What are some common pitfalls and best practices for MCMC convergence diagnostics?

1 Why convergence matters

MCMC convergence means that your chains have reached a stationary distribution that approximates the true posterior distribution of your model parameters. Convergence is important because it guarantees that your MCMC estimates are consistent, unbiased, and reliable. If your chains have not converged, you may end up with misleading or inaccurate results, such as wrong confidence intervals, posterior predictions, or model comparisons.

2 How to diagnose convergence

Diagnosing MCMC convergence is not foolproof or definitive, but there are several methods to try. Visual inspection is a popular option, where you can plot chains and look for signs of non-convergence such as drifts, trends, cycles, or divergence. Additionally, summary statistics like the potential scale reduction factor (PSRF), the effective sample size (ESS), or the Gelman-Rubin statistic can be calculated to compare the within-chain and between-chain variability of your chains. Formal tests such as the Geweke test, the Heidelberger-Welch test, or the Raftery-Lewis test can also be applied to compare different segments or subsamples of your chains and test for stationarity, convergence, or accuracy.

3 What to do if convergence fails

If your convergence diagnostics suggest that your chains have not converged, you should not trust your MCMC estimates or draw any conclusions from them. Instead, you should attempt to identify and fix the source of the issue. Poor initialization, inadequate tuning, and model misspecification are all potential causes. To address poor initialization, you could use better initial values such as the output of a simpler model, a prior sample, or a random draw from a wide distribution. To address inadequate tuning, you could adjust parameters such as the step size, the proposal distribution, or the adaptation scheme. You could also use adaptive or self-tuning algorithms for automatic adjustment. To address model misspecification, you could revise assumptions such as the likelihood, the priors, or the constraints; then check for model adequacy, fit, and sensitivity.

4 How to improve convergence

To improve the convergence and efficiency of your MCMC chains, you can take several steps. Running multiple chains from different initial values can help detect and avoid problems like local modes or label switching, while combining the output of multiple chains can increase the effective sample size and reduce the Monte Carlo error. Additionally, discarding some initial iterations (burn-in) and subsequent iterations (thinning) may reduce autocorrelation and storage, though this is not always necessary or beneficial. Finally, you can use better algorithms such as Metropolis-Hastings, Gibbs sampling, Hamiltonian Monte Carlo, or No-U-Turn Sampler, which have different strengths and weaknesses depending on speed, accuracy, flexibility, or scalability. Hybrid or composite algorithms are also available that combine different methods or update different parameters separately.

5 How to learn more

MCMC convergence diagnostics are an essential part of statistical programming, but they are not easy to master or apply. You need to have a good understanding of the theory and practice of MCMC methods, as well as the specific features and challenges of your model and data. You also need to use appropriate software and tools to implement and visualize your MCMC chains and diagnostics. There are many resources available online and offline to help you learn more about MCMC convergence diagnostics, such as books, courses, tutorials, blogs, podcasts, or forums.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?