Likelihood-free Density Ratio Acquisition Functions are not Equivalent to Expected Improvements

Bayesian Optimization (BO) is one of the most effective black-box optimization methods, yet the need to ensure analytical tractability in the posterior predictive makes it challenging to apply BO to large-scale problems with high-dimensional observations. For these problems, likelihood-free methods present a promising avenue since they can work with more expressive models and are often more efficient. Previous papers have claimed that density ratios acquired from the likelihood-free inference are equivalent to the widely popular expected improvement acquisition function, allowing us to perform BO without expensive exact posterior inference. Unfortunately, we show in this paper that the claim is false; we identify errors in their reasoning and illustrate a counter-example where density ratios are inversely correlated to expected improvements. Our results suggest that additional care is needed when interpreting and applying density ratio acquisition functions from likelihood-free inference.