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An autocorrelation time measures how long an observable retains dependence on its past. It depends on both the process and the chosen observable. For a stationary sequence with finite positive variance and autocorrelation function $\rho (k)$, the integrated autocorrelation time is ${\tau }_{\mathrm{int}}=1+2{\sum }_{k\ge 1}\rho (k)$ when the series is absolutely summable. It controls the leading variance of a long sample mean.

An exponential autocorrelation time instead describes tail decay, for example $\rho (k)\approx C{e}^{-k/{\tau }_{\exp }}$. These are different summaries. A stationary autoregressive process of order one with coefficient $0<a<1$ has $\rho (k)=a^k$, so ${\tau }_{\mathrm{int}}=(1+a)/(1-a)$ and ${\tau }_{\exp }=-1/\log a$. For a Markov chain with several decay modes, different observables can emphasize different timescales.

For a stationary Markov chain observable $h$ with positive variance $v$ and absolutely summable autocorrelation function $\rho (k)$,If the limiting integrated autocorrelation time ${\tau }_{\mathrm{int}}>0$, the leading variance is $v{\tau }_{\mathrm{int}}/N$. Matching this to the sample-mean variance $v/N_{\mathrm{eff}}$ of independent observations defines $N_{\mathrm{eff}}\approx N/{\tau }_{\mathrm{int}}$.

This is an observable-specific variance comparison, not a literal count of independent retained states. Positive serial dependence reduces effective sample size; negative dependence can make it exceed $N$. For a stationary AR(1) observable with coefficient $-1/2$, ${\tau }_{\mathrm{int}}=1/3$ and the asymptotic effective sample size is $3N$. Estimated effective sample size also requires reliable estimation of the long-run covariance; it does not prove that all modes have been explored.

Markov chain Monte Carlo estimates expectations under a target probability distribution $\pi$ by simulating a Markov chain that preserves $\pi$. For an integrable observable $h$, the assumptions of a Markov-chain law of large numbers justifyalmost surely. A Gibbs sampler preserves the target through full conditional distributions; a Metropolis–Hastings algorithm proposes a move and accepts or rejects it using a ratio of target and proposal densities. The unknown normalization constant cancels from that ratio, which makes unnormalized Bayesian posterior densities usable.

Successive states are generally dependent, so Monte Carlo precision depends on the observable's integrated autocorrelation time, rather than the raw iteration count alone. Burn-in can reduce initial-state effects but is not a proof of convergence. A correct stationary distribution also does not ensure rapid exploration: isolated modes or strong dependence can produce a long mixing time. Markov chain Monte Carlo convergence diagnostics investigate these limitations.

Tune the proposal scale using pilot runs on a grid of candidate ${\sigma }_q$ values. After allowing for initial transients, compare the effective sample size of a Markov chain per unit computation time for representative functions, such as both coordinates and a quadratic function. Equivalently inspect their integrated autocorrelation time, using $\mathrm{ESS}\approx n/(1+2{\sum }_{k\ge 1}\mathrm{Corr}(h(Z_0),h(Z_k)))$. Choose a useful scale and keep it fixed for the production run. Another pilot criterion is the average squared accepted displacement, possibly in coordinates standardized by the target covariance, which balances move size and rejection.

Acceptance alone is not an efficiency criterion. As ${\sigma }_q\downarrow 0$, proposals are almost the current state, so acceptance tends to one while exploration becomes arbitrarily slow. Very large scales propose mostly low-density states and produce long sequences of identical values. The proposal scale and random-walk Metropolis efficiency trade-off requires both movement and acceptance; neither maximizing nor merely reporting the acceptance probability guarantees a useful sample. With strong correlation, an isotropic proposal also struggles with the target's narrow direction; covariance-scaled proposals can improve this geometry beyond tuning one scalar variance.

A convenient general-state-space ergodic theorem is the following. For a positive Harris recurrent Markov chain with invariant probability measure $\pi$, and a measurable function $h$ with $\pi |h|<\infty$,Here $\pi h=\int hd\pi$ is the stationary expected value. Harris recurrence means that every set of positive irreducibility measure is visited almost surely from every state; positive recurrence supplies an invariant probability rather than only an infinite invariant measure. The ergodic theorem for a positive Harris recurrent Markov chain holds from any starting state under these Harris hypotheses. aperiodicity is not necessary just for averages.

A useful sufficient form of the central limit theorem for a geometrically ergodic Markov chain adds aperiodicity, geometric ergodicity, and $\pi (|h{|}^{2+\delta })<\infty$ for some $\delta >0$. It givesThese are sufficient hypotheses, not a claim that irreducibility alone ensures a central limit theorem. The Markov chain Monte Carlo asymptotic variance uses stationary covariances ${\gamma }_h(k)={\mathrm{Cov}}_{\pi }(h(X_0),h(X_k))$, with ${\gamma }_h(0)={\mathrm{Var}}_{\pi }h$. The series is absolutely convergent under the stated sufficient assumptions. If ${\gamma }_h(0)>0$, write ${\rho }_h(k)={\gamma }_h(k)/{\gamma }_h(0)$ and $v_h={\gamma }_h(0){\tau }_{\mathrm{int}}$, where the integrated autocorrelation time is ${\tau }_{\mathrm{int}}=1+2{\sum }_{k\ge 1}{\rho }_h(k)$. When $v_h>0$, the effective sample size of a Markov chain is approximately $n/{\tau }_{\mathrm{int}}$. Negative correlations can reduce the asymptotic variance; a zero asymptotic variance gives a degenerate normal limit. For constant $h$, variance is zero and the autocorrelation normalization is undefined.

A log-flat improper prior $p(v)\propto 1/v$ is not appropriate when all known $r_s={\sigma }_{m,s}^2$ are positive: the integrated likelihood in (iii) has a positive finite limit as $v\downarrow 0$, so its integral against $dv/v$ diverges. One defensible noninformative choice is the scalar Jeffreys prior for an additive variance component, calculated holding the mean parameters fixed:Indeed the normal variance score has information $I_{vv}=\frac{1}{2}{\sum }_s(v+r_s{)}^{-2}$. This is a scalar conditional-information prior, not a claim that its product with the flat mean prior is the joint Jeffreys prior. It is bounded near zero and is $O(v^{-1})$ at infinity. The integrated likelihood is uniformly bounded near zero and is bounded by a constant times $v^{-(N-1)/2}$ at infinity, independently of the mean-shape parameters, since $e^{-Q/2}\le 1$. Thus $N>1$ and a proper external prior on the physically admissible $(w,{\Omega }_M)$ ensure posterior propriety. Proper weak scale priors are another option. If known errors vanish, the boundary argument and appropriate prior need separate reconsideration.

For an efficient parameterization, use $\theta =(\beta ,u,w,{\Omega }_M)$ with $u=\log v$ and $\beta =M_0+h(H_0)$. A chain on this joint reduced parameter space targetsThe factor $e^u$ is the Jacobian determinant. At each iteration propose ${\theta }^{\prime }=\theta +\epsilon$, with $\epsilon \sim N(0,{\Sigma }_q)$ for a fixed nonsingular proposal covariance, and accept with probability $\min \{1,\tilde{\pi }({\theta }^{\prime })/\tilde{\pi }(\theta )\}$. Proposals outside the physical prior domain have target zero. Tune ${\Sigma }_q$ during warmup and freeze it for the retained random-walk Metropolis algorithm. To obtain samples of the full original parameter vector, independently draw $H_0$ from its retained positive prior and reconstruct $M_0=\beta -h(H_0)$; the resulting vector is $(M_0,v,H_0,w,{\Omega }_M)$. Analytically marginalizing $\beta$ using (iii) would also be valid.

Use several dispersed chains, trace plots, rank-normalized split $\hat{R}$, and the effective sample size of a Markov chain for each parameter and for $w,w^2$. These Markov chain Monte Carlo convergence diagnostics reveal poor mixing and disagreement but do not prove convergence. Estimate the integrated autocorrelation time ${\tau }_{\mathrm{int}}$; with $B$ retained draws, $B_{\mathrm{eff}}\approx B/{\tau }_{\mathrm{int}}$. Default to no thinning of a Markov chain, so the thinning factor is $1$. If storage requires thinning, choose a spacing after inspecting the autocorrelation and verify the retained-chain effective sample size; no finite spacing guarantees independent draws.

Using the retained $w_b$ samples, computeThese are posterior summaries, provided the corresponding moments exist, not uncertainties of the numerical estimates. A proper prior alone does not ensure finite second moments; choose or check the external prior accordingly. The Monte Carlo standard error of the posterior mean is approximately $\widehat{\mathrm{SD}}(w)/\sqrt{B_{\mathrm{eff}}}$.