#StackBounty: #bayesian #kullback-leibler #variational-bayes #approximate-inference #variational Estimating the posterior distributions…

Bounty: 100

In the following graphical model

enter image description here

the generative model of the mixture of the gamma distribution is given as
begin{equation}
begin{split}
p(z|pi)&=prod_{k=1}^Kpi_k^{z_k}\
p(gamma|z)&=prod_{k=1}^Kmathrm{Gamma}(gamma|a,b)^{z_k}
end{split}
end{equation}

so the joint distribution of $boldsymbol{gamma}$ and $mathrm{z}$ is
$$p(boldsymbol{gamma},mathrm{z}|pi,a,b)=prod_{i=1}^Nprod_{k=1}^Kpi_k^{z_{ik}}mathrm{Gamma}(gamma_i|a,b)^{z_{ik}}$$
How can I use variational message passing to compute messages from $pi$, $a$, $b$ and $z$ ?


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