Autocorrelation times and effective sample size

For an observable O recorded at equally spaced update intervals, subtract its equilibrium expectation and define the covariance at lag t. Normalize by the zero-lag variance to obtain rho. We use integrated autocorrelation time equal to one half plus the positive-lag sum. Independent measurements then have tau equal to one half, not one. Some authors instead call twice this quantity the autocorrelation factor; copying a quoted number without its definition creates a factor-of-two error in the reported uncertainty.

The exponential relaxation time measures an asymptotic decay rate of slow modes, whereas the integrated time weights their amplitudes in this particular observable. A small-amplitude slow tail may scarcely affect a short plot yet matter for a precise error estimate. Negative correlations can lower the integrated time below one half and give a variance-based effective size above the number stored. That mathematical possibility does not create additional independent configurations or justify ignoring stationarity and tail-estimation assumptions.

The exact finite-length variance of the mean contains a triangularly weighted sum over all lags. When the run is much longer than its relevant correlation scale, replacing those weights by one gives the common asymptotic variance formula. Effective sample size is defined by comparing this variance with that of independent measurements having the same marginal variance. It is therefore a diagnostic for estimating this observable’s mean, rather than a universal property attached permanently to a saved file.

Increasing the measurement spacing can reduce correlation between stored entries, but it also discards information already paid for unless storage or measurement cost is the bottleneck. Efficiency should be compared using uncertainty per total computational cost. A longer run reduces statistical uncertainty only after thermalization and mixing are credible. An apparently small estimated autocorrelation time from a chain trapped in one sector cannot account for the missing sector; within-sector fluctuations alone underestimate the full uncertainty.

A practical autocorrelation sum needs a window: including too few lags loses the tail, while including arbitrarily many mostly noisy lags destabilizes the estimate. Vary the window and examine stability, ideally using independent replicas as additional checks. Blocking offers another diagnostic: combine consecutive measurements into blocks longer than the relevant correlation scale and estimate the uncertainty from their means. Increasing the block size leaves fewer blocks, so a seeming plateau with only a handful remaining can itself be statistically weak.

For a nonlinear ratio such as a Binder cumulant or a reweighted mean, its ingredients fluctuate together. A block jackknife or bootstrap must resample the same blocks for all ingredients and recompute the complete ratio or fit each time. Resampling individual correlated measurements as if independent defeats the purpose. These methods control sampling uncertainty and some finite-sample bias; they do not repair an incorrect action or an unsampled phase. The exact Markov lab illustrates a relaxation eigenmode without generating a stochastic history from which to estimate an error.