베이지안에서 많이 사용하는 분포

Univariate	pdf	Multivariate	pdf
Binomial$(n,p)$	$\binom{n}{x}p^x(1-p)^{n-x}$	Multinomial$(n,p_1,\cdots,p_k)$	$\frac{n!}{x_1!\cdots x_k!}p_1^{x_1}\cdots p_k^{x_k}$, $\sum x_i = n$
Normal$(\mu,\sigma^2)$	$(2\pi\sigma^2)^{-1/2}\exp\left\{-\frac{(x-\mu)^2}{2\sigma^2}\right\}$	Multivariate Normal$(\mu, \Sigma)$	$(2\pi)^{-\frac{k}{2}}|\Sigma|^{-\frac{1}{2}}\exp\left\{(x-\mu)'\Sigma^{-1}(x-\mu)\right\}$
Gamma$(\alpha,\beta)$	$\frac{x^{\alpha-1}e^{-x/\beta}}{\Gamma(\alpha)\beta^\alpha}$	Wishart$(n,\Sigma)$
Beta$(\alpha, \beta)$	$\frac{x^{\alpha-1}(1-x)^{\beta-1}}{\text{B}(\alpha,\beta)}$	Dirichlet$(\alpha_1,\cdots,\alpha_{k+1})$

• Exponential$(\beta)$: Gamma$(\alpha=1, \beta)$
• Chi-squared$(k)$: Gamma$(\alpha=k/2, \beta=1/2)$
• Uniform$(0,1)$: Beta$(\alpha=1, \beta=1)$

역감마분포 Inverse-gamma distribution

$1/X$가 Gamma 분포를 따르면 $X$는 Inverse-gamma 분포를 따른다고 한다.
\[
f(x) = \frac{\beta^\alpha)}{\Gamma(\alpha)} x^{-(\alpha+1)} e^{-\beta/x}
\]

디리클레 분포 Dirichlet distribution

디리클레 분포의 pdf:
\[
f(x_1,\cdots,x_{k};\alpha_1,\cdots,\alpha_{k+1}) = \frac{\Gamma(\sum_{i=1}^{k+1}\alpha_i)}{\prod_{i=1}^{k+1}\Gamma(\alpha_i)}\prod_{i=1}^{k+1} x_i^{\alpha_i -1 }
\]

위샤트 분포 Wishart distribution

다변량정규분포를 따르는 $n$개의 iid 확률벡터 $X_1,\cdots,X_n \stackrel{iid}{\sim} N(0,\Sigma)$가 있을 때
\[
\sum_{i=1}^n X_iX_i^T \sim \text{Wishart}(n,\Sigma)
\]
이다.

역위샤트 분포 Inverse-Wishart distribution