On Monte Carlo methods for Bayesian inference

Bayesian methods are experiencing increased use for probabilistic ecological modelling. Most Bayesian inference requires the numerical approximation of analytically intractable integrals. Two methods based on Monte Carlo simulation have appeared in the ecological/environmental modelling literature. Though they sound similar, the Bayesian Monte Carlo (BMC) and Markov Chain Monte Carlo (MCMC) methods are very different in their efficiency and effectiveness in providing useful approximations for accurate inference in Bayesian applications. We compare these two methods using a low-dimensional biochemical oxygen demand decay model as an example. We demonstrate that the BMC is extremely inefficient because the prior parameter distribution, from which the Monte Carlo sample is drawn, is often a poor surrogate for the posterior parameter distribution, particularly if the parameters are highly correlated. In contrast, MCMC generates a chain that converges, in distribution, on the posterior parameter distribution, that can be regarded as a sample from the posterior distribution. The inefficiency of the BMC can lead to marginal posterior parameter distributions that appear irregular and may be highly misleading because the important region of the posterior distribution may never be sampled. We also point out that a priori specification of the model error variance can strongly influence the estimation of the principal model parameters. Although the BMC does not require that the model error variance be specified, most published applications have treated this variance as a known constant. Finally, we note that most published BMC applications have chosen a uniform prior distribution, making the BMC more similar to a likelihood-based inference rather than a Bayesian method because the posterior is unaffected by the prior. Though other prior distributions could be applied, the treatment of Monte Carlo samples with any other choice of prior distribution has not been discussed in the BMC literature.

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