Differentiating Metropolis-Hastings to Optimize Intractable Densities

When performing inference on probabilistic models, target densities often become intractable, necessitating the use of Monte Carlo samplers. We develop a methodology for unbiased differentiation of the Metropolis-Hastings sampler, allowing us to differentiate through probabilistic inference. By fusing recent advances in stochastic differentiation with Markov chain coupling schemes, the procedure can be made unbiased, low-variance, and automatic. This allows us to apply gradient-based optimization to objectives expressed as expectations over intractable target densities. We demonstrate our approach by finding an ambiguous observation in a Gaussian mixture model and by maximizing the specific heat in an Ising model.