# #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?
begin{align}
alpha = frac{P(x’)}{P(x^{(0)})} = frac{0.4 cdot ldots}{0.6 cdot ldots} = z
end{align
}
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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