternatives | Imagine testing $\mu=0$. You do your calculations and find that $\bar{x}=99$ and your z-statistic (or t-stat) is 123.
I would have serious doubts about hypothesis 1 and very much believe hypothesis 2. | Testing Hypothesis with different alternatives | Imagine testing $\mu=0$. You do your calculations and find that $\bar{x}=99$ and your z-statistic (or t-stat) is 123.
I would have serious doubts about hypothesis 1 and very much believe hypothesis 2. | Testing Hypothesis with different alternatives
Imagine testing $\mu=0$. You do your calculations and find that $\bar{x}=99$ and your z-statistic (or t-stat) is 123.
I would have serious doubts about hypothesis 1 and very much believe hypothesis 2. | Testing Hypothesis with different alternatives
Imagine testing $\mu=0$. You do your calculations and find that $\bar{x}=99$ and your z-statistic (or t-stat) is 123.
I would have serious doubts about hypothesis 1 and very much believe hypothesis 2. |
53,545 | Testing Hypothesis with different alternatives | The research question is king. The job of the hypothesis test is to answer the research question. The job of the data (and statistics) is to help you perform the hypothesis test.
I think you are somewhat confused about how to set up your hypotheses. Your hypotheses should be formed from your research question and befor... | Testing Hypothesis with different alternatives | The research question is king. The job of the hypothesis test is to answer the research question. The job of the data (and statistics) is to help you perform the hypothesis test.
I think you are somew | Testing Hypothesis with different alternatives
The research question is king. The job of the hypothesis test is to answer the research question. The job of the data (and statistics) is to help you perform the hypothesis test.
I think you are somewhat confused about how to set up your hypotheses. Your hypotheses should ... | Testing Hypothesis with different alternatives
The research question is king. The job of the hypothesis test is to answer the research question. The job of the data (and statistics) is to help you perform the hypothesis test.
I think you are somew |
53,546 | Testing Hypothesis with different alternatives | Yes, it is possible that test 1 fails to reject $H_0$ and testing 2 rejects $H_0$. (Consider for example a t-test with level 1% on $n = 9$ datapoints where $s^2 = 1$, $\overline{X} = \mu_0 + 3$ ).
In such a case, you have to... wonder what is the relevant alternative, and this depends on what you want to test.
Keep in ... | Testing Hypothesis with different alternatives | Yes, it is possible that test 1 fails to reject $H_0$ and testing 2 rejects $H_0$. (Consider for example a t-test with level 1% on $n = 9$ datapoints where $s^2 = 1$, $\overline{X} = \mu_0 + 3$ ).
In | Testing Hypothesis with different alternatives
Yes, it is possible that test 1 fails to reject $H_0$ and testing 2 rejects $H_0$. (Consider for example a t-test with level 1% on $n = 9$ datapoints where $s^2 = 1$, $\overline{X} = \mu_0 + 3$ ).
In such a case, you have to... wonder what is the relevant alternative, and ... | Testing Hypothesis with different alternatives
Yes, it is possible that test 1 fails to reject $H_0$ and testing 2 rejects $H_0$. (Consider for example a t-test with level 1% on $n = 9$ datapoints where $s^2 = 1$, $\overline{X} = \mu_0 + 3$ ).
In |
53,547 | How are artificially balanced datasets corrected for? | I have practical experience with training classifiers from imbalanced training sets. There are problems with this. Basically, the variances of the parameters associated with the less frequent classes - these variances grow large. The more uneven the prior distribution is in the training set, the more volatile your clas... | How are artificially balanced datasets corrected for? | I have practical experience with training classifiers from imbalanced training sets. There are problems with this. Basically, the variances of the parameters associated with the less frequent classes | How are artificially balanced datasets corrected for?
I have practical experience with training classifiers from imbalanced training sets. There are problems with this. Basically, the variances of the parameters associated with the less frequent classes - these variances grow large. The more uneven the prior distributi... | How are artificially balanced datasets corrected for?
I have practical experience with training classifiers from imbalanced training sets. There are problems with this. Basically, the variances of the parameters associated with the less frequent classes |
53,548 | How are artificially balanced datasets corrected for? | With fewer equations: Ideally, to make a decision, we need to know the probability that the input vector $x$ belongs to class $i$, using Bayes rule,
$p_t(C_i|x) = \frac{p_t(x|C_i)p_t(C_i)}{p_t(X)}$
where the $t$ subscript represents the conditions given in the training set. Now if the training set is representative o... | How are artificially balanced datasets corrected for? | With fewer equations: Ideally, to make a decision, we need to know the probability that the input vector $x$ belongs to class $i$, using Bayes rule,
$p_t(C_i|x) = \frac{p_t(x|C_i)p_t(C_i)}{p_t(X)}$
w | How are artificially balanced datasets corrected for?
With fewer equations: Ideally, to make a decision, we need to know the probability that the input vector $x$ belongs to class $i$, using Bayes rule,
$p_t(C_i|x) = \frac{p_t(x|C_i)p_t(C_i)}{p_t(X)}$
where the $t$ subscript represents the conditions given in the trai... | How are artificially balanced datasets corrected for?
With fewer equations: Ideally, to make a decision, we need to know the probability that the input vector $x$ belongs to class $i$, using Bayes rule,
$p_t(C_i|x) = \frac{p_t(x|C_i)p_t(C_i)}{p_t(X)}$
w |
53,549 | How are artificially balanced datasets corrected for? | The accepted answer from Match Maker EE seems right, but because I've had a hard time following the step from $P(class = j | \mathbf{x})$ to $P'(class=j|\mathbf{x})$, I've decided to write my own derivation. Further used notation is more Bishop like.
Firstly let's state that our new (balanced) dataset was created by ra... | How are artificially balanced datasets corrected for? | The accepted answer from Match Maker EE seems right, but because I've had a hard time following the step from $P(class = j | \mathbf{x})$ to $P'(class=j|\mathbf{x})$, I've decided to write my own deri | How are artificially balanced datasets corrected for?
The accepted answer from Match Maker EE seems right, but because I've had a hard time following the step from $P(class = j | \mathbf{x})$ to $P'(class=j|\mathbf{x})$, I've decided to write my own derivation. Further used notation is more Bishop like.
Firstly let's s... | How are artificially balanced datasets corrected for?
The accepted answer from Match Maker EE seems right, but because I've had a hard time following the step from $P(class = j | \mathbf{x})$ to $P'(class=j|\mathbf{x})$, I've decided to write my own deri |
53,550 | How are artificially balanced datasets corrected for? | I know this is a late reply and you probably do not need this answer, however I believe that I can add valuable information for future Pattern Recognition and Machine Learning by Christopher Bishop readers.
Answers from Match Maker EE and Dikran Marsupial provide good explanation on the logic behind the formula of the ... | How are artificially balanced datasets corrected for? | I know this is a late reply and you probably do not need this answer, however I believe that I can add valuable information for future Pattern Recognition and Machine Learning by Christopher Bishop re | How are artificially balanced datasets corrected for?
I know this is a late reply and you probably do not need this answer, however I believe that I can add valuable information for future Pattern Recognition and Machine Learning by Christopher Bishop readers.
Answers from Match Maker EE and Dikran Marsupial provide go... | How are artificially balanced datasets corrected for?
I know this is a late reply and you probably do not need this answer, however I believe that I can add valuable information for future Pattern Recognition and Machine Learning by Christopher Bishop re |
53,551 | How to multiply a likelihood by a prior? | Perhaps the multiplication of 'prior' by 'likelihood' to obtain 'posterior' will be clearer if we make a careful comparison of (a) a familiar elementary application of Bayes' Theorem
for a finite partition with (b) the use of a continuous version of Bayes' Theorem
for inference on a parameter.
Bayes' Theorem with a fin... | How to multiply a likelihood by a prior? | Perhaps the multiplication of 'prior' by 'likelihood' to obtain 'posterior' will be clearer if we make a careful comparison of (a) a familiar elementary application of Bayes' Theorem
for a finite part | How to multiply a likelihood by a prior?
Perhaps the multiplication of 'prior' by 'likelihood' to obtain 'posterior' will be clearer if we make a careful comparison of (a) a familiar elementary application of Bayes' Theorem
for a finite partition with (b) the use of a continuous version of Bayes' Theorem
for inference ... | How to multiply a likelihood by a prior?
Perhaps the multiplication of 'prior' by 'likelihood' to obtain 'posterior' will be clearer if we make a careful comparison of (a) a familiar elementary application of Bayes' Theorem
for a finite part |
53,552 | How to multiply a likelihood by a prior? | Bruce's answer is correct if—and only if—the prior and the likelihood contain no overlapping information. When that is true, Bayesian evidence combination is done by pointwise product of densities in the continuous case, the pointwise product of masses in the discrete case, etc. This is called product of experts by G... | How to multiply a likelihood by a prior? | Bruce's answer is correct if—and only if—the prior and the likelihood contain no overlapping information. When that is true, Bayesian evidence combination is done by pointwise product of densities in | How to multiply a likelihood by a prior?
Bruce's answer is correct if—and only if—the prior and the likelihood contain no overlapping information. When that is true, Bayesian evidence combination is done by pointwise product of densities in the continuous case, the pointwise product of masses in the discrete case, etc... | How to multiply a likelihood by a prior?
Bruce's answer is correct if—and only if—the prior and the likelihood contain no overlapping information. When that is true, Bayesian evidence combination is done by pointwise product of densities in |
53,553 | Green poop, leafy greens and probability of disease, how can I formalize this reasoning? | I use the following binary variables:
Poop is green: G
Am sick: D
Ate leafy greens: L
First, let's see how you can reach $P(D=1|G=1) = 0.8$. While you "knew" that you had eaten leafy greens and that it could cause green poop, when you thought about it first, you only considered a disease as a potential cause. That is... | Green poop, leafy greens and probability of disease, how can I formalize this reasoning? | I use the following binary variables:
Poop is green: G
Am sick: D
Ate leafy greens: L
First, let's see how you can reach $P(D=1|G=1) = 0.8$. While you "knew" that you had eaten leafy greens and that | Green poop, leafy greens and probability of disease, how can I formalize this reasoning?
I use the following binary variables:
Poop is green: G
Am sick: D
Ate leafy greens: L
First, let's see how you can reach $P(D=1|G=1) = 0.8$. While you "knew" that you had eaten leafy greens and that it could cause green poop, whe... | Green poop, leafy greens and probability of disease, how can I formalize this reasoning?
I use the following binary variables:
Poop is green: G
Am sick: D
Ate leafy greens: L
First, let's see how you can reach $P(D=1|G=1) = 0.8$. While you "knew" that you had eaten leafy greens and that |
53,554 | Green poop, leafy greens and probability of disease, how can I formalize this reasoning? | It seems to me you are looking at the Bayes's theorem and in particular at the prior probability.
Your data ($green\;poop, \; etc$) is the same before and after checking the internet. However, initially, your prior probability is either neutral or in favour of disease since green poop is odd. After checking the interne... | Green poop, leafy greens and probability of disease, how can I formalize this reasoning? | It seems to me you are looking at the Bayes's theorem and in particular at the prior probability.
Your data ($green\;poop, \; etc$) is the same before and after checking the internet. However, initial | Green poop, leafy greens and probability of disease, how can I formalize this reasoning?
It seems to me you are looking at the Bayes's theorem and in particular at the prior probability.
Your data ($green\;poop, \; etc$) is the same before and after checking the internet. However, initially, your prior probability is e... | Green poop, leafy greens and probability of disease, how can I formalize this reasoning?
It seems to me you are looking at the Bayes's theorem and in particular at the prior probability.
Your data ($green\;poop, \; etc$) is the same before and after checking the internet. However, initial |
53,555 | Green poop, leafy greens and probability of disease, how can I formalize this reasoning? | This kind of problem can be handled using Bayesian analysis, but it requires a bit of care. The tricky bit here is that there is a distinction between the conditioning event "ate leafy greens" and the other conditioning event "information showing that eating leafy greens causes green poo". You already know you ate le... | Green poop, leafy greens and probability of disease, how can I formalize this reasoning? | This kind of problem can be handled using Bayesian analysis, but it requires a bit of care. The tricky bit here is that there is a distinction between the conditioning event "ate leafy greens" and th | Green poop, leafy greens and probability of disease, how can I formalize this reasoning?
This kind of problem can be handled using Bayesian analysis, but it requires a bit of care. The tricky bit here is that there is a distinction between the conditioning event "ate leafy greens" and the other conditioning event "inf... | Green poop, leafy greens and probability of disease, how can I formalize this reasoning?
This kind of problem can be handled using Bayesian analysis, but it requires a bit of care. The tricky bit here is that there is a distinction between the conditioning event "ate leafy greens" and th |
53,556 | Green poop, leafy greens and probability of disease, how can I formalize this reasoning? | $$statistics \neq mathematics$$
We can mathematically express probabilities (like you did two times) but they are not the real probabilities and instead only probabilities according to some model.
So a probability expression has a "probability" to fail. By how much... that depends on the quality of the model.
If your m... | Green poop, leafy greens and probability of disease, how can I formalize this reasoning? | $$statistics \neq mathematics$$
We can mathematically express probabilities (like you did two times) but they are not the real probabilities and instead only probabilities according to some model.
So | Green poop, leafy greens and probability of disease, how can I formalize this reasoning?
$$statistics \neq mathematics$$
We can mathematically express probabilities (like you did two times) but they are not the real probabilities and instead only probabilities according to some model.
So a probability expression has a ... | Green poop, leafy greens and probability of disease, how can I formalize this reasoning?
$$statistics \neq mathematics$$
We can mathematically express probabilities (like you did two times) but they are not the real probabilities and instead only probabilities according to some model.
So |
53,557 | introductory machine learning concept questions [closed] | The `1 regularization cannot shrink parameters to zero, hence it can
be used for the purpose of feature selection
Yes. You can refer to this answer.
Deep Neural Networks
Many other hyperparameters, like embedding dimension, layer dimension, input length, parameter sharing, reused layers in transfer learning, early s... | introductory machine learning concept questions [closed] | The `1 regularization cannot shrink parameters to zero, hence it can
be used for the purpose of feature selection
Yes. You can refer to this answer.
Deep Neural Networks
Many other hyperparameters, | introductory machine learning concept questions [closed]
The `1 regularization cannot shrink parameters to zero, hence it can
be used for the purpose of feature selection
Yes. You can refer to this answer.
Deep Neural Networks
Many other hyperparameters, like embedding dimension, layer dimension, input length, param... | introductory machine learning concept questions [closed]
The `1 regularization cannot shrink parameters to zero, hence it can
be used for the purpose of feature selection
Yes. You can refer to this answer.
Deep Neural Networks
Many other hyperparameters, |
53,558 | introductory machine learning concept questions [closed] | #2
A covariance matrix cannot have eigenvalues less than zero, as it is a real, symmetric matrix. However, there is no such restriction on positive/negative/zero in the matrix itself.
As you note, covariance can be less than zero. This happens when variables have correlation less than zero. Therefore, there can be numb... | introductory machine learning concept questions [closed] | #2
A covariance matrix cannot have eigenvalues less than zero, as it is a real, symmetric matrix. However, there is no such restriction on positive/negative/zero in the matrix itself.
As you note, cov | introductory machine learning concept questions [closed]
#2
A covariance matrix cannot have eigenvalues less than zero, as it is a real, symmetric matrix. However, there is no such restriction on positive/negative/zero in the matrix itself.
As you note, covariance can be less than zero. This happens when variables have... | introductory machine learning concept questions [closed]
#2
A covariance matrix cannot have eigenvalues less than zero, as it is a real, symmetric matrix. However, there is no such restriction on positive/negative/zero in the matrix itself.
As you note, cov |
53,559 | introductory machine learning concept questions [closed] | Re: L1 regularization, I think that's a trick question. The conclusion is true, but the antecedent is false -- L1 regularization can shrink parameters to zero, and that's why it can be used for variable selection (any features associated with zero parameters are effectively cut out of the model).
However, as you know, ... | introductory machine learning concept questions [closed] | Re: L1 regularization, I think that's a trick question. The conclusion is true, but the antecedent is false -- L1 regularization can shrink parameters to zero, and that's why it can be used for variab | introductory machine learning concept questions [closed]
Re: L1 regularization, I think that's a trick question. The conclusion is true, but the antecedent is false -- L1 regularization can shrink parameters to zero, and that's why it can be used for variable selection (any features associated with zero parameters are ... | introductory machine learning concept questions [closed]
Re: L1 regularization, I think that's a trick question. The conclusion is true, but the antecedent is false -- L1 regularization can shrink parameters to zero, and that's why it can be used for variab |
53,560 | Interpreting and troubleshooting nls in R with quadratic plateau model | We need better starting values. Fit a non-plateau model, model0, and use the parameters from that to fit all the data points giving model and then use a and b from that and a grid of values for clx (due to its problematic nature) giving model.Ab and model.La. (Note that it will not be able to produce fits from some o... | Interpreting and troubleshooting nls in R with quadratic plateau model | We need better starting values. Fit a non-plateau model, model0, and use the parameters from that to fit all the data points giving model and then use a and b from that and a grid of values for clx ( | Interpreting and troubleshooting nls in R with quadratic plateau model
We need better starting values. Fit a non-plateau model, model0, and use the parameters from that to fit all the data points giving model and then use a and b from that and a grid of values for clx (due to its problematic nature) giving model.Ab an... | Interpreting and troubleshooting nls in R with quadratic plateau model
We need better starting values. Fit a non-plateau model, model0, and use the parameters from that to fit all the data points giving model and then use a and b from that and a grid of values for clx ( |
53,561 | Interpreting and troubleshooting nls in R with quadratic plateau model | Additionally, if anyone out there is good at interpreting formulas, can you help me by writing up this code into a readable formula?
function(x, a, b, clx) {
ifelse(x < clx, a + b * x + (-0.5*b/clx) * x * x,
a + b * clx + (-0.5*b/clx) * clx * clx)}
$$
f(x, a, b, x_{cl}) =
\begin{cases}
a + bx... | Interpreting and troubleshooting nls in R with quadratic plateau model | Additionally, if anyone out there is good at interpreting formulas, can you help me by writing up this code into a readable formula?
function(x, a, b, clx) {
ifelse(x < clx, a + b * x + (-0.5*b/clx | Interpreting and troubleshooting nls in R with quadratic plateau model
Additionally, if anyone out there is good at interpreting formulas, can you help me by writing up this code into a readable formula?
function(x, a, b, clx) {
ifelse(x < clx, a + b * x + (-0.5*b/clx) * x * x,
a + b * clx + (-0.5*b/clx) *... | Interpreting and troubleshooting nls in R with quadratic plateau model
Additionally, if anyone out there is good at interpreting formulas, can you help me by writing up this code into a readable formula?
function(x, a, b, clx) {
ifelse(x < clx, a + b * x + (-0.5*b/clx |
53,562 | Predicting proportions with Machine Learning | You have what is called compositional-data. There is quite some literature on how to model this. Take a look through the tag, or search for the term.
Typically, one would choose a reference category and work with log ratios, or similar. One paper I personally know about predicting compositional data is Snyder at al. (2... | Predicting proportions with Machine Learning | You have what is called compositional-data. There is quite some literature on how to model this. Take a look through the tag, or search for the term.
Typically, one would choose a reference category a | Predicting proportions with Machine Learning
You have what is called compositional-data. There is quite some literature on how to model this. Take a look through the tag, or search for the term.
Typically, one would choose a reference category and work with log ratios, or similar. One paper I personally know about pred... | Predicting proportions with Machine Learning
You have what is called compositional-data. There is quite some literature on how to model this. Take a look through the tag, or search for the term.
Typically, one would choose a reference category a |
53,563 | Predicting proportions with Machine Learning | Answering my past self... One elegant solution is to use the cross-entropy with "soft-targets" as loss. This means that your targets will not be in one-hot-encodding format, but they will still sum to one. The original cross-entropy formula formula applies.
The cross-entropy loss with soft targets is widely used in the... | Predicting proportions with Machine Learning | Answering my past self... One elegant solution is to use the cross-entropy with "soft-targets" as loss. This means that your targets will not be in one-hot-encodding format, but they will still sum to | Predicting proportions with Machine Learning
Answering my past self... One elegant solution is to use the cross-entropy with "soft-targets" as loss. This means that your targets will not be in one-hot-encodding format, but they will still sum to one. The original cross-entropy formula formula applies.
The cross-entropy... | Predicting proportions with Machine Learning
Answering my past self... One elegant solution is to use the cross-entropy with "soft-targets" as loss. This means that your targets will not be in one-hot-encodding format, but they will still sum to |
53,564 | What is the variance of $X^2$ (without assuming normality)? | The general form of this variance depends on the first four moments of the distribution. To facilitate our analysis, we suppose that $X$ has mean $\mu$, variance $\sigma^2$, skewness $\gamma$ and kurtosis $\kappa$. The variance of interest exists if $\kappa < \infty$ and does not exist otherwise. Using the relations... | What is the variance of $X^2$ (without assuming normality)? | The general form of this variance depends on the first four moments of the distribution. To facilitate our analysis, we suppose that $X$ has mean $\mu$, variance $\sigma^2$, skewness $\gamma$ and kur | What is the variance of $X^2$ (without assuming normality)?
The general form of this variance depends on the first four moments of the distribution. To facilitate our analysis, we suppose that $X$ has mean $\mu$, variance $\sigma^2$, skewness $\gamma$ and kurtosis $\kappa$. The variance of interest exists if $\kappa ... | What is the variance of $X^2$ (without assuming normality)?
The general form of this variance depends on the first four moments of the distribution. To facilitate our analysis, we suppose that $X$ has mean $\mu$, variance $\sigma^2$, skewness $\gamma$ and kur |
53,565 | Maximum possible number of random variables with the same correlation? | The answer depends on what $\rho$ is. For $-1 \leq \rho < -\frac 12$, the answer is two random variables. More generally, the maximum number of random variables that can have common correlation $\rho$ is $n$ for $\rho$ in the range $\left[-\frac{1}{n-1}, -\frac{1}{n}\right)$. For $\rho \geq 0$, the number of random var... | Maximum possible number of random variables with the same correlation? | The answer depends on what $\rho$ is. For $-1 \leq \rho < -\frac 12$, the answer is two random variables. More generally, the maximum number of random variables that can have common correlation $\rho$ | Maximum possible number of random variables with the same correlation?
The answer depends on what $\rho$ is. For $-1 \leq \rho < -\frac 12$, the answer is two random variables. More generally, the maximum number of random variables that can have common correlation $\rho$ is $n$ for $\rho$ in the range $\left[-\frac{1}{... | Maximum possible number of random variables with the same correlation?
The answer depends on what $\rho$ is. For $-1 \leq \rho < -\frac 12$, the answer is two random variables. More generally, the maximum number of random variables that can have common correlation $\rho$ |
53,566 | Maximum possible number of random variables with the same correlation? | To supplement Dilip Sarwate's answer, if you take $I_0 \sim Bernoulli(p_0)$ and any number of independent $I_i \sim Bernoulli(p)$, all independent then
$$cor(I_0 + I_i, I_0 + I_j) = \frac 1 {1 + \frac {p (1-p)} {p_0(1-p_0)}},$$
so you can choose $p_0$ and $p$ to get any $\rho$ in the interval $(0,1)$. | Maximum possible number of random variables with the same correlation? | To supplement Dilip Sarwate's answer, if you take $I_0 \sim Bernoulli(p_0)$ and any number of independent $I_i \sim Bernoulli(p)$, all independent then
$$cor(I_0 + I_i, I_0 + I_j) = \frac 1 {1 + \frac | Maximum possible number of random variables with the same correlation?
To supplement Dilip Sarwate's answer, if you take $I_0 \sim Bernoulli(p_0)$ and any number of independent $I_i \sim Bernoulli(p)$, all independent then
$$cor(I_0 + I_i, I_0 + I_j) = \frac 1 {1 + \frac {p (1-p)} {p_0(1-p_0)}},$$
so you can choose $p_... | Maximum possible number of random variables with the same correlation?
To supplement Dilip Sarwate's answer, if you take $I_0 \sim Bernoulli(p_0)$ and any number of independent $I_i \sim Bernoulli(p)$, all independent then
$$cor(I_0 + I_i, I_0 + I_j) = \frac 1 {1 + \frac |
53,567 | Frameworks for modeling prior knowledge other than Bayesian statistics | There are alternatives, for example, you can use constrained optimization, or regularization. Notice however, that in most cases those approaches can be thought as Bayesian inference in disguise. For example, constraining range of the parameter during optimization, is the same as using flat prior over this range. Using... | Frameworks for modeling prior knowledge other than Bayesian statistics | There are alternatives, for example, you can use constrained optimization, or regularization. Notice however, that in most cases those approaches can be thought as Bayesian inference in disguise. For | Frameworks for modeling prior knowledge other than Bayesian statistics
There are alternatives, for example, you can use constrained optimization, or regularization. Notice however, that in most cases those approaches can be thought as Bayesian inference in disguise. For example, constraining range of the parameter duri... | Frameworks for modeling prior knowledge other than Bayesian statistics
There are alternatives, for example, you can use constrained optimization, or regularization. Notice however, that in most cases those approaches can be thought as Bayesian inference in disguise. For |
53,568 | Frameworks for modeling prior knowledge other than Bayesian statistics | One way in which prior information can be incorporated into the estimator is through the likelihood (or model, depending on how you look at it). That is to say, when we build a standard parametric model, we are constraining ourselves to say that we are going to allow the model to follow a very specific form, that we kn... | Frameworks for modeling prior knowledge other than Bayesian statistics | One way in which prior information can be incorporated into the estimator is through the likelihood (or model, depending on how you look at it). That is to say, when we build a standard parametric mod | Frameworks for modeling prior knowledge other than Bayesian statistics
One way in which prior information can be incorporated into the estimator is through the likelihood (or model, depending on how you look at it). That is to say, when we build a standard parametric model, we are constraining ourselves to say that we ... | Frameworks for modeling prior knowledge other than Bayesian statistics
One way in which prior information can be incorporated into the estimator is through the likelihood (or model, depending on how you look at it). That is to say, when we build a standard parametric mod |
53,569 | We flip a coin 20 times and observe 12 heads. What is the probability that the coin is fair? | You appear to be using a Beta(1,1) prior on $\theta$. Since this is a continuous distribution, the prior (and posterior) probability of the event that the coin is exactly fair, $\theta=1/2$, is zero.
What would perhaps be a more sensible prior (see Lindley 1957 pp. 188-189 for a discussion of similar examples) would ... | We flip a coin 20 times and observe 12 heads. What is the probability that the coin is fair? | You appear to be using a Beta(1,1) prior on $\theta$. Since this is a continuous distribution, the prior (and posterior) probability of the event that the coin is exactly fair, $\theta=1/2$, is zero. | We flip a coin 20 times and observe 12 heads. What is the probability that the coin is fair?
You appear to be using a Beta(1,1) prior on $\theta$. Since this is a continuous distribution, the prior (and posterior) probability of the event that the coin is exactly fair, $\theta=1/2$, is zero.
What would perhaps be a m... | We flip a coin 20 times and observe 12 heads. What is the probability that the coin is fair?
You appear to be using a Beta(1,1) prior on $\theta$. Since this is a continuous distribution, the prior (and posterior) probability of the event that the coin is exactly fair, $\theta=1/2$, is zero. |
53,570 | Admissible Empirical Bayes Examples | The question has no clear answer because the empirical Bayes
formulation does not & cannot specify how the hyperparameter is
estimated.
Take the simplest Normal mean estimation problem. When$$X\sim\mathcal N_p(\theta,I_p)\qquad\qquad\theta\sim\mathcal N_p(0,\sigma^2 I_p)$$the Bayes estimator of $\theta$ is$$\delta^\pi... | Admissible Empirical Bayes Examples | The question has no clear answer because the empirical Bayes
formulation does not & cannot specify how the hyperparameter is
estimated.
Take the simplest Normal mean estimation problem. When$$X\sim\m | Admissible Empirical Bayes Examples
The question has no clear answer because the empirical Bayes
formulation does not & cannot specify how the hyperparameter is
estimated.
Take the simplest Normal mean estimation problem. When$$X\sim\mathcal N_p(\theta,I_p)\qquad\qquad\theta\sim\mathcal N_p(0,\sigma^2 I_p)$$the Bayes ... | Admissible Empirical Bayes Examples
The question has no clear answer because the empirical Bayes
formulation does not & cannot specify how the hyperparameter is
estimated.
Take the simplest Normal mean estimation problem. When$$X\sim\m |
53,571 | Multivariate bayesian inference: learning about the mean of a variable by observing another variable | I think it would be good to have the notations cleared first.
$$
\begin{aligned}
\vec{\mu} &\sim \mathcal{N}(\vec{\mu}_0, \Sigma_0) &&\textrm{prior},\\
\vec{x}_i \vert \vec{\mu} &\sim \mathcal{N}(\vec{\mu}, \Sigma) &&\textrm{likelihood},\\
\vec{\mu} \vert \{\vec{x}_i\} &\sim \mathcal{N}(\vec{\mu}_n, \Sigma_n) &&\textrm... | Multivariate bayesian inference: learning about the mean of a variable by observing another variable | I think it would be good to have the notations cleared first.
$$
\begin{aligned}
\vec{\mu} &\sim \mathcal{N}(\vec{\mu}_0, \Sigma_0) &&\textrm{prior},\\
\vec{x}_i \vert \vec{\mu} &\sim \mathcal{N}(\vec | Multivariate bayesian inference: learning about the mean of a variable by observing another variable
I think it would be good to have the notations cleared first.
$$
\begin{aligned}
\vec{\mu} &\sim \mathcal{N}(\vec{\mu}_0, \Sigma_0) &&\textrm{prior},\\
\vec{x}_i \vert \vec{\mu} &\sim \mathcal{N}(\vec{\mu}, \Sigma) &&\t... | Multivariate bayesian inference: learning about the mean of a variable by observing another variable
I think it would be good to have the notations cleared first.
$$
\begin{aligned}
\vec{\mu} &\sim \mathcal{N}(\vec{\mu}_0, \Sigma_0) &&\textrm{prior},\\
\vec{x}_i \vert \vec{\mu} &\sim \mathcal{N}(\vec |
53,572 | Multivariate bayesian inference: learning about the mean of a variable by observing another variable | This might be slightly counterintuitive at first, but the fact that two variables are correlated doesn't mean that you can learn about the mean of one from another.
Suppose for example that $x_1$ is normally distributed with mean $0$ and $x_2 = x_1 + 100$. $x_1$ and $x_2$ are clearly maximally correlated, but you can l... | Multivariate bayesian inference: learning about the mean of a variable by observing another variable | This might be slightly counterintuitive at first, but the fact that two variables are correlated doesn't mean that you can learn about the mean of one from another.
Suppose for example that $x_1$ is n | Multivariate bayesian inference: learning about the mean of a variable by observing another variable
This might be slightly counterintuitive at first, but the fact that two variables are correlated doesn't mean that you can learn about the mean of one from another.
Suppose for example that $x_1$ is normally distributed... | Multivariate bayesian inference: learning about the mean of a variable by observing another variable
This might be slightly counterintuitive at first, but the fact that two variables are correlated doesn't mean that you can learn about the mean of one from another.
Suppose for example that $x_1$ is n |
53,573 | Incorporating Prior Information Into Time Series Prediction | Great Question !
"Could I combine my ARIMA forecast with this prior information somehow to form an ensemble forecast?"
I have been involved with a commercial time series forecasting package called AUTOBOX and have incorporated delphi-type predictor series where the user provides probabilities of intervals and this is t... | Incorporating Prior Information Into Time Series Prediction | Great Question !
"Could I combine my ARIMA forecast with this prior information somehow to form an ensemble forecast?"
I have been involved with a commercial time series forecasting package called AUT | Incorporating Prior Information Into Time Series Prediction
Great Question !
"Could I combine my ARIMA forecast with this prior information somehow to form an ensemble forecast?"
I have been involved with a commercial time series forecasting package called AUTOBOX and have incorporated delphi-type predictor series wher... | Incorporating Prior Information Into Time Series Prediction
Great Question !
"Could I combine my ARIMA forecast with this prior information somehow to form an ensemble forecast?"
I have been involved with a commercial time series forecasting package called AUT |
53,574 | Incorporating Prior Information Into Time Series Prediction | I think a good way to do this is via Bayesian Structural Time Series (BSTS). I found out about this approach via these 2 sites (1, 2). I would still be interested in other approaches.
Here is the example done with the bsts package in R. I use a time series component and a regression component. The regression component... | Incorporating Prior Information Into Time Series Prediction | I think a good way to do this is via Bayesian Structural Time Series (BSTS). I found out about this approach via these 2 sites (1, 2). I would still be interested in other approaches.
Here is the exa | Incorporating Prior Information Into Time Series Prediction
I think a good way to do this is via Bayesian Structural Time Series (BSTS). I found out about this approach via these 2 sites (1, 2). I would still be interested in other approaches.
Here is the example done with the bsts package in R. I use a time series co... | Incorporating Prior Information Into Time Series Prediction
I think a good way to do this is via Bayesian Structural Time Series (BSTS). I found out about this approach via these 2 sites (1, 2). I would still be interested in other approaches.
Here is the exa |
53,575 | How does R's "poisson.test" function work, mathematically? | When you say "another Poisson rate" ... if that other Poisson rate is derived from data then you are comparing data with data.
I'll assume you mean against some prespecified/theoretical rate (i.e. that you're performing a one-sample test).
You didn't state whether you were doing a one-tailed or two-tailed test. I'll d... | How does R's "poisson.test" function work, mathematically? | When you say "another Poisson rate" ... if that other Poisson rate is derived from data then you are comparing data with data.
I'll assume you mean against some prespecified/theoretical rate (i.e. tha | How does R's "poisson.test" function work, mathematically?
When you say "another Poisson rate" ... if that other Poisson rate is derived from data then you are comparing data with data.
I'll assume you mean against some prespecified/theoretical rate (i.e. that you're performing a one-sample test).
You didn't state whe... | How does R's "poisson.test" function work, mathematically?
When you say "another Poisson rate" ... if that other Poisson rate is derived from data then you are comparing data with data.
I'll assume you mean against some prespecified/theoretical rate (i.e. tha |
53,576 | How does R's "poisson.test" function work, mathematically? | Glen's answer notes that you can check the code for this function, but I'm not sure if you know how to do this, so I'll augment his answer by showing you how. To check the code, just load the relevant library and type in the function name without any arguments:
library(stats)
poisson.test
function (x, T = 1, r = 1, a... | How does R's "poisson.test" function work, mathematically? | Glen's answer notes that you can check the code for this function, but I'm not sure if you know how to do this, so I'll augment his answer by showing you how. To check the code, just load the relevan | How does R's "poisson.test" function work, mathematically?
Glen's answer notes that you can check the code for this function, but I'm not sure if you know how to do this, so I'll augment his answer by showing you how. To check the code, just load the relevant library and type in the function name without any arguments... | How does R's "poisson.test" function work, mathematically?
Glen's answer notes that you can check the code for this function, but I'm not sure if you know how to do this, so I'll augment his answer by showing you how. To check the code, just load the relevan |
53,577 | Possible that one model is better than two? | Here's a perspective: the two model approach is more constrained, hence is always going to result in an inferior model. Consider the 2m (two-model) model - it looks like:
$$ f_{2m}(\mathbf{x}) = 1.5 (\mathbf{c_1} \cdot \mathbf{x}^T) + 1.0 (\mathbf{c_2} \cdot \mathbf{x}^T)$$
where $\mathbf{c}_i$ were trained in separate... | Possible that one model is better than two? | Here's a perspective: the two model approach is more constrained, hence is always going to result in an inferior model. Consider the 2m (two-model) model - it looks like:
$$ f_{2m}(\mathbf{x}) = 1.5 ( | Possible that one model is better than two?
Here's a perspective: the two model approach is more constrained, hence is always going to result in an inferior model. Consider the 2m (two-model) model - it looks like:
$$ f_{2m}(\mathbf{x}) = 1.5 (\mathbf{c_1} \cdot \mathbf{x}^T) + 1.0 (\mathbf{c_2} \cdot \mathbf{x}^T)$$
w... | Possible that one model is better than two?
Here's a perspective: the two model approach is more constrained, hence is always going to result in an inferior model. Consider the 2m (two-model) model - it looks like:
$$ f_{2m}(\mathbf{x}) = 1.5 ( |
53,578 | Possible that one model is better than two? | Impossible that one predictive model is better than two?
Rather than getting into the weeds on your specific models, let's just step back and view this question in a more general setting. If we consider an arbitrary series of observable values, then it is possible that a model could give a perfect prediction of those... | Possible that one model is better than two? | Impossible that one predictive model is better than two?
Rather than getting into the weeds on your specific models, let's just step back and view this question in a more general setting. If we cons | Possible that one model is better than two?
Impossible that one predictive model is better than two?
Rather than getting into the weeds on your specific models, let's just step back and view this question in a more general setting. If we consider an arbitrary series of observable values, then it is possible that a mo... | Possible that one model is better than two?
Impossible that one predictive model is better than two?
Rather than getting into the weeds on your specific models, let's just step back and view this question in a more general setting. If we cons |
53,579 | Possible that one model is better than two? | As the previous answers have indicated, simply adding a model that is wrong can decrease performance. However, there are clever ways around this issue.
Generalized stacking algorithms (super learner is one example) is an alternative strategy to aggregating the results of multiple models. It has the advantage of discard... | Possible that one model is better than two? | As the previous answers have indicated, simply adding a model that is wrong can decrease performance. However, there are clever ways around this issue.
Generalized stacking algorithms (super learner i | Possible that one model is better than two?
As the previous answers have indicated, simply adding a model that is wrong can decrease performance. However, there are clever ways around this issue.
Generalized stacking algorithms (super learner is one example) is an alternative strategy to aggregating the results of mult... | Possible that one model is better than two?
As the previous answers have indicated, simply adding a model that is wrong can decrease performance. However, there are clever ways around this issue.
Generalized stacking algorithms (super learner i |
53,580 | How can I calibrate my point-by-point variances for Gaussian process regression? | Standard Gaussian process (GP) regression assumes constant noise variance, whereas it seems you want to allow it to vary. So, this is a heteroscedastic GP regression problem. Similar problems have been addressed in the literature (see references below). For example, Goldberg et al. (1998) treat the noise variance as an... | How can I calibrate my point-by-point variances for Gaussian process regression? | Standard Gaussian process (GP) regression assumes constant noise variance, whereas it seems you want to allow it to vary. So, this is a heteroscedastic GP regression problem. Similar problems have bee | How can I calibrate my point-by-point variances for Gaussian process regression?
Standard Gaussian process (GP) regression assumes constant noise variance, whereas it seems you want to allow it to vary. So, this is a heteroscedastic GP regression problem. Similar problems have been addressed in the literature (see refe... | How can I calibrate my point-by-point variances for Gaussian process regression?
Standard Gaussian process (GP) regression assumes constant noise variance, whereas it seems you want to allow it to vary. So, this is a heteroscedastic GP regression problem. Similar problems have bee |
53,581 | How can I calibrate my point-by-point variances for Gaussian process regression? | In the end, although I appreciated the answer from user20160, I found it impractical to implement. Instead, I went ahead with the idea I mentioned in my question, and problems didn't materialize. Specifically:
I began with the GP model I already had, and a roughly plausible mapping of confidence scores to variances.
I... | How can I calibrate my point-by-point variances for Gaussian process regression? | In the end, although I appreciated the answer from user20160, I found it impractical to implement. Instead, I went ahead with the idea I mentioned in my question, and problems didn't materialize. Spec | How can I calibrate my point-by-point variances for Gaussian process regression?
In the end, although I appreciated the answer from user20160, I found it impractical to implement. Instead, I went ahead with the idea I mentioned in my question, and problems didn't materialize. Specifically:
I began with the GP model I ... | How can I calibrate my point-by-point variances for Gaussian process regression?
In the end, although I appreciated the answer from user20160, I found it impractical to implement. Instead, I went ahead with the idea I mentioned in my question, and problems didn't materialize. Spec |
53,582 | How can I calibrate my point-by-point variances for Gaussian process regression? | I assume by sigmas you mean the variances of the components of the kernel functions. If you want to choose them in the light of your CS, then that sounds Bayesian to me. How are you setting the length scales?
There is a literature on priors for Gaussian process parameters, eg Trangucci, Betancourt, Vehtari (2016). | How can I calibrate my point-by-point variances for Gaussian process regression? | I assume by sigmas you mean the variances of the components of the kernel functions. If you want to choose them in the light of your CS, then that sounds Bayesian to me. How are you setting the length | How can I calibrate my point-by-point variances for Gaussian process regression?
I assume by sigmas you mean the variances of the components of the kernel functions. If you want to choose them in the light of your CS, then that sounds Bayesian to me. How are you setting the length scales?
There is a literature on prior... | How can I calibrate my point-by-point variances for Gaussian process regression?
I assume by sigmas you mean the variances of the components of the kernel functions. If you want to choose them in the light of your CS, then that sounds Bayesian to me. How are you setting the length |
53,583 | Mixed Effects, Doctors & Operations: predicting on new data containing previously unobserved levels, and updating our confidence accordingly | What you describe is the concept of dynamic predictions from mixed models.
Initially, when you have no information for a doctor you only use the fixed effects in the prediction, i.e., you put his/her ability level equal to the average ($\alpha_j = 0$).
But, as extra information is recorded you can update your predict... | Mixed Effects, Doctors & Operations: predicting on new data containing previously unobserved levels, | What you describe is the concept of dynamic predictions from mixed models.
Initially, when you have no information for a doctor you only use the fixed effects in the prediction, i.e., you put his/her | Mixed Effects, Doctors & Operations: predicting on new data containing previously unobserved levels, and updating our confidence accordingly
What you describe is the concept of dynamic predictions from mixed models.
Initially, when you have no information for a doctor you only use the fixed effects in the prediction, ... | Mixed Effects, Doctors & Operations: predicting on new data containing previously unobserved levels,
What you describe is the concept of dynamic predictions from mixed models.
Initially, when you have no information for a doctor you only use the fixed effects in the prediction, i.e., you put his/her |
53,584 | Mixed Effects, Doctors & Operations: predicting on new data containing previously unobserved levels, and updating our confidence accordingly | I think a Bayesian approach might be beneficial. When predicting for an unobserved category in a Bayesian mixed model, you generate posterior predictions by sampling the $\alpha_j$ from the fitted $N(0, \sigma^2_\alpha)$ (aside from sampling the fixed effects from the fitted posterior). This way, you will see high unce... | Mixed Effects, Doctors & Operations: predicting on new data containing previously unobserved levels, | I think a Bayesian approach might be beneficial. When predicting for an unobserved category in a Bayesian mixed model, you generate posterior predictions by sampling the $\alpha_j$ from the fitted $N( | Mixed Effects, Doctors & Operations: predicting on new data containing previously unobserved levels, and updating our confidence accordingly
I think a Bayesian approach might be beneficial. When predicting for an unobserved category in a Bayesian mixed model, you generate posterior predictions by sampling the $\alpha_j... | Mixed Effects, Doctors & Operations: predicting on new data containing previously unobserved levels,
I think a Bayesian approach might be beneficial. When predicting for an unobserved category in a Bayesian mixed model, you generate posterior predictions by sampling the $\alpha_j$ from the fitted $N( |
53,585 | Can someone provide a brief explanation as to why reproducing kernel Hilbert space is so popular in machine learning? | The typical way to give some intuition for reproducing kernel spaces (and, in particular, the kernel trick), is the application area of support vector machines. The aim is to linearly separate two classes of points in $\mathbb R^n$, which works fine if they actually are linearly separable.
If they are not, the kernel t... | Can someone provide a brief explanation as to why reproducing kernel Hilbert space is so popular in | The typical way to give some intuition for reproducing kernel spaces (and, in particular, the kernel trick), is the application area of support vector machines. The aim is to linearly separate two cla | Can someone provide a brief explanation as to why reproducing kernel Hilbert space is so popular in machine learning?
The typical way to give some intuition for reproducing kernel spaces (and, in particular, the kernel trick), is the application area of support vector machines. The aim is to linearly separate two class... | Can someone provide a brief explanation as to why reproducing kernel Hilbert space is so popular in
The typical way to give some intuition for reproducing kernel spaces (and, in particular, the kernel trick), is the application area of support vector machines. The aim is to linearly separate two cla |
53,586 | What does 'km' transform in cox.zph function mean? | km stands for Kaplan-Meier estimator.
$$\hat{S}(t) = \prod_{i: t_i \le t}\left(1-\frac{d_i}{n_i} \right)$$
with $t_{i}$ a time when at least one event happened, $d_i$ the number of events (i.e., deaths) that happened at time
$t_{i}$ and ${\displaystyle n_{i}}$ the individuals known to have survived (have not yet had a... | What does 'km' transform in cox.zph function mean? | km stands for Kaplan-Meier estimator.
$$\hat{S}(t) = \prod_{i: t_i \le t}\left(1-\frac{d_i}{n_i} \right)$$
with $t_{i}$ a time when at least one event happened, $d_i$ the number of events (i.e., death | What does 'km' transform in cox.zph function mean?
km stands for Kaplan-Meier estimator.
$$\hat{S}(t) = \prod_{i: t_i \le t}\left(1-\frac{d_i}{n_i} \right)$$
with $t_{i}$ a time when at least one event happened, $d_i$ the number of events (i.e., deaths) that happened at time
$t_{i}$ and ${\displaystyle n_{i}}$ the ind... | What does 'km' transform in cox.zph function mean?
km stands for Kaplan-Meier estimator.
$$\hat{S}(t) = \prod_{i: t_i \le t}\left(1-\frac{d_i}{n_i} \right)$$
with $t_{i}$ a time when at least one event happened, $d_i$ the number of events (i.e., death |
53,587 | What does 'km' transform in cox.zph function mean? | Like the original poster, I also wondered what, exactly, is the transformation "based on the Kaplan-Meier estimate" doing? Tracking this down proved to be more difficult than you would expect, as pretty much every source I found used some variant of the "based on Kaplan-Meier" language without further explanation. I ... | What does 'km' transform in cox.zph function mean? | Like the original poster, I also wondered what, exactly, is the transformation "based on the Kaplan-Meier estimate" doing? Tracking this down proved to be more difficult than you would expect, as pre | What does 'km' transform in cox.zph function mean?
Like the original poster, I also wondered what, exactly, is the transformation "based on the Kaplan-Meier estimate" doing? Tracking this down proved to be more difficult than you would expect, as pretty much every source I found used some variant of the "based on Kapl... | What does 'km' transform in cox.zph function mean?
Like the original poster, I also wondered what, exactly, is the transformation "based on the Kaplan-Meier estimate" doing? Tracking this down proved to be more difficult than you would expect, as pre |
53,588 | Spurious Regressions (Random Walk) | Consider what random walks are: each new value is just a small perturbation of the old value.
When an explanatory variable $x_t$ and a synchronous response $y_t$ are both random walks, the pair of points $(x_t,y_t)$ is a random walk in the plane with similar properties: each new point is a small random step (in a rando... | Spurious Regressions (Random Walk) | Consider what random walks are: each new value is just a small perturbation of the old value.
When an explanatory variable $x_t$ and a synchronous response $y_t$ are both random walks, the pair of poi | Spurious Regressions (Random Walk)
Consider what random walks are: each new value is just a small perturbation of the old value.
When an explanatory variable $x_t$ and a synchronous response $y_t$ are both random walks, the pair of points $(x_t,y_t)$ is a random walk in the plane with similar properties: each new point... | Spurious Regressions (Random Walk)
Consider what random walks are: each new value is just a small perturbation of the old value.
When an explanatory variable $x_t$ and a synchronous response $y_t$ are both random walks, the pair of poi |
53,589 | Spurious Regressions (Random Walk) | It's not always a problem when both dependent and independent variables are random walk. If two variables are cointegrated then you still can run OLS. It won't be the best choice, but it will retain its good properties.
So why is it a problem sometimes? In a random walk you get the variance increasing with time proport... | Spurious Regressions (Random Walk) | It's not always a problem when both dependent and independent variables are random walk. If two variables are cointegrated then you still can run OLS. It won't be the best choice, but it will retain i | Spurious Regressions (Random Walk)
It's not always a problem when both dependent and independent variables are random walk. If two variables are cointegrated then you still can run OLS. It won't be the best choice, but it will retain its good properties.
So why is it a problem sometimes? In a random walk you get the va... | Spurious Regressions (Random Walk)
It's not always a problem when both dependent and independent variables are random walk. If two variables are cointegrated then you still can run OLS. It won't be the best choice, but it will retain i |
53,590 | Difference between Cross validation,GridSearchCV and does cross validation refer the train test split? | Cross Validation(CV) or K-Fold Cross Validation (K-Fold CV) is very similar to what you already know as train-test split. When people refer to cross validation they generally mean k-fold cross validation. In k-fold cross validation what you do is just that you have multiple(k) train-test sets instead of 1. This basical... | Difference between Cross validation,GridSearchCV and does cross validation refer the train test spli | Cross Validation(CV) or K-Fold Cross Validation (K-Fold CV) is very similar to what you already know as train-test split. When people refer to cross validation they generally mean k-fold cross validat | Difference between Cross validation,GridSearchCV and does cross validation refer the train test split?
Cross Validation(CV) or K-Fold Cross Validation (K-Fold CV) is very similar to what you already know as train-test split. When people refer to cross validation they generally mean k-fold cross validation. In k-fold cr... | Difference between Cross validation,GridSearchCV and does cross validation refer the train test spli
Cross Validation(CV) or K-Fold Cross Validation (K-Fold CV) is very similar to what you already know as train-test split. When people refer to cross validation they generally mean k-fold cross validat |
53,591 | Matching with Multiple Treatments | I recommend taking a look at Lopez & Gutman (2017), who clearly describe the issues at hand and the methods used to solve them.
Based on your description, it sounds like you want the average treatment effect in the control group (ATC) for several treatments. For each treatment level, this answers the question, "For th... | Matching with Multiple Treatments | I recommend taking a look at Lopez & Gutman (2017), who clearly describe the issues at hand and the methods used to solve them.
Based on your description, it sounds like you want the average treatmen | Matching with Multiple Treatments
I recommend taking a look at Lopez & Gutman (2017), who clearly describe the issues at hand and the methods used to solve them.
Based on your description, it sounds like you want the average treatment effect in the control group (ATC) for several treatments. For each treatment level, ... | Matching with Multiple Treatments
I recommend taking a look at Lopez & Gutman (2017), who clearly describe the issues at hand and the methods used to solve them.
Based on your description, it sounds like you want the average treatmen |
53,592 | Iterated expectations and variances examples | Your calculation is correct, and is a good way I think. One other approach might be just using the PDF of $X$, using uniform PDF, $\Pi(x)$:
$$f_X(x)=\frac{1}{2}\Pi(x)+\frac{1}{2}\Pi(x-3)$$
Expected value can be fairly easy via both method, we just need $E[X^2]$:
$$E[X^2]=\frac{1}{2}\int_0^{1}x^2dx+\frac{1}{2}\int_3^4x^... | Iterated expectations and variances examples | Your calculation is correct, and is a good way I think. One other approach might be just using the PDF of $X$, using uniform PDF, $\Pi(x)$:
$$f_X(x)=\frac{1}{2}\Pi(x)+\frac{1}{2}\Pi(x-3)$$
Expected va | Iterated expectations and variances examples
Your calculation is correct, and is a good way I think. One other approach might be just using the PDF of $X$, using uniform PDF, $\Pi(x)$:
$$f_X(x)=\frac{1}{2}\Pi(x)+\frac{1}{2}\Pi(x-3)$$
Expected value can be fairly easy via both method, we just need $E[X^2]$:
$$E[X^2]=\fr... | Iterated expectations and variances examples
Your calculation is correct, and is a good way I think. One other approach might be just using the PDF of $X$, using uniform PDF, $\Pi(x)$:
$$f_X(x)=\frac{1}{2}\Pi(x)+\frac{1}{2}\Pi(x-3)$$
Expected va |
53,593 | Iterated expectations and variances examples | This problem can be simplified substantially by decomposing the random variable of interest as a sum of two independent parts:
$$X = U+3V
\quad \quad \quad \quad U \sim \text{U}(0,1)
\quad \quad \quad \quad V \sim \text{Bern}(\tfrac{1}{2}).$$
Using this decomposition we have mean:
$$\begin{equation} \begin{aligned}
\ma... | Iterated expectations and variances examples | This problem can be simplified substantially by decomposing the random variable of interest as a sum of two independent parts:
$$X = U+3V
\quad \quad \quad \quad U \sim \text{U}(0,1)
\quad \quad \quad | Iterated expectations and variances examples
This problem can be simplified substantially by decomposing the random variable of interest as a sum of two independent parts:
$$X = U+3V
\quad \quad \quad \quad U \sim \text{U}(0,1)
\quad \quad \quad \quad V \sim \text{Bern}(\tfrac{1}{2}).$$
Using this decomposition we have... | Iterated expectations and variances examples
This problem can be simplified substantially by decomposing the random variable of interest as a sum of two independent parts:
$$X = U+3V
\quad \quad \quad \quad U \sim \text{U}(0,1)
\quad \quad \quad |
53,594 | Iterated expectations and variances examples | There are generally two ways to approach these types of problems: by (1) Finding the second stage expectation $E(X)$ with the theorem
of total expectation; or by (2) Finding the second stage expectation
$E(X)$, using $f_{X}(x)$. These are equivalent methods, but you
might find one easier to comprehend, so I present the... | Iterated expectations and variances examples | There are generally two ways to approach these types of problems: by (1) Finding the second stage expectation $E(X)$ with the theorem
of total expectation; or by (2) Finding the second stage expectati | Iterated expectations and variances examples
There are generally two ways to approach these types of problems: by (1) Finding the second stage expectation $E(X)$ with the theorem
of total expectation; or by (2) Finding the second stage expectation
$E(X)$, using $f_{X}(x)$. These are equivalent methods, but you
might fi... | Iterated expectations and variances examples
There are generally two ways to approach these types of problems: by (1) Finding the second stage expectation $E(X)$ with the theorem
of total expectation; or by (2) Finding the second stage expectati |
53,595 | Iterated expectations and variances examples | Comment: Here is a brief simulation, comparing
approximate simulated results with theoretical results derived in this Q and A. Everything below matches within the margin of simulation error.
Also see Wikipedia on Mixture Distributions, under Moments, for some relevant formulas.
set.seed(420) # for reproducibility
u1 =... | Iterated expectations and variances examples | Comment: Here is a brief simulation, comparing
approximate simulated results with theoretical results derived in this Q and A. Everything below matches within the margin of simulation error.
Also see | Iterated expectations and variances examples
Comment: Here is a brief simulation, comparing
approximate simulated results with theoretical results derived in this Q and A. Everything below matches within the margin of simulation error.
Also see Wikipedia on Mixture Distributions, under Moments, for some relevant formul... | Iterated expectations and variances examples
Comment: Here is a brief simulation, comparing
approximate simulated results with theoretical results derived in this Q and A. Everything below matches within the margin of simulation error.
Also see |
53,596 | Does data normalization and transformation change the Pearson's correlation? | Pearson's correlation measures the linear component of association. So you
are correct that linear transformations of data will not affect the correlation between them. However, nonlinear transformations will generally have an effect.
Here is a demonstration: Generate right-skewed, correlated data vectors x and y. Pea... | Does data normalization and transformation change the Pearson's correlation? | Pearson's correlation measures the linear component of association. So you
are correct that linear transformations of data will not affect the correlation between them. However, nonlinear transformati | Does data normalization and transformation change the Pearson's correlation?
Pearson's correlation measures the linear component of association. So you
are correct that linear transformations of data will not affect the correlation between them. However, nonlinear transformations will generally have an effect.
Here is ... | Does data normalization and transformation change the Pearson's correlation?
Pearson's correlation measures the linear component of association. So you
are correct that linear transformations of data will not affect the correlation between them. However, nonlinear transformati |
53,597 | Does a log transform always bring a distribution closer to normal? | For purely positive quantities a log-transformation is indeed the standard first transformation to try and is very frequently used. It is also done if for regression you want a multiplicative interpretation of coefficients (e.g. doubling/ halving of blood cholesterol).
Of course it will not always make a distribution m... | Does a log transform always bring a distribution closer to normal? | For purely positive quantities a log-transformation is indeed the standard first transformation to try and is very frequently used. It is also done if for regression you want a multiplicative interpre | Does a log transform always bring a distribution closer to normal?
For purely positive quantities a log-transformation is indeed the standard first transformation to try and is very frequently used. It is also done if for regression you want a multiplicative interpretation of coefficients (e.g. doubling/ halving of blo... | Does a log transform always bring a distribution closer to normal?
For purely positive quantities a log-transformation is indeed the standard first transformation to try and is very frequently used. It is also done if for regression you want a multiplicative interpre |
53,598 | R: GLMM for unbalanced zero-inflated data (glmmTMB) | A1: "All in all, I have about 33% of the dates having counts of zero, which makes me think the data is zero inflated." -> this is a common misconception - zero-inflation != lots of zeros. Zero-inflation means you have more zeros than you would expect, given your fitted model. Without having fit a model, you can't know ... | R: GLMM for unbalanced zero-inflated data (glmmTMB) | A1: "All in all, I have about 33% of the dates having counts of zero, which makes me think the data is zero inflated." -> this is a common misconception - zero-inflation != lots of zeros. Zero-inflati | R: GLMM for unbalanced zero-inflated data (glmmTMB)
A1: "All in all, I have about 33% of the dates having counts of zero, which makes me think the data is zero inflated." -> this is a common misconception - zero-inflation != lots of zeros. Zero-inflation means you have more zeros than you would expect, given your fitte... | R: GLMM for unbalanced zero-inflated data (glmmTMB)
A1: "All in all, I have about 33% of the dates having counts of zero, which makes me think the data is zero inflated." -> this is a common misconception - zero-inflation != lots of zeros. Zero-inflati |
53,599 | Why can't t-SNE capture a simple parabola structure? | Three general remarks:
t-SNE is excellent at preserving cluster structure but is not very good at preserving continuous "manifold structure". One famous toy example is the Swiss roll data set, and it is well-known that t-SNE has trouble "unrolling" it. In fact, one can use t-SNE to unroll it, but one has to be really ... | Why can't t-SNE capture a simple parabola structure? | Three general remarks:
t-SNE is excellent at preserving cluster structure but is not very good at preserving continuous "manifold structure". One famous toy example is the Swiss roll data set, and it | Why can't t-SNE capture a simple parabola structure?
Three general remarks:
t-SNE is excellent at preserving cluster structure but is not very good at preserving continuous "manifold structure". One famous toy example is the Swiss roll data set, and it is well-known that t-SNE has trouble "unrolling" it. In fact, one ... | Why can't t-SNE capture a simple parabola structure?
Three general remarks:
t-SNE is excellent at preserving cluster structure but is not very good at preserving continuous "manifold structure". One famous toy example is the Swiss roll data set, and it |
53,600 | Explanation for Additive Property of Variance? | It doesn't!
In general:
Var(A+B) = Var(A) + Var(B) + Cov(A, B)
The additive property only holds if the two random variables have no covariation. This is almost a circular statement, since a legitimate definition of the covariation could be:
Cov(A, B) = Var(A) + Var(B) - Var(A + B)
This means that the covariance meas... | Explanation for Additive Property of Variance? | It doesn't!
In general:
Var(A+B) = Var(A) + Var(B) + Cov(A, B)
The additive property only holds if the two random variables have no covariation. This is almost a circular statement, since a legitima | Explanation for Additive Property of Variance?
It doesn't!
In general:
Var(A+B) = Var(A) + Var(B) + Cov(A, B)
The additive property only holds if the two random variables have no covariation. This is almost a circular statement, since a legitimate definition of the covariation could be:
Cov(A, B) = Var(A) + Var(B) - ... | Explanation for Additive Property of Variance?
It doesn't!
In general:
Var(A+B) = Var(A) + Var(B) + Cov(A, B)
The additive property only holds if the two random variables have no covariation. This is almost a circular statement, since a legitima |
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