  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) or
importance sampling methods, with the precision of the marginal likelihood
estimator determined by the number of samples. The new acquisition strategy
gives the optimizer the option to explore the function with cheap noisy
evaluations and therefore find the optimum faster. The method is applied to
estimating the prior hyperparameters in two popular models on US macroeconomic
time series data: the steady-state Bayesian vector autoregressive (BVAR) and
the time-varying parameter BVAR with stochastic volatility. The proposed method
is shown to find the optimum much quicker than traditional Bayesian
optimization or grid search.
