Annealed Importance Sampling (AIS) [27,18] is the gold standard for estimating partition functions or marginal likelihoods, corresponding to importance sampling over a path of distributions between a tractable base and an unnormalized target. While AIS yields an unbiased estimator for any path, existing literature has been primarily limited to the geometric mixture or moment-averaged paths associated with the exponential family and KL divergence [13]. We explore AIS using qpaths, which include the geometric path as a special case and are related to the homogeneous power mean, deformed exponential family, and α-divergence [3].
Many common machine learning methods involve the geometric annealing path, a sequence of intermediate densities between two distributions of interest constructed using the geometric average. While alternatives such as the moment-averaging path have demonstrated performance gains in some settings, their practical applicability remains limited by exponential family endpoint assumptions and a lack of closed form energy function. In this work, we introduce q-paths, a family of paths which is derived from a generalized notion of the mean, includes the geometric and arithmetic mixtures as special cases, and admits a simple closed form involving the deformed logarithm function from nonextensive thermodynamics. Following previous analysis of the geometric path, we interpret our q-paths as corresponding to a qexponential family of distributions, and provide a variational representation of intermediate densities as minimizing a mixture of α-divergences to the endpoints. We show that small deviations away from the geometric path yield empirical gains for Bayesian inference using Sequential Monte Carlo and generative model evaluation using Annealed Importance Sampling.
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