Q19. Assessing a Metropolis–Hastings sampler

Topic: Markov chains, Gibbs sampling, and Metropolis–Hastings · more questions on this topic
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[4 marks]

A researcher runs a random-walk Metropolis–Hastings sampler using a Normal proposal centred at the current state. The proposal standard deviation is very small, and approximately 99% of proposals are accepted. The researcher concludes that the sampler is highly efficient.

Explain why the acceptance rate alone does not justify this conclusion. Describe the behaviour you would expect from this sampler and identify one additional feature of its output that you would examine.

Definitions. The acceptance rate is the fraction of proposed moves accepted. Autocorrelation measures dependence between draws at different lags. For a chosen posterior summary, effective sample size is the number of independent target draws giving comparable Monte Carlo precision. Efficiency concerns that precision per unit of computation, not the acceptance fraction alone.

An acceptance rate of 99% does not by itself demonstrate efficiency. With a very small proposal standard deviation, proposed values lie close to the current state. The target density changes little across such a short move, so most proposals are accepted; however, each accepted move advances the chain only a small distance.

The expected consequences are slow exploration of the posterior and strong autocorrelation between successive draws. A large recorded sample may then contain information equivalent to a much smaller independent sample.

In addition to acceptance rate, useful diagnostics include:

  • a trace plot, to assess whether the chain explores the full target region;

  • the autocorrelation function, to measure serial dependence; and

  • the effective sample size, to quantify the independent information represented by the chain.

Thus a very high acceptance rate may indicate that the proposal scale is too small, not that the sampler is highly efficient.

An exact independent draw from the target would also have acceptance probability one and would be efficient. High acceptance is therefore neither sufficient evidence of efficiency nor inherently a fault; the movement and dependence matter.