MCMC & Detailed Balance

Markov Chain Monte Carlo (MCMC) builds a Markov chain whose stationary distribution is the target $\pi$ and samples along the chain. Metropolis-Hastings draws a candidate $j$ from a proposal $q(j \mid i)$ each step and accepts it with rate

otherwise staying at $i$. Writing $r = \pi(j) q(i \mid j) / (\pi(i) q(j \mid i))$: when $r \ge 1$, $\alpha(i,j) = 1$ and the reverse acceptance is $1/r$, so pointwise substitution yields the detailed balance of the MH chain, $\pi(i) q(j \mid i) \alpha(i,j) = \pi(j) q(i \mid j) \alpha(j,i)$, hence $\pi$ is stationary; moreover $\pi$ only needs to be known up to a constant (it cancels in the ratio). Gibbs sampling is a special case: each step updates one coordinate $x_k$ by drawing from the conditional $\pi(x_k \mid x_{\setminus k})$, whose definition forces the acceptance rate to be 1 — no rejection needed.

Bayesian posterior inference, latent-variable models (LDA, Gaussian mixtures), sampling high-dimensional joint distributions; interviews often ask to derive the MH acceptance rate and relate Gibbs to MH.

rejection sampling collapses as $1/M$ and importance weights explode in high dimension, while MCMC trades correlated samples for a positive acceptance rate — the only general-purpose high-dimensional sampler; the cost is autocorrelation: burn-in must discard the non-stationary head of the chain, and convergence is diagnosed via acceptance rate (rule of thumb 0.2–0.5) and the Gelman-Rubin $\hat R$ statistic.

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