Tractable Expected Information Gains for Exponential Family Posteriors

We investigate which models admit a collapse of the expected information gain (EIG) and its derivative from a doubly intractable to a singly intractable expression. We prove that a sufficient condition is that the posterior distribution belongs to an exponential family (EF) and depends on the experimental design and data only through its natural parameters, and derive corresponding singly intractable and unbiased estimators for the $\operatorname{EIG}$ and its (reparameterized) gradient. We further show that this is achieved when using a likelihood of an analogous form and any arbitrary prior. This is complemented by a theoretical analysis of certain degenerate behaviors that may arise when optimizing the $\operatorname{EIG}$ for EF-modeled experiments. Finally, we empirically demonstrate the benefits of using our singly intractable estimators, showing superior convergence rates, and substantial performance gains for sequential design problems compared to using standard nested estimators.