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Annealed Importance Sampling with q-Paths

Rob Brekelmans, Vaden Masrani, Thang Bui, Frank Wood, Aram Galstyan, Greg Ver Steeg, Frank Nielsen

TL;DR

This work addresses the efficiency of annealed importance sampling (AIS) for partition-function estimation by introducing $q$-paths, a generalization of the standard geometric path based on power means. The authors connect $q$-paths to the deformed logarithm ($\ln_q$), the $q$-exponential family, and $\alpha$-divergences, deriving both deterministic representations and variational characterizations that extend beyond the conventional exponential-family framework. They show that $q$-paths recover the geometric path as $q\to1$ and provide a variational formulation via $D_{\alpha}$ that unifies several intuition sources, including moment-matching scenarios. Empirical results on Gaussian and Student-$t$ endpoints demonstrate that certain $q$-paths (notably $q\approx0.9$) can yield tighter log-ratio estimates with fewer intermediate steps, suggesting practical benefits for partition function estimation in Bayesian settings and highlighting connections to non-extensive thermodynamics and information geometry.

Abstract

Annealed importance sampling (AIS) 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. We explore AIS using $q$-paths, which include the geometric path as a special case and are related to the homogeneous power mean, deformed exponential family, and $α$-divergence.

Annealed Importance Sampling with q-Paths

TL;DR

This work addresses the efficiency of annealed importance sampling (AIS) for partition-function estimation by introducing -paths, a generalization of the standard geometric path based on power means. The authors connect -paths to the deformed logarithm (), the -exponential family, and -divergences, deriving both deterministic representations and variational characterizations that extend beyond the conventional exponential-family framework. They show that -paths recover the geometric path as and provide a variational formulation via that unifies several intuition sources, including moment-matching scenarios. Empirical results on Gaussian and Student- endpoints demonstrate that certain -paths (notably ) can yield tighter log-ratio estimates with fewer intermediate steps, suggesting practical benefits for partition function estimation in Bayesian settings and highlighting connections to non-extensive thermodynamics and information geometry.

Abstract

Annealed importance sampling (AIS) 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. We explore AIS using -paths, which include the geometric path as a special case and are related to the homogeneous power mean, deformed exponential family, and -divergence.

Paper Structure

This paper contains 31 sections, 2 theorems, 45 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Interpretations of the Geometric Path
  3. Log Mixture
  4. Exponential Family
  5. Variational Representation
  6. Interpretations of the $q$-Path
  7. Power Means
  8. Deformed Log Mixture
  9. $q$-Exponential Family
  10. Variational Representation using the $\alpha$-Divergence
  11. Moment-Matching Procedures
  12. Experiments
  13. Conclusion
  14. Abstract Mean is Invariant to Affine Transformations
  15. Derivations of the $q$-Path
  16. ...and 16 more sections

Key Result

Lemma 1

Sum identity

Figures (5)

  • Figure 1: Intermediate densities between $\mathcal{N}(-4, 3)$ and $\mathcal{N}(4,1)$ for various $q$-paths and 10 equally spaced $\beta$. The path approaches a mixture of Gaussians with weight $\beta$ at $q=0$. For the geometric mixture ($q=1$), intermediate $\pi_{\beta}$ stay within the exponential family since both $\pi_0$, $\pi_T$ are Gaussian.
  • Figure 2: lower and upper bound estimates of $\log Z_T/Z_0$ by $q$-path order and number of intermediate distributions ($T$), for annealing between $\mathcal{N}(-4,3) \to \mathcal{N}(4,1)$.
  • Figure 3: Intermediate densities between $\mathcal{N}(-4, 3)$ and $\mathcal{N}(4,1)$ for various $q$-paths and 10 equally spaced $\beta$. The path approaches a mixture of Gaussians with weight $\beta$ at $q=0$. For the geometric mixture ($q=1$), intermediate $\pi_{\beta}$ stay within the exponential family since both $\pi_0$, $\pi_T$ are Gaussian.
  • Figure 4: Intermediate densities between Student-$t$ distributions, $t_{\nu = 1}(-4, 3)$ and $t_{\nu = 1}(4,1)$ for various $q$-paths and 10 equally spaced $\beta$, Note that $\nu=1$ corresponds to $q=2$, so that the $q=2$ path stays within the $q$-exponential family. We provide code to reproduce experiments at https://github.com/vmasrani/q_paths.
  • Figure : Annealed IS

Theorems & Definitions (5)

  • proof
  • Lemma 1
  • Lemma 2
  • proof
  • proof