Trace, Running-Mean, and Autocorrelation Diagnostics

Three Views of One MCMC Output

An MCMC run is a time-ordered sequence, not a collection of independent observations. A trace plot preserves that order; a running mean shows how an estimate accumulates; and an autocorrelation plot summarizes linear dependence at different lags. Each view can reveal features that are difficult to notice from a histogram or a single final estimate. None, by itself, certifies that the chain has converged to its target distribution or explored all relevant parts of the state space.

Let \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space, and let \((X_n)_{n\ge 0}\) be a discrete-time Markov chain on a state space \(S\), with its natural filtration \(\mathcal{F}_n=\sigma(X_0,\ldots,X_n)\). For a scalar quantity of interest, choose a measurable function \(f:S\to\mathbb{R}\) and write \(Y_n=f(X_n)\). Diagnostics are usually made from a finite retained sequence \(y_1,\ldots,y_N\) of these scalar values. If the state itself is scalar, one can take \(f(x)=x\). For a vector-valued state, a trace of one coordinate or one scientifically meaningful function is only a view of the full state.

The empirical formula uses the overall sample mean \(\bar y_N\), even when only \(N-k\) pairs contribute to its numerator. This convention makes \(r_0=1\) whenever the observations are not all equal. Other software conventions exist, so comparisons should use the convention actually reported. A population autocorrelation describes a stationary process; a finite-sample \(r_k\) is a descriptive estimate and does not make a nonstationary sequence stationary.

How to Read the Diagnostics

A trace plot is primarily a record of order and movement. Look for sustained trends, long stretches at similar values, abrupt shifts, repeated patterns, and unusually large excursions. Such features may be evidence of slow exploration, dependence, or a changing distribution over the displayed interval. Their absence is not proof that the chain is sampling its target correctly: a trace can look stable while remaining in a region it has not left.

The running mean addresses a different question: how much is the estimated average changing as observations are added? It is cumulative, so early observations continue to affect it, with decreasing weight. A stable-looking tail of running means is reassuring about the observed average over that run, but does not establish that the target expectation has been estimated accurately. In particular, if the chain has not visited an important region, the running mean may settle to a misleading value.

An autocorrelation plot displays \(r_k\) against lag \(k\). A slowly decaying positive pattern indicates that nearby observations tend to remain similar across multiple iterations; this is a warning that the chain moves through the observed quantity slowly. Negative autocorrelation indicates a tendency for deviations from the mean to alternate in sign. The pattern across lags matters: one value, such as \(r_1\), is not a complete description of dependence. Autocorrelation is also specific to the selected function \(f\). A coordinate can appear to move well while another function of the state moves poorly.

Proof. By definition, \(n\bar y_n=\sum_{i=1}^n y_i\) and \((n-1)\bar y_{n-1}=\sum_{i=1}^{n-1}y_i\). Subtracting the second equality from the first gives \(n\bar y_n-(n-1)\bar y_{n-1}=y_n\). Rearranging yields \(n(\bar y_n-\bar y_{n-1})=y_n-\bar y_{n-1}\). Division by the positive integer \(n\) proves the identity. \(\square\)

This identity explains why a running mean often changes less near the end of a run: the same-sized deviation is divided by a larger \(n\). Small late changes are therefore not, by themselves, strong evidence of stability. The new observations may simply be receiving less weight.

Proof. Put \(z_i=y_i-\bar y_N\) and \(D=\sum_{i=1}^N z_i^2\), so \(D>0\). For \(1\le k<N\), the numerator is \(\sum_{i=1}^{N-k}z_i z_{i+k}\). By the Cauchy–Schwarz inequality, its absolute value is at most \(\sqrt{(\sum_{i=1}^{N-k}z_i^2)(\sum_{i=1}^{N-k}z_{i+k}^2)}\). Each sum inside the square root is at most \(D\), since it includes only some of the nonnegative terms in \(D\). The numerator’s absolute value is therefore at most \(D\), giving \(|r_k|\le1\) after division by \(D\). At \(k=0\), the numerator equals \(D\), so \(r_0=1\). \(\square\)

Worked Examples

Limitations and a Practical Reading Order

These plots are exploratory diagnostics, not convergence tests. They describe the selected output and selected function; they do not prove that the chain reached the intended distribution, that its long-run average equals the target expectation, or that other coordinates and functions behave similarly. Even a weakly stationary process can produce a finite segment with an apparent trend by chance, while an apparently stable segment can conceal slow movement on a longer time scale. The population autocorrelation definition requires stationarity and finite positive variance; the empirical formula can be calculated without assuming stationarity, but then it is only a summary of that finite sequence.

A useful reading order is to inspect the trace first for visible shifts, plateaus, or unusual movement; inspect the running mean next for changes in the cumulative estimate; then inspect autocorrelation over several lags for persistence or alternation. Return to the trace when an autocorrelation pattern seems important: the time-ordered record can show whether that summary corresponds to blocks, oscillation, or another structure. Repeat the inspection for relevant functions of the state, not only the most convenient coordinate.

Common mistake. It is tempting to say, “The running mean is nearly flat, so the chain has converged.” That conclusion does not follow. The update identity gives \(\bar y_n-\bar y_{n-1}=(y_n-\bar y_{n-1})/n\), so division by a large \(n\) can make changes small even when new values still differ substantially from the current mean. A flat trace has a separate ambiguity, as the constant-output example shows. The correction is to treat visual stability as a reason to examine the output further, not as proof; assess trace, cumulative behavior, and dependence together, and keep their limitations explicit.