Distributions
James Totterdell
2021-03-24
Source:vignettes/web_only/99-distributions.Rmd
99-distributions.RmdIntroduction
This document provides a reference for relevant terms for useful distributions and their relationships in the context of variational inference.
Gamma and Inverse-Gamma
Gamma
\[ \begin{aligned} y &\sim \text{Gamma}(\alpha, \beta) \\ f(y|\alpha,\beta) &= \frac{\beta^\alpha}{\Gamma(\alpha)}e^{-y\beta}y^{\alpha-1},\quad y,\alpha,\beta\in\mathbb R^+ \\ \ln f(y|\alpha,\beta) &= \alpha\ln\beta-\ln\Gamma(\alpha)-y\beta+(\alpha-1)\ln y\\ \mathbb E[y] &= \alpha\beta \\ \mathbb V[y] &= \alpha\beta^2 \\ \mathbb H[y] &= \alpha-\ln\beta+\ln\Gamma(a)+(1-\alpha)\psi(\alpha) \\ \mathbb E[\ln y] &= \psi(\alpha)-\ln(\beta) \end{aligned} \]
Inverse-Gamma
If then \(y^{-1}\sim\text{Inverse-Gamma}(\alpha,\beta)\).
\[ \begin{aligned} y &\sim \text{Inverse-Gamma}(\alpha,\beta) \\ f(y|\alpha,\beta) &= \frac{\beta^\alpha}{\Gamma(\alpha)}e^{-y/\beta}y^{-\alpha-1},\quad y,\alpha,\beta\in\mathbb R^+ \\ \ln f(y|\alpha,\beta) &= \alpha\ln\beta-\ln\Gamma(\alpha)-y/\beta-(\alpha+1)\ln(y) \\ \mathbb E[y] &= \frac{\beta}{\alpha-1},\quad \alpha>1 \\ \mathbb E[y^{-1}] &= \frac{\alpha}{\beta} \\ \mathbb V[y] &= \frac{\beta^2}{(\alpha-1)^2(\alpha-2)},\quad \alpha>2 \\ \mathbb H[y] &= \alpha+\ln\beta+\ln\Gamma(a)-(1+\alpha)\psi(\alpha) \\ \mathbb E[\ln y] &= \ln\beta-\psi(\alpha) \end{aligned} \]
Relations
\[ \begin{align} Y &\sim \text{Gamma}(\alpha,\beta) &\iff Y &\sim \text{Inv-Gamma}(\alpha, \beta) \\ Y &\sim \text{Gamma}(\nu/2,1/2) &\iff Y &\sim \text{Chi-square}(\nu)\\ Y &\sim \text{Inv-Gamma}(\alpha,1/2) &\iff Y &\sim \text{Inv-Chi-square}(2\alpha) \\ Y &\sim \text{Inv-Gamma}(\alpha, \beta) &\iff Y&\sim \text{Inv-Wishart}(2\alpha,2\beta) \end{align} \]
\[ \begin{aligned} Y^2|X&\sim\text{Inv-Gamma}(\nu/2,\nu/X)\\ X&\sim\text{Inv-Gamma}(1/2,1/A^2) \\ \implies Y&\sim\text{Half-}t(\nu,A) \\ Y^2|X&\sim\text{Inv-Gamma}(1/2,1/X)\\ X&\sim\text{Inv-Gamma}(1/2,1/A^2) \\ \implies Y&\sim\text{Half-Cauchy}(A) \end{aligned} \]
Chi-squared, Inverse-Chi-squared, and Scaled-Chi-squared
Chi-squared
\[ \begin{aligned} y &\sim \text{Chi-squared}(\nu) \\ f(y|\nu) &= \frac{1}{2^{n/2}\Gamma(n/2)}e^{-\nu/2}y^{n/2-1},\quad y,\nu\in\mathbb R^+ \\ \ln f(y|\nu) &= -(n/2)\ln(2)-\ln\Gamma(n/2)-\nu/2+(n/2-1)\ln(y)\\ \mathbb E[y] &= \nu \\ \mathbb V[y] &= 2\nu \\ \mathbb H[y] &= \psi(n/2) + \ln(2) \end{aligned} \]
Inverse-Chi-squared
\[ \begin{aligned} y &\sim \text{Inverse-Chi-squared}(\nu) \\ f(y|\nu) &= \frac{2^{-\nu/2}}{\Gamma(\nu/2)}y^{-\nu/2-1}e^{-1/(2y)},\quad y,\nu\in\mathbb R^{+} \\ \ln f(y|\nu) &= -(\nu/2)\ln(2)-\ln\Gamma(\nu/2)-(\nu/2+1)\ln(y)-1/(2y) \\ \mathbb E[y] &= \frac{1}{\nu-2},\quad\nu>2\\ \mathbb V[y] &= \frac{2}{(\nu-2)^2(\nu-4)},\quad \nu>4 \\ \mathbb H[y] &= \nu/2+\ln\left(\frac{\nu}{2}\Gamma(\nu/2)\right)-(\nu/2+1)\psi(\nu/2) \end{aligned} \]
Scale-inverse-Chi-squared
\[ \begin{aligned} y &\sim \text{Scaled-inverse-Chi-squared}(\nu,\tau^2) \\ f(y|\nu,\tau^2) &= \frac{(\tau^2\nu/2)^{\nu/2}}{\Gamma(\nu/2)} \frac{\exp\left(\frac{-\nu\tau^2}{2y}\right)}{y^{1+\nu/2}} \\ \ln f(y,\tau^2) &= (\nu/2)\ln(\nu\tau^2/2)-\ln\Gamma(\nu/2)-\frac{\nu\tau^2}{2y}-(1+\nu/2)\ln(y) \\ \mathbb E[y] &= \frac{\nu\tau^2}{\nu-2},\quad \nu>2\\ \mathbb V[y] &= \frac{2\nu^2\tau^4}{(\nu-2)^2(\nu-4)},\quad \nu>4\\ \mathbb H[y] &= \nu/2+\ln\left(\frac{\nu\tau^2}{2}\Gamma(\nu/2)\right)-(1+\nu/2)\psi(\nu/2) \end{aligned} \]
Wishart, Inverse-Wishart, G-Wishart
Wishart
\[ \begin{aligned} \Sigma &\sim \text{Wishart}_d(\xi,\Sigma) \\ f(\Sigma|\xi,\lambda) &= \frac{1}{2^{\xi d/2}|\Lambda|^{\xi/2}\Gamma_d(\xi/2)}|\Sigma|^{(\xi-d-1)/2}e^{-\text{tr}(\Lambda^{-1}\Sigma)/2},\quad \xi>d-1,\Lambda>0 \\ \ln f(\Sigma|\xi,\lambda) &= -(\xi d/2)-(\xi/2)\ln|\Lambda|-\ln\Gamma_d(\xi/2)+(\xi-d-1)/2\ln|\Sigma|-\text{tr}(\Lambda^{-1}\Sigma)/2\\ \mathbb E[\Sigma] &= \xi\Lambda \\ \mathbb V[\Sigma]_{ij} &= \\ \mathbb H[\Sigma] &= \frac{d+1}{2}\ln|\Lambda|+\frac{d(d+1)}{2}\ln(2)+\ln\Gamma_d(\xi/2)-\frac{\xi-d-1}{2}\psi_d(\xi/2)+\frac{\xi d}{2} \\ \mathbb E[\ln|\Sigma|] &= \psi_d(\xi/2)+d\ln(2)+\ln|\Lambda| \\ \end{aligned} \]
Inverse-Wishart
If \(\Sigma\sim\text{Wishart}_d(\xi,\Lambda)\) then \(\Sigma^{-1}\sim\text{Inverse-Wishart}_d(\xi,\Lambda^{-1})\)
\[ \begin{aligned} \Sigma &\sim \text{Inverse-Wishart}_d(\xi,\Sigma) \\ f(\Sigma|\xi,\Lambda) &= \frac{|\Lambda|^{\xi/2}}{2^{\xi d/2}\Gamma_d(\xi/2)}|\Sigma|^{-(\xi+d+1)/2}e^{-\text{tr}(\Lambda\Sigma^{-1})/2},\quad \xi>d-1,\Lambda>0 \\ \ln f(\Sigma|\xi,\Lambda) &= \xi/2\ln|\Lambda|-(\xi d)/2\ln(2)-\ln\Gamma_d(\xi/2)-(\xi+d+1)/2\ln|\Sigma|-\text{tr}(\Lambda\Sigma^{-1})\\ \mathbb E[\Sigma] &= \frac{\Lambda}{\xi-d-1} \\ \mathbb E[\Sigma^{-1}] &= \xi\Lambda^{-1} \\ \mathbb V[\Sigma]_{ij} &= \frac{(\xi-d+1)\lambda_{ij}^2+(\xi-d-1)\lambda_{ii}\lambda_{jj}}{(\xi-d)(\xi-d-1)^2(\xi-d-3)} \\ \mathbb H[\Sigma] &= -\frac{\xi}{2}\ln|\Lambda|+(\xi+d+1)\mathbb E[\ln|\Sigma|]+\frac{\xi d}{2}\ln(2) +\ln\Gamma_d(\xi/2) + \frac{\xi d}{2}\\ \mathbb E[\ln|\Sigma|] &= \ln|\tfrac{1}{2}\Lambda|-\psi_d(\xi-d+1) \end{aligned} \]
Relations
\[ \begin{aligned} \text{Wishart}_1(\xi,\Lambda) &\sim \text{Gamma}(\xi/2,\Lambda/2) \\ \text{inv-Wishart}_1(\xi,\Lambda) &\sim \text{Inv-Gamma}(\xi/2,\Lambda/2) \end{aligned} \]
\[ \begin{aligned} \Sigma|X_1,...,X_p&\sim\text{Inv-Wishart}_p(\nu+p-1,2\nu\text{diag}(1/X_1,...,1/X_p))\\ X_j&\overset{\text{ind}}{\sim}\text{Inv-Gamma}(1/2,1/A_j^2) \\ \implies \sigma_{j}&\sim\text{Half-}t(\nu,A_j) \\ \rho_{ij} &\propto (1-\rho_{ij}^2)^{\nu/2-1} \end{aligned} \] where \(\Sigma_{ij} = \rho_{ij}\sigma_i\sigma_j\)
Identities and Definitions
\[ \begin{aligned} \Gamma_d(x) &= \text{multivariate gamma function}(x) \\ &= \pi^{d(d-1)/4}\prod_{j=1}^d \Gamma\left[x+(1-j)/2\right]\\ \psi_d(x) &= \text{multivariate digamma function}(x) \\ &= \sum_{j=1}^d \psi\left[x+(1-j)/2\right] \end{aligned} \]
\[ \left(\text{bdiag}(A_1,...,A_p)\right)^{-1}=\text{bdiag}(A_1^{-1},...,A_p^{-1}) \]
\[ \mathbb E[\text{tr}(AX)] = \text{tr}(A\mathbb E[X]) \]