Chains, Iterations, and Effective Sample Size

Summary

How many chains, how long, and how to know when to stop. The section’s most useful corrective is arithmetic: 10 independent draws already give a posterior-mean estimate only $\sqrt{1.1}=1.05$ times as uncertain as infinite draws. “Just 10 simulations are enough to obtain posterior accuracy within 5% of the computational ideal.” The reason we run thousands is not accuracy but diagnostics — “we don’t usually need a lot of simulation draws, but we do want them to approximately come from the target distribution.” Recommended defaults: at least four chains, run until $\hat{R}<1.01$ for all quantities of interest, with $\hat{R}\approx 1.1$ acceptable during exploration.

Overview

The two challenges of MCMC (Ch. 11.4, p. 197)

1. Forgetting the initial values. “The distribution of simulations depends on initial values, and the user needs to run MCMC long enough that initial values have essentially been forgotten. Even if the initial value would be within the typical set, a single draw is not representative of the distribution.”
2. Monte Carlo error. “Inferences are based on simulations rather than deterministic estimates; as a result the user must account for Monte Carlo error or else average over enough simulation draws that such error is negligible. Markov chain Monte Carlo iterations have dependence, which makes the Monte Carlo error usually larger than with the same number of independent draws.”

Convergence vs. mixing time

“With increasing number of MCMC iterations, the draws should collectively represent the posterior better and better. Theoretically, convergence of Markov chains can be quantified by measuring some discrepancy such as total variation compared to the distribution represented by the draws from the target distribution. The convergence can be made faster by discarding the initial transition phase iterations to reduce the initial value bias.

If a chain is initialized with a draw from the target distribution, the time needed for a chain to explore the distribution sufficiently can be called mixing time. As the target distribution is unknown, we estimate the convergence and mixing time by comparing several independent Markov chains.”

Main Content

Intuition from independent draws

Figure 11.2 — how noisy is a 95% interval? (Ch. 11.4, pp. 197-198)

Five replications of a 95% interval for a unit normal parameter, from 100 and from 1000 independent draws. The correct answer is $(-1.96,1.96)$.

100 draws	1000 draws
$(-1.79,1.69)$	$(-1.83,1.97)$
$(-1.80,1.85)$	$(-2.01,2.04)$
$(-1.64,2.15)$	$(-2.10,2.13)$
$(-2.08,2.38)$	$(-1.97,1.95)$
$(-1.68,2.10)$	$(-2.10,1.97)$

Two readings of the same table, both correct:

• “From one perspective, these estimates are pretty bad: even with 1000 simulations, either bound can easily be off by more than 0.1, and the entire interval width can easily be off by 10%.”
• “On the other hand, for the goal of inference about the parameter, even the far-off estimates above aren’t so bad: the interval $(-2.08,2.38)$ has 97% probability coverage, and $(-1.79,1.69)$ has 92% coverage.”

Which reading applies depends on the goal: “If the goal is to precisely determine the endpoints of the interval to determine if a coefficient is ‘statistically significant’ or simply to present a replicable value for publication, then many simulations are required — even in this extremely easy problem, 1000 independent draws are not enough to pin down the interval endpoints to one decimal place. However, if the goal is to get an interval for the mean with approximate 95% coverage in the target distribution, even 100 draws are reasonable.”

The 10-draw calculation

“For many purposes even 10 simulation draws would be enough. The Monte Carlo standard error of the average of 10 independent draws of $\theta$ is $1/\sqrt{10}$ times the standard deviation of $\theta$ in the target distribution; thus inference about the posterior mean based on 10 simulations is $\sqrt{1.1}=1.05$ times as uncertain as from infinite draws. Just 10 simulations are enough to obtain posterior accuracy within 5% of the computational ideal.

This does not tell the whole story — the posterior mean is not the only summary of interest, and 5% isn’t nothing — but it does convey that just a few simulations are enough to get most of the way there, which is one reason that we focus on monitoring convergence. We don’t usually need a lot of simulation draws, but we do want them to approximately come from the target distribution.”

And the qualification that explains the actual defaults: “While 10 independent simulation draws might be sufficient, 10 or even 100 MCMC iterations are rarely sufficient, as diagnosing convergence and measuring the sampling efficiency usually themselves require more iterations to do well.”

The logic is worth internalizing: the iteration count is driven by the diagnostics, not by the estimate. The estimate converged long ago; you are paying for the ability to know that.

Monitoring convergence with multiple chains

$\hat{R}$, the potential scale reduction factor (Gelman and Rubin 1992)

”$\hat{R}$ is computed for each scalar quantity of interest, as the standard deviation of that quantity from all the chains included together (after the warmup steps have been discarded), divided by the root mean square of the separate within-chain standard deviations.

The idea is that [if] the simulations from multiple chains have not mixed well, the variance of all the chains mixed together should be higher than the variance of individual chains.

At convergence … “the ratio $\hat{R}$ should equal 1. If $\hat{R}$ is greater than 1, this implies that the chains have not fully mixed and that further simulation might increase the precision of inferences.”

Robustness improvements: “For quantities of interest that are far from normally distributed, it can make sense to first perform a rank-normal transformation” (Vehtari, Gelman, Simpson, et al. 2021); a nested-$\hat{R}$ variant for “a large number of short chains” is given by Margossian, Hoffman, et al. (2025).

The multivariate companion: ”$\hat{R}$ is used for univariate marginal quantities, and $R^{\ast }$ (Lambert and Vehtari 2022) can detect convergence issues jointly for all parameters.”

The thresholds, with their caveats (Ch. 11.4, p. 198)

• Run at least four independent chains by default. “Multiple chains are more likely to reveal multimodality and poor adaptation or mixing. Complex, misspecified, or nonidentifiable models arise in our work all the time.”
• $\hat{R}<1.01$ for all parameters and quantities of interest for final results.
• $\hat{R}\approx 1.1$ is acceptable in early model exploration (illustrated in Ch. 27).

Two honest caveats attached to the 1.01 rule:

1. “We recognize that this rule can declare convergence prematurely, which is one reason why we always recommend comparing results to estimates from simpler models.”
2. ”$\hat{R}$ itself is also stochastic, and if we examine $\hat{R}$ for many quantities, it is possible just by chance that sometimes $\hat{R}$ will be bigger than 1.01, even if the chains are mixing well.”

Why not a hypothesis test: “We never reach exact convergence, and so it does not make sense to try to check convergence using statistical hypothesis tests of the null hypothesis of perfect mixing. Instead, we use statistical estimation — post-processing of simulation results — to summarize the mixing of the simulated chains, comparing variation within and between chains.”

A nonparametric alternative

“Mixing of chains can also be monitored nonparametrically, for example by computing the 80% (say) central interval from each chain and then determining its coverage with respect to the empirical distribution of all the other chains combined together. … At convergence, the average coverage of these 80% separate-chain intervals should be 80%; a much lesser value indicates poor mixing” (Brooks and Gelman 1998; Brooks, Giudici, and Phillipe 2003).

Traceplots — when they help

“When convergence is poor, it can help to look at traceplots — time series of the progress of multiple chains, looking at one or two parameters or quantities of interest at a time. … Traceplots can be helpful for gaining insight for the convergence issues but are rarely interesting if the chains have mixed well.”

Examples in Failure Modes and Steps Forward (§12.3).

How long to run

The two questions (Ch. 11.4, p. 199)

“1. How many chains and iterations do we need to assess that the algorithm is sampling from something close to the target distribution? 2. How many draws do we need so that the reported posterior summaries would not change in important ways if the inference would be repeated?

The first question requires assessment of mixing of Markov chains; the second question is more general and arises with any stochastic algorithm.”

Against premature optimization of the sampler

“It might seem like a safe and conservative choice to run MCMC until the effective sample size is in the thousands or Monte Carlo standard error is tiny in comparison to the required precision — but if this takes a long time, it limits the number of models that can be fit in the exploration stage.

More often than not, we often have model problems (including coding errors) that become apparent after running only a few iterations, so that the remaining computation is wasted. In this respect, running many iterations for a newly-written model is similar to premature optimization in software engineering.

For the final model, the required number of iterations depends on the desired Monte Carlo accuracy for the quantities of interest.”

The direct statement of the Fit Fast, Fail Fast principle, applied to iteration counts.

Where to spend parallelism

“Another choice in computation is how to best make use of available parallelism, beyond the default of running 4 or 8 separate chains on multiple cores. Instead of increasing the number of iterations, effective variance reduction can also be obtained by increasing the number of parallel chains” (Hoffman and Ma 2020; Margossian, Hoffman, et al. 2025).

When the algorithm is trustworthy, diagnostics become unnecessary: “In general, if we would know that the inference algorithm is working perfectly and producing independent draws from the posterior, we would not need convergence diagnostics … When we don’t trust the algorithm, it turns out that often we need more iterations to be confident about approximate convergence.”

Examples

Exercise 11.5 — a case $\hat{R}$ catches, and one it misses (Ch. 11.8, p. 206)

(a) Simulate two chains, one from a unit normal and one from a $t_3$ distribution (independent draws), and compute $\hat{R}$ as implemented in the posterior package. “This should correctly reveal poor mixing. Explain how this happens.”

(b) “Create data from two chains that do not mix well, but where $\hat{R}$ is less than 1.01, so this poor mixing is not revealed. Explain the problem and suggest an improved $\hat{R}$ that would catch this problem.”

Part (b) is the more instructive half: $\hat{R}$ compares variances, so two chains with matching variances but different shapes or locations-in-shape can slip through. This is precisely what the rank-normalization and $R^{\ast }$ extensions address.

Exercise 11.1 — how many draws to pin down a quantile (Ch. 11.8, p. 205)

“Suppose $\theta$ is approximately normally distributed in a posterior summarized by $n$ independent draws. How large does $n$ have to be so that the 2.5% and 97.5% quantiles of $\theta$ are specified to an accuracy of $0.1\text{sd}(\theta |y)$? (a) Figure this out mathematically, without simulation. (b) Check your answer using simulation.”

Connections

• The “diagnostics drive the iteration count” argument is the reason Effective Sample Size and Monte Carlo Standard Error is a reporting tool rather than a stopping rule on its own.
• $\hat{R}$’s stochasticity and its false negatives are why the book pairs it with the completely different check of SBC — a diagnostic of the algorithm, not of the chains.
• The four-chain default and the multimodality caveat connect to Challenge of Multimodality - Differential Equation for Planetary Motion.

See Also

• Effective Sample Size and Monte Carlo Standard Error — quantifying the dependence between draws
• How Many Digits to Report — turning MCSE into a reporting decision
• Failure Modes and Steps Forward — what to do when $\hat{R}$ will not come down
• Efficient MCMC — BDA3 background on convergence assessment