Doubly intractable distributions arise in many settings, for example in Markov models for point processes and exponential random graph models for networks. Bayesian inference for these models is challenging because they involve intractable normalizing "constants" that are actually functions of the parameters of interest. Although several clever computational methods have been developed for these models, each method suffers from computational issues that makes it computationally burdensome or even infeasible for many problems. I will discuss a framework for understanding existing algorithms, as well as a proposal for a new algorithm that replaces Monte Carlo approximations to the normalizing function with a Gaussian process-based approximation. This is joint work with Jaewoo Park.