#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 $$q(theta)=mathrm{Cat}(beta)$$?

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