Metropolis–Hastings and detailed balance

At current state $x$ with positive target density, draw a candidate $z$ from proposal density $q(z\mid x)$. The acceptance probability compares target-times-proposal flow in the two directions. A symmetric proposal makes the proposal factors cancel; an asymmetric proposal generally does not. If a proposed move has positive forward flow but zero reverse target-proposal flow, its acceptance is zero. Work with log ratios for numerical stability, comparing a log-uniform random draw with the smaller of zero and the log acceptance ratio.

Rejection means recording the current state again. Those repeated states are part of the transition kernel and of averages taken over iteration time. Discarding all rejected iterations usually changes the distribution of the retained jump sequence toward states with higher departure rates, unless a separate correction is applied. Repeated states are therefore neither missing values nor harmless duplicates to delete. Their frequency can reveal poor proposals, but a high acceptance rate alone is also insufficient: tiny steps can be accepted almost always while exploring the target extremely slowly.

For distinct states, the accepted probability flow is the minimum of the two directional target-proposal flows. This expression is symmetric when the directions are swapped, establishing detailed balance. Summing or integrating that identity, together with the self-transition created by rejection, establishes target invariance. Detailed balance is sufficient for invariance, not necessary: nonreversible samplers can also preserve a target. The derivation assumes a legitimate proposal kernel and a proper normalized target exists behind the unnormalized density.

In a finite state space, irreducibility makes the invariant probability unique. Adding aperiodicity ensures convergence of the time-n marginal distribution from any start. Invariance alone supplies neither condition: a chain that never moves preserves every distribution, while deterministic alternation preserves a balanced distribution yet oscillates forever from a fixed start. For general continuous state spaces, appropriate recurrence and irreducibility conditions replace the simple finite argument. Do not transfer a finite-state convergence proof to an arbitrary posterior without checking its hypotheses.

Even a valid ergodic kernel can remain near one mode for a very long time. Run multiple chains from dispersed valid starting points, inspect traces and between-chain agreement, and estimate Monte Carlo error for the quantities actually needed. Warmup can tune proposal parameters, but a proof for one fixed kernel does not automatically justify arbitrary ongoing adaptation. Standard workflows adapt during warmup and then fix tuning for retained draws; more general adaptive methods need their own conditions.

Separate uncertainty in the posterior from error in estimating its summaries. A narrow posterior interval can be computed poorly, and many correlated draws can contain less information about a mean than a much smaller independent sample. Autocorrelation-based effective sample size refers to a particular function or summary and is different from importance-weight ESS. Diagnostics can reveal trouble but cannot prove that an unseen mode does not exist. Analytical special cases, alternative parameterizations, and independent implementations provide useful additional checks of both the model and the sampler.