MCMC and Metropolis-Hastings

MCMC builds a Markov chain whose stationary distribution is the posterior \(p(X \given y)\). We estimate any \(\mathbb{E}[r(X)]\) by averaging \(r\) along the chain, using only ratios of the unnormalized target \(p(y, X)\), never the evidence \(p(y)\).

Metropolis-Hastings is one recipe for such a chain. From the current state \(X\):

1. propose \(X' \sim q(X' \given X)\);
2. accept (\(X \leftarrow X'\)) with probability \(\min(1, \alpha)\), otherwise keep \(X\), where \[\alpha = \frac{p(X' \given y)\, q(X \given X')}{p(X \given y)\, q(X' \given X)}.\]

Question 1

In the likelihood-weighting activity you measured the ESS with a prior sitting far from the data:

(let [mu (sample (normal 0 1))] (observe (normal mu 1) 8.0) mu)

There run(program, seed) returns (value, log_w) with \(\log W = \log p(y \given X)\) (only observe contributes, since the prior is the proposal).

In the same notebook, implement independent MH and run it here. The chain barely moves: explain why, in terms of what MH does with a proposal that likelihood weighting would merely down-weight.

Independent MH

def independent_mh(program, S=20000, seed=0):
    rng = np.random.default_rng(seed)
    x, logw = run(program, seed=0)        # current run
    chain = []
    accept = 0
    for i in range(S):
        x2, logw2 = run(program, seed=i+1)  # propose a fresh run
        if np.log(rng.random()) < logw2 - logw:                  # accept w.p. min(1, W'/W)
            x, logw = x2, logw2
            accept += 1
        chain.append(x)
    print("acceptance rate:", accept / S)
    return np.array(chain)

• Proposing from the prior cancels prior against proposal: \(\alpha = p(y \given X')/p(y \given X) = W'/W\), already computed by run.