Bayesian Optimization of Hyperparameters when the Marginal Likelihood is Estimated by MCMC

Bayesian models often involve a small set of hyperparameters determined by maximizing the marginal likelihood. Bayesian optimization is a popular iterative method where a Gaussian process posterior of the underlying function is sequentially updated by new function evaluations. An acquisition strategy uses this posterior distribution to decide where to place the next function evaluation. We propose a novel Bayesian optimization framework for situations where the user controls the computational effort, and therefore the precision of the function evaluations. This is a common situation in econometrics where the marginal likelihood is often computed by Markov Chain Monte Carlo (MCMC) methods, with the precision of the marginal likelihood estimate determined by the number of MCMC draws. The proposed acquisition strategy gives the optimizer the option to explore the function with cheap noisy evaluations and therefore finds the optimum faster. Prior hyperparameter estimation in the steady-state Bayesian vector autoregressive (BVAR) model on US macroeconomic time series data is used for illustration. The proposed method is shown to find the optimum much quicker than traditional Bayesian optimization or grid search.