Does thinning an MCMC chain to every 10th draw improve the posterior estimates?

## The intuition that motivates thinning, and why it misleads MCMC draws are autocorrelated. Keeping only every tenth one visibly reduces that correlation, and the retained sequence starts to resemble the independent sample that textbook Monte Carlo formulas assume. It feels like a cleanup step. The mistake is assuming that "looks more independent" means "is more informative". It does not. Consider what you are doing: you generated 10,000 draws, paid for all of them, and are now choosing to compute your posterior mean from 1,000 of them. The 9,000 you discarded were correlated with their neighbours, but correlated is not the same as redundant — each still carries some information about where posterior mass lies. Averaging over all 10,000 uses that information; averaging over 1,000 does not. The standard result is that, for a fixed number of iterations, the estimator using every draw has variance no larger than the thinned estimator, and usually smaller. Thinning is an information-discarding operation dressed as a variance-reduction one. ## What actually changes when you thin - **Stored draws**: down by the thinning factor, by construction. - **Effective sample size**: down as well, though by less than the thinning factor. The *ratio* of effective to stored draws improves, which is exactly why thinning looks good on a diagnostic dashboard while making the estimate slightly worse. - **Monte Carlo error of the posterior mean**: unchanged at best, larger in practice. - **Bias from a chain that has not converged**: unchanged. Every tenth draw from a stuck chain is still from a stuck chain. That third and fourth line are the ones interviewers probe. Thinning is a bookkeeping change, not a statistical fix. ## When thinning is genuinely the right call Thinning is a *resource* decision, and there are real cases: - **Memory and storage.** A model with tens of thousands of parameters times tens of thousands of iterations produces a draw matrix that will not fit anywhere convenient. Storing a tenth of it is a reasonable trade of a little precision for feasibility. - **Expensive per-draw downstream work.** If each retained draw drives a costly simulation, a per-draw forecast over a long horizon, or a human-reviewed artefact, the cost is per *retained* draw. Thinning buys the compute budget directly. - **Transport and reproducibility.** Shipping a draw archive to collaborators or storing it long-term has its own economics. In each case you are trading a small, quantifiable loss of precision for a resource you actually lack. That is a defensible engineering trade. "It makes the autocorrelation plot look nicer" is not. ## Thinning versus discarding warm-up These two are constantly confused because both discard draws, but they address opposite problems: - **Discarding warm-up** removes draws that came from the *wrong distribution* — states dominated by the starting value, or produced while the sampler was still tuning itself and therefore not targeting the posterior at all. Keeping them biases the estimate, and no amount of extra sampling later removes that bias. Discarding them genuinely improves the estimate. - **Thinning** removes draws that came from the *right distribution* and were merely correlated with their neighbours. Keeping them helps; removing them costs. If an interviewer asks which of the two actually improves your posterior estimate, the answer is: only the warm-up discard. Thinning at best breaks even. ## How to answer the question well Say no, explain that correlated draws are still informative and that the all-draws estimator dominates for a fixed run length, then name the resource cases where you would thin anyway. Add that thinning does not raise effective sample size and does not repair a convergence failure. Finish with what you would do instead when the chain is sticky: reparameterise the model, use a sampler that takes longer informed moves, or run more chains in parallel — all of which raise the information in the sample rather than rearranging it.