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How are MCMC diagnostics used in Markov chain Monte Carlo?
Markov Chain Monte Carlo (MCMC) diagnostics are tools that can be used to check whether the quality of a sample generated with an MCMC algorithm is sufficient to provide an accurate approximation of the target distribution. In particular, MCMC diagnostics are used to check:
What happens when a chain converges in MCMC?
When the chain converges slowly, a large portion of our MCMC sample might be made up of observations drawn from distributions that are significantly different from the target distribution. If we are able to spot this kind of problem, we can try to fix it by: discarding a large chunk of initial observations (the so-called burn-in sample);
Why is chain 3 slow in MCMC diagnostics?
In the third plot (Chain 3), there is a lot of serial correlation between successive draws. The chain is very slow in exploring the sample space. The sample space has been explored only few times. In other words, there seems to be few independent observations in our sample.
What is the effective sample size of MCMC?
The dimension of this sub-sample, which we call effective sample size, is . Roughly speaking, the MCMC sample is equivalent to a sample of independent observations having size . The slower the decay of the dependence between the terms of the Markov chain is, the larger , and the smaller the effective sample size is.
Which is the vignette plotting package for MCMC?
Plots of parameter estimates from MCMC draws are covered in the separate vignette Plotting MCMC draws, and graphical posterior predictive model checking is covered in the Graphical posterior predictive checks vignette. Note that most of these plots can also be browsed interactively using the shinystan package.
What kind of problem do MCMC diagnostics try to spot?
When we run an MCMC algorithm for periods, we get an MCMC sample made up of the first realizations of the chain: and then we use the empirical distribution of the sample to approximate the target distribution. What kind of problems do MCMC diagnostics try to spot?
How to diagnose divergent transitions in MCMC?
This vignette focuses on MCMC diagnostic plots, in particular on diagnosing divergent transitions and on the n_eff and Rhat statistics that help you determine that the chains have mixed well.
Which is a diagnostic plot of a Markov chain?
With the centered parameterization the chains never could make it into the neck of funnel and we see a clustering of divergences and no draws in the tail of the distribution. Another useful diagnostic plot is the trace plot, which is a time series plot of the Markov chains.
How is MCMC different from maximum likelihood Monte Carlo?
Simulation-based methods, such as MCMC methods, offer a fundamentally different approach from maximum-likelihood–based methods for obtaining the distribution of parameters of interest. Transitioning from maximum-likelihood to simulation-based methods requires some explanation of their similarities and differences.
How to measure serial correlation in a Markov chain?
To measure serial correlation precisely, we can use so-called ACF (autocorrelation function) plots, also called correlograms. From these plots you can see how the sample autocorrelation between the terms of the chain decreases as a function of their lag (see the lecture on autocorrelation for more details).
How are MCMC diagnostics used in the real world?
In particular, MCMC diagnostics are used to check: whether a large portion of the MCMC sample has been drawn from distributions that are significantly different from the target distribution; whether the size of the generated sample is too small.
Which is the Gelman-Rubin diagnostic for MCMC?
Gelman-Rubin diagnostic ( ) • Compute mindependent Markov chains • Compares variance of each chain to pooled variance • If initial states (θ 1j ) are overdispersed, then approaches unity from above • Provides estimate of how much variance could be reduced by running chains longer • It is an estimate!