Why does an MCMC run with 10,000 stored draws report an effective sample size of only 200?

## Dependent draws are worth less Ordinary Monte Carlo reasoning assumes independent samples. MCMC gives you a *chain*: each state is generated from the previous one, so consecutive draws resemble each other. If the sampler takes small steps, draw 4,001 is nearly a copy of draw 4,000, and the pair together tells you barely more than either alone. The stored draw count is therefore a bookkeeping number, not an information count. **Effective sample size (ESS)** converts the stored count into the honest one. Its definition is operational: the number of *independent* draws that would give an estimate with the same variance as the one your correlated chain produced. ## The formula For a chain of `N` draws with autocorrelation `rho_k` at lag `k`: ``` ESS = N / (1 + 2 * (rho_1 + rho_2 + rho_3 + ...)) ``` Every lag contributes. Positive autocorrelation makes the denominator larger than 1 and pushes ESS below `N`. Two consequences worth stating in an interview: - The relevant quantity is the *sum over all lags*, not the lag-1 value alone. Two chains can share a lag-1 correlation of 0.98 and have quite different ESS if one's correlation decays quickly after lag 1 and the other's decays slowly. - Negative autocorrelation is possible — some samplers deliberately produce antithetic behaviour — and then ESS can exceed `N`. It is not a contradiction; it means the chain over-corrects and averages more efficiently than independent draws would. For the idealised case where correlations decay geometrically as `rho_k = rho^k`, the sum has a closed form and `ESS = N * (1 - rho) / (1 + rho)`. At `rho = 0.98` that is about one percent of `N` — 10,000 draws worth roughly a hundred. Real chains are messier, which is why a reported ESS of 200 from 10,000 draws is entirely plausible. ## Why it matters: precision The reason to care is the Monte Carlo error of your reported numbers. The standard error of a posterior mean estimated from a chain behaves like ``` MCSE = posterior_sd / sqrt(ESS) ``` with `ESS`, not `N`, in the denominator. So the ESS of 200 means your posterior mean is roughly `sqrt(10000 / 200)` — about seven times — less precise than the stored count would suggest. If you report a posterior mean to three significant figures on the strength of 10,000 draws while the effective count is 200, most of those figures are noise you have dressed up as inference. A useful sanity rule: you want ESS in the low hundreds at minimum for stable means and intervals, and more for anything in the tails. ## Bulk versus tail Modern reporting splits ESS in two. **Bulk-ESS** governs how well the centre of the distribution is estimated — means, medians. **Tail-ESS** governs the extreme quantiles used for interval endpoints. Tail-ESS is usually the smaller of the two, because the chain visits the tails rarely and those visits are strongly clustered. A fit with an acceptable bulk-ESS can still have a 95% interval whose endpoints wobble noticeably between reruns. If your deliverable is an interval, tail-ESS is the number that gates it. ## What actually fixes a low ESS - **Reparameterise.** Correlated or badly scaled parameters force the sampler to take tiny steps. Recentring, rescaling, or restating a hierarchy in a form with flatter geometry often buys an order of magnitude. - **Use a sampler that takes longer informed moves.** Gradient-informed trajectories traverse the distribution in one transition where a small random-walk step needs hundreds. - **Run longer.** ESS grows roughly linearly in iterations for a fixed chain, so this always works and is always the expensive answer. - **Run more chains.** ESS is additive across chains, and chains run in parallel, so wall-clock cost is often flat. ## What does not fix it **Thinning does not raise ESS.** Keeping every tenth draw removes nine tenths of the draws and roughly a corresponding share of the effective ones; the ratio ESS/stored-draws improves, but the ESS itself does not, and it usually falls slightly. Thinning changes the bookkeeping, not the information. Likewise, discarding more warm-up removes start-up bias but leaves the correlation structure of the retained draws untouched. ## Reading it next to convergence ESS and a between-chain convergence statistic answer different questions. A convergence statistic asks *are the chains describing the same distribution*; ESS asks *how much information do the draws carry*. A chain can be beautifully converged and nearly useless, or highly informative within a region it should never have been trapped in. Both are checked, per parameter, before any number is reported.