Stochastic Gradient Annealed Importance Sampling for Efficient Online Marginal Likelihood Estimation

Abstract

We consider estimating the marginal likelihood in settings with independent and identically distributed (i.i.d.) data. We propose estimating the predictive distributions in a sequential factorization of the marginal likelihood in such settings by using stochastic gradient Markov Chain Monte Carlo techniques. This approach is far more efficient than traditional marginal likelihood estimation techniques such as nested sampling and annealed importance sampling due to its use of mini-batches to approximate the likelihood. Stability of the estimates is provided by an adaptive annealing schedule. The resulting stochastic gradient annealed importance sampling (SGAIS) technique, which is the key contribution of our paper, enables us to estimate the marginal likelihood of a number of models considerably faster than traditional approaches, with no noticeable loss of accuracy. An important benefit of our approach is that the marginal likelihood is calculated in an online fashion as data becomes available, allowing the estimates to be used for applications such as online weighted model combination. = ∏ n p(y n |θ), as is common in many parametric models. This restriction is to ensure that a central limit theorem applies to the stochastic likelihood approximation. This can be weakened to any factorization that exhibits a central limit theorem, such as conditionally Markov data and autoregressive models. The posterior distribution over a set of models is proportional to their MLs, and so approximations to ML are sometimes used for model comparison and weighted model averaging [1] (chapter 12). The above integral is typically analytically intractable for any but the simplest models, so one must resort to numerical approximation methods.