#StackBounty: #sampling #mcmc #bayesian-network #metropolis-hastings Metropolis sampling for Bayesian networks

Bounty: 100

Gibbs sampling is a profound and popular technique for creating samples of Bayesian networks (BNs). Metropolis sampling is another popular technique, though – in my opinion – a less accessible method. Since concepts as transfer probabilities of your Markov chain and a (symmetric) proposal distribution $g$ appear, I have difficulties with understanding Metropolis sampling in the context of BNs correctly.

My approach:

Suppose we deal with the below Student BN structure and corresponding conditional probability tables (CPTs). For the topological ordering $(D, I, G, S, L)$, consider the initial state $x^{(0)} = (D = d^1, I = I^0, G = g^1, S = s^0, L = l^0)$. Based on $x^{(0)}$, if we consider $g$ to be a random walk proposal, which new states $x’$ can we consider?

If we change one variable (like Gibbs sampling) according to the fixed ordening at a time, we could obtain $x’ = (D = d^0, I = I^0, G = g^1, S = s^0, L = l^0)$. Then, can we determine the acceptance ratio $alpha$ according to the CPTs?
alpha = frac{P(x’)}{P(x^{(0)})} = frac{0.4 cdot ldots}{0.6 cdot ldots} = z
Consecutively, with the help of a random integer $u$, we accept $x’$ as $x^{(1)}$ if $u > z$. We repeat this procedure.

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