#StackBounty: #expected-value #gamma-distribution #variational-bayes #dirichlet-distribution computing expected value of Dirichlet dist…

Bounty: 150

If one defines a Dirichlet distribution as following
$$pisimmathrm{Dir}(alpha G_0)$$
where $alpha$ is a scalar and $G_0$ is a base distribution which is defined to be a Categorical distribution. The hyper prior over $alpha$ is a Gamma distribution
begin{align}
G_0=Theta&simmathrm{Cat}(boldsymbol{theta}|mathbf{C},boldsymbol{s})=prod_{i=1}^{K}prod_{j=1}^{N}c_{ij}^{theta_i s_j}\
alpha&simmathrm{Gamma}(a,b)
end{align}

in order to do inference using variational Bayes we should compute the following
$$q(pi)proptoexpBig(mathbb{E}_{sim q(pi)}big[log P(X,lambda)big]Big)$$
where $X$ is the observed data and $lambda$ is all set of unknown variables including $pi$, $alpha$ and $theta$.

My question is how can I compute $mathbb{E}{ q(alpha)q(theta)}big[log P(pi|alpha,theta)big]$? It will be
begin{equation}
langlelogbig(frac{Gamma(alphaoverbrace{sum_ktheta_k}^{1})}{prod_kGamma(alphatheta_k)}prod_kpi_k^{alphatheta_k-1}big)rangle
{q(alpha)q(theta)}
end{equation}
I assume that the approximate distributions for $alpha$ and $theta$ are $q(alpha)=mathrm{Gamma}(m,n)$ and $q(theta)=mathrm{Cat}(beta)$?


Get this bounty!!!

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