| \text{Bernoulli}(p) |
\text{Beta}(\alpha,\beta) |
{\displaystyle \text{Beta}\left( \alpha +\sum _{i=1}^{n}x_{i},\,\beta +n-\sum _{i=1}^{n}x_{i}\right)} |
{\displaystyle \mathbb{P}r({\tilde {x}}=1)={\frac {\alpha '}{\alpha '+\beta '}}} |
| \text{Binomial}(trials=m,p) |
\text{Beta}(\alpha,\beta) |
{\displaystyle \text{Beta}\left(\alpha +\sum _{i=1}^{n}x_{i},\,\beta +\sum _{i=1}^{n}N_{i}-\sum _{i=1}^{n}x_{i}\right)} |
{\displaystyle \operatorname {BetaBin} ({\tilde {x}}|\alpha ',\beta ')} |
| \text{NegBinomial}(fails=r) |
\text{Beta}(\alpha,\beta) |
{\displaystyle \text{Beta}\left( \alpha +rn,\beta + \sum _{i=1}^{n} x_{i}\right)} |
{\displaystyle \operatorname {BetaNegBin} ({\tilde {x}}|\alpha ',\beta ')} |
| \text{Poisson}(rate=\lambda) |
\text{Gamma}(k,\theta) |
{\displaystyle \text{Gamma}\left( k+\sum _{i=1}^{n}x_{i},\ {\frac {\theta }{n\theta +1}}\!\right)} |
{\displaystyle \operatorname {NB} \left({\tilde {x}}\mid k',{\frac {1}{\theta '+1}}\right)} |
| \text{Poisson}(rate=\lambda) |
\text{Gamma}(\alpha,\beta) |
{\displaystyle\text{Gamma}\left( \alpha +\sum _{i=1}^{n}x_{i},\ \beta +n\!\right)} |
{\displaystyle \operatorname {NB} \left({\tilde {x}}\mid \alpha ',{\frac {\beta '}{1+\beta '}}\right)} |
| \text{Categorical}(probs=p,cats=k) |
\text{Dir}(\alpha_k)\mathbb{I}_{k\ge1} |
{\displaystyle \text{Dir}\left({ {\boldsymbol {\alpha }}+(c_{1},\ldots ,c_{k})}\right)} |
{\displaystyle {\begin{aligned}\mathbb{P}r({\tilde {x}}=i)&={\frac {{\alpha _{i}}'}{\sum _{i}{\alpha _{i}}'}} \\ &={\frac {\alpha _{i}+c_{i}}{\sum _{i}\alpha _{i}+n}}\end{aligned}}} |
| \text{Multinomial}(probs=p,cats=k) |
\text{Dir}(\alpha_k)\mathbb{I}_{k\ge1} |
{\displaystyle \text{Dir}\left({ {\boldsymbol {\alpha }}+\sum _{i=1}^{n}\mathbf {x} _{i}\!}\right)} |
{\displaystyle \operatorname {DirMult} ({\tilde {\mathbf {x} }}\mid {\boldsymbol {\alpha }}')} |
| \text{Hypergeometric}(pop=n) |
\text{BetaBinomial}(\alpha,\beta,n=N) |
{\displaystyle \text{BetaBinomial}\left({\displaystyle \alpha +\sum _{i=1}^{n}x_{i},\,\beta +\sum _{i=1}^{n}N_{i}-\sum _{i=1}^{n}x_{i}}\right)} |
|
| \text{Geometric}(p) |
\text{Beta}(\alpha,\beta) |
{\displaystyle\text{Beta}\left( \alpha +n,\,\beta +\sum _{i=1}^{n}x_{i}\right)} |
|