Bossvercell

illustration

- LDA总结一共轭分布.

Further conjugate priors may give intuition by more transparently showing how a likelihood function updates a prior distribution.

. Otherwise numerical integration may be necessary.

A conjugate prior is an algebraic convenience giving a closed-form expression for the posterior.

$\begin{aligned} \pi(\theta \mid t) &=\frac{\left(\begin{array}{c} n \\ t \end{array}\right) \theta^{t}(1-\theta)^{n-t} \cdot 1}{\int_{0}^{1}\left(\begin{array}{c} n \\ t \end{array}\right) \theta^{t}(1-\theta)^{n-t} \cdot 1 d \theta}=\frac{\theta^{t}(1-\theta)^{n-t}}{\int_{0}^{1} \theta^{t}(1-\theta)^{n-t} d \theta} \\ &=\frac{1}{B(t+1, n-t+1)} \theta^{t}(1-\theta)^{n-t} \\ &=\frac{\Gamma(n+2)}{\Gamma(t+1) \Gamma(n-t+1)} \theta^{(t+1)-1}(1-\theta)^{(n-t+1)-1} \\ & \sim \text { beta }(t+1, n-t+1) \end{aligned}\\$

$\begin{array}{|ccc|} \hline \text { 总体分布 } & \text { 参数 } & \text { 共轩先验分布 } \\ \hline \text { 二项分布 } & \text { 成功概率 } & \text { beta分布 } \\ \text { 泊松分布 } & \text { 均值 } & \text { gamma分布 } \\ \text { 指数分布 } & \text { 均值 } & \text { 倒gamma分布 } \\ \text { 指数分布 } & \text { 均值倒数 } & \text { gamma分布 } \\ \text { 正态分布(方差已知) } & \text { 均值 } & \text { 正态分布 } \\ \text { 正态分布(均值已知) } & \text { 方差 } & \text { 倒gamma分布 } \\ \hline \end{array}\\$

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