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MCMC diagnostics

TL;DR

MENoBiS microcanonical MCMC routes use kernels with exactly the target distribution as their stationary law. Finite runs still require burn-in and mixing assessment: the stationary law guarantees the limit, not the finite-run quality. Inspect acceptance, autocorrelation, and effective sample size (ESS) per statistic.

Exact target

A kernel is exact stationary MCMC when its stationary distribution is exactly the target constrained measure. Sample exactness is a property of the kernel, not of how the initial state was found. See Validation for how exactness is verified.

Initialization

The constructor provides one feasible state on the constraint fiber. Initialization:

  • must satisfy the hard constraints;
  • need not satisfy detailed balance;
  • contributes initial-state bias that burn-in removes.

Burn-in

Burn-in discards early draws while the chain forgets the initial state. Its required length depends on the route and instance; there is no universal "more sweeps is enough" rule — measure the diagnostics below.

Mixing

Mixing describes how fast the chain explores the target state space. Fast-mixing chains visit diverse configurations quickly; slow mixing shows up as high autocorrelation and small ESS.

Autocorrelation

For a statistic \(g\) evaluated on chain states \(t_1,\ldots,t_R\), the lag- \(\tau\) autocorrelation is

\[ \rho_g(\tau) = \operatorname{Corr}(g(t_r),g(t_{r+\tau})). \]

Effective sample size

\[ ESS_g \approx \frac{R} {1+2\sum_{\tau\ge 1}\rho_g(\tau)}. \]

ESS is statistic-specific: a statistic that is insensitive to the slow directions of the chain can have a larger ESS than one that probes them. Estimate tails of \(g\) only with enough effective samples (see Ensemble statistics).

Diagnostics to monitor

  • acceptance rate (too low → wasteful proposals; too high → small moves);
  • effective movement rate;
  • support change rate (how often the occupied-pair set changes);
  • trace returns for trace-based kernels;
  • cost ESS for cost-influenced routes;
  • repeated-chain agreement (independent chains, same seed offset);
  • autocorrelation of the high-level metrics you will report.

A note on convergence language

These kernels have exactly the target distribution as their stationary law, so "MCMC does not converge" is inaccurate. The accurate concern is finite- run mixing in tight or heterogeneous fibers:

The kernel has the correct stationary target, but finite-run mixing can be slow in tight or heterogeneous fibers.

Distinguish throughout: deterministic/iterative fit convergence, stochastic gamma-fit convergence, stationarity, burn-in, mixing, and Monte Carlo error — they are different things (see Validation).