The \widehat{R} page compared different chains. We now use the correlation concept introduced earlier to describe dependence across iterations within one Markov chain. Suppose one chain contains saved draws for a consonant-duration parameter, where duration is the elapsed acoustic time between consistently defined segment boundaries and is thus a measurement rather than a phonological category:
.10,.12,.13,.15,.14,.18,\ldots
Adjacent values are similar because the sampler moves through the posterior gradually. The question is how to measure that within-chain resemblance at different separations. The autocorrelation measures the association between a sequence and a shifted copy of itself.
At lag one, align every draw except the last with the following draw:
| draw at s | draw at s+1 |
|---|---|
| .10 | .12 |
| .12 | .13 |
| .13 | .15 |
| .15 | .14 |
| .14 | .18 |
The correlation between these columns is the empirical lag-one autocorrelation.
Let \operatorname{Corr}(U,V) denote the correlation between random variables U and V under the stationary distribution. For a stationary chain, the population autocorrelation at lag t is
\rho_t = \operatorname{Corr} (\Theta_s,\Theta_{s+t}).
Lag one compares adjacent draws, while lag ten compares draws separated by ten transitions. A chain of length 1,000 supplies 999 lag-one pairs and 990 lag-ten pairs.
The sequence
\rho_1,\rho_2,\rho_3,\ldots
is the autocorrelation function (ACF). Rapid decay toward zero suggests that dependence weakens over a small number of transitions. Slow decay means that information about one draw persists across many later draws.
An autoregressive sequence supplies a controlled illustration:
X_s=.85X_{s-1}+\varepsilon_s.
Its theoretical autocorrelation is \rho_t=.85^t.
set.seed(829)
draws <- as.numeric(
arima.sim(model = list(ar = .85), n = 5000)
)
empirical_acf <- as.numeric(
acf(draws, lag.max = 10, plot = FALSE)$acf
)
stopifnot(abs(empirical_acf[2] - .85) < .05)
data.frame(
lag = 0:10,
empirical = empirical_acf,
theoretical = .85^(0:10)
)The entry at lag zero is one because a sequence is perfectly correlated with itself. The following entries estimate dependence across increasing separation.
High autocorrelation does not by itself imply the wrong stationary distribution, since a transition can preserve the correct posterior while moving through it slowly. The problem is efficiency for a finite run.
When adjacent draws are nearly repetitions, 5,000 saved values contain less information about a posterior mean than 5,000 independent target draws. Strong positive autocorrelation thus increases Monte Carlo error for a fixed raw draw count.
Negative autocorrelation can instead make alternating draws offset one another and may improve precision for some averages. The complete autocorrelation pattern matters, not only the lag-one value.
Autocorrelation does not by itself make a stationary chain invalid. For an ergodic chain initialized away from stationarity, the starting state can still contribute finite-run bias, and autocorrelation changes the variance of Monte Carlo estimates. Failure to reach a target region is a different problem.
Keeping every tenth draw reduces visible autocorrelation in the stored sequence, but it also discards information. Unless storage or downstream computation is limiting, retaining draws and accounting for their dependence is usually more informative than routine thinning.
The effective-sample-size page translates the dependence pattern into an information-equivalent draw count.