Eﬀective number of simulation draws#

Part 3 · Stage 9 · ⛓️ MCMC: Gibbs, Metropolis & HMC · Lesson 073 of 144 · intermediate

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Important

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Correlated draws carry less information#

An MCMC chain’s draws are dependent by construction: each is a perturbation of the last. A thousand such draws therefore say less about the posterior than a thousand independent ones. The effective sample size measures how much less.

The definition#

If the chain’s autocorrelation at lag \(t\) is \(\rho_t\), then

\[S_{\text{eff}} = \frac{S}{1 + 2 \sum_{t=1}^{\infty} \rho_t} .\]

Independent draws (\(\rho_t = 0\)) give \(S_{\text{eff}} = S\). Positively autocorrelated draws — the usual case for random-walk Metropolis and Gibbs — give \(S_{\text{eff}} \ll S\). The estimate truncates the sum once the estimated \(\rho_t\) becomes noise, and in practice pools information across chains.

Better than independent#

A pleasing surprise: the sum can be negative. When successive draws are anticorrelated, the effective sample size exceeds the number of iterations, \(S_{\text{eff}} > S\). Hamiltonian Monte Carlo, and the NUTS sampler in particular, routinely achieves this for parameters whose posterior is close to Gaussian with little dependence on the others — the chain deliberately overshoots, so consecutive draws sit on opposite sides of the mode. For such problems MCMC beats independent sampling.

Bulk and tail#

One number is not enough. bulk-ESS governs the precision of the mean and standard deviation; tail-ESS governs the extreme quantiles, and a chain that mixes briskly through the centre can crawl in the tails. Report both, and require both above roughly 400 before trusting any summary — including \(\hat{R}\) itself, whose reliability depends on adequate ESS.

import arviz as az
az.summary(idata)                    # ess_bulk, ess_tail, mcse_mean, mcse_sd, r_hat
az.ess(idata, method="tail")         # tail-ESS explicitly
az.plot_autocorr(idata, var_names=["tau"])   # where does rho_t decay to zero?

Don’t thin#

A persistent habit deserves retiring. Thinning — keeping every tenth draw — does not improve inference. Discarding draws throws away information; the effective sample size of the thinned chain is at best what the full chain already carried, and if the draws are anticorrelated, thinning actively destroys the advantage. The only legitimate reason to thin is memory. If ESS is too low, the answer is a better sampler, a reparameterisation, or a longer run — never a smaller one.

Hint