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Distribution Gallery: Add HalfStudentT (arviz-devs#541)
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--- | ||
jupytext: | ||
text_representation: | ||
extension: .md | ||
format_name: myst | ||
kernelspec: | ||
display_name: Python 3 | ||
language: python | ||
name: python3 | ||
--- | ||
# Half-Student's t Distribution | ||
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<audio controls> <source src="../../_static/halfstudentt.mp3" type="audio/mpeg"> This browser cannot play the pronunciation audio file for this distribution. </audio> | ||
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The Half-Student's t distribution, also known as the half-t distribution, is a continuous probability distribution that is derived from the Student's t distribution but is restricted to only positive values. It is characterized by two parameters: the degrees of freedom ($\nu$) and the scale parameter ($\sigma$), which determines the width of the distribution. The smaller the value of $\nu$, the heavier the tails of the distribution. | ||
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In Bayesian statistics, the Half-Student's t distribution is often used as a prior for scale parameters. | ||
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## Parametrization | ||
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The Half-Student's t distribution has 2 alternative parameterizations. In terms of $\nu$ and $\sigma$, or in terms of $\nu$ and $\lambda$. | ||
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The link between the 2 alternatives is given by: | ||
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$$ | ||
\begin{align*} | ||
\lambda & = \frac{1}{\sigma^2} | ||
\end{align*} | ||
$$ | ||
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where $\sigma$ is the standard deviation as $\nu$ increases, and $\lambda$ is the precision as $\nu$ increases. | ||
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## Probability Density Function (PDF): | ||
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::::::{tab-set} | ||
:class: full-width | ||
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:::::{tab-item} Parameters $\nu$ and $\sigma$ | ||
:sync: nu_sigma | ||
```{jupyter-execute} | ||
:hide-code: | ||
from preliz import HalfStudentT, style | ||
style.use('preliz-doc') | ||
nus = [2., 5., 5.] | ||
sigmas = [1., 1., 2.] | ||
for nu, sigma in zip(nus, sigmas): | ||
HalfStudentT(nu, sigma).plot_pdf(support=(0, 5)) | ||
``` | ||
::::: | ||
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:::::{tab-item} Parameters $\nu$ and $\lambda$ | ||
:sync: nu_lambda | ||
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```{jupyter-execute} | ||
:hide-code: | ||
lambdas = [1., 1., 0.25] | ||
for nu, lam in zip(nus, lambdas): | ||
HalfStudentT(nu, lam=lam).plot_pdf(support=(0, 5)) | ||
``` | ||
::::: | ||
:::::: | ||
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## Cumulative Distribution Function (CDF): | ||
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::::::{tab-set} | ||
:class: full-width | ||
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:::::{tab-item} Parameters $\nu$ and $\sigma$ | ||
:sync: nu_sigma | ||
```{jupyter-execute} | ||
:hide-code: | ||
for nu, sigma in zip(nus, sigmas): | ||
HalfStudentT(nu, sigma).plot_cdf(support=(0, 5)) | ||
``` | ||
::::: | ||
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:::::{tab-item} Parameters $\nu$ and $\lambda$ | ||
:sync: nu_lambda | ||
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```{jupyter-execute} | ||
:hide-code: | ||
for nu, lam in zip(nus, lambdas): | ||
HalfStudentT(nu, lam=lam).plot_cdf(support=(0, 5)) | ||
``` | ||
::::: | ||
:::::: | ||
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## Key properties and parameters: | ||
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```{eval-rst} | ||
======== ========================================== | ||
Support :math:`x \in [0, \infty)` | ||
Mean .. math:: | ||
2\sigma\sqrt{\frac{\nu}{\pi}}\ | ||
\frac{\Gamma\left(\frac{\nu+1}{2}\right)} | ||
{\Gamma\left(\frac{\nu}{2}\right)(\nu-1)}\, \text{for } \nu > 2 | ||
Variance .. math:: | ||
\sigma^2\left(\frac{\nu}{\nu - 2}-\ | ||
\frac{4\nu}{\pi(\nu-1)^2}\left(\frac{\Gamma\left(\frac{\nu+1}{2}\right)} | ||
{\Gamma\left(\frac{\nu}{2}\right)}\right)^2\right) \text{for } \nu > 2\, \infty\ | ||
\text{for } 1 < \nu \le 2\, \text{otherwise undefined} | ||
======== ========================================== | ||
``` | ||
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**Probability Density Function (PDF):** | ||
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$$ | ||
f(x \mid \sigma,\nu) = | ||
\frac{2\;\Gamma\left(\frac{\nu+1}{2}\right)} | ||
{\Gamma\left(\frac{\nu}{2}\right)\sqrt{\nu\pi\sigma^2}} | ||
\left(1+\frac{1}{\nu}\frac{x^2}{\sigma^2}\right)^{-\frac{\nu+1}{2}} | ||
$$ | ||
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where $\Gamma$ is the [gamma function](https://en.wikipedia.org/wiki/Gamma_function). | ||
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**Cumulative Distribution Function (CDF):** | ||
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$$ | ||
F(x \mid \sigma,\nu) = | ||
\begin{cases} | ||
\frac{1}{2} \cdot I_{\frac{\nu}{x^2 + \nu}}\left(\frac{\nu}{2}, \frac{1}{2}\right) & \text{if } x \geq 0 \\ | ||
0 & \text{if } x < 0 | ||
\end{cases} | ||
$$ | ||
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where $I_x(a, b)$ denotes the [regularized incomplete beta function](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function). | ||
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```{seealso} | ||
:class: seealso | ||
**Common Alternatives:** | ||
- [Half-Cauchy](halfcauchy.md) - The Half-Cauchy distribution is a special case of the Half-Student's t distribution with $\nu = 1$. | ||
- [Half-Normal](halfnormal.md) - As $\nu \to \infty$, the Half-Student's t distribution approaches the Half-Normal distribution. | ||
**Related Distributions:** | ||
- [Student's t](studentt.md) - The Student's t distribution is the parent distribution from which the Half-Student's t distribution is derived. | ||
``` | ||
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## References | ||
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- [Wikipedia - Folded-t and Half-t Distributions](https://en.wikipedia.org/wiki/Folded-t_and_half-t_distributions) | ||
- [Wikipedia - Student's t-distribution](https://en.wikipedia.org/wiki/Student%27s_t-distribution) |
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"Exponential", | ||
"HalfCauchy", | ||
"HalfNormal", | ||
"HalfStudentT", | ||
"Hurdle", | ||
"InverseGamma", | ||
"Logistic", | ||
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