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Exact and Approximate MCMC for Doubly-intractable Probabilistic Graphical Models Leveraging the Underlying Independence Model

Yujie Chen, Antik Chakraborty, Anindya Bhadra

Abstract

Bayesian inference for doubly-intractable pairwise exponential graphical models typically involves variations of the exchange algorithm or approximate Markov chain Monte Carlo (MCMC) samplers. However, existing methods for both classes of algorithms require either perfect samplers or sequential samplers for complex models, which are often either not available, or suffer from poor mixing, especially in high dimensions. We develop a method that does not require perfect or sequential sampling, and can be applied to both classes of methods: exact and approximate MCMC. The key to our approach is to utilize the tractable independence model underlying the intractable probabilistic graphical model for the purpose of constructing a finite sample unbiased Monte Carlo (and not MCMC) estimate of the Metropolis--Hastings ratio. This innovation turns out to be crucial for scalability in high dimensions. The method is demonstrated on the Ising model. Gradient-based alternatives to construct a proposal, such as Langevin and Hamiltonian Monte Carlo approaches, also arise as a natural corollary to our general procedure, and are demonstrated as well.

Exact and Approximate MCMC for Doubly-intractable Probabilistic Graphical Models Leveraging the Underlying Independence Model

Abstract

Bayesian inference for doubly-intractable pairwise exponential graphical models typically involves variations of the exchange algorithm or approximate Markov chain Monte Carlo (MCMC) samplers. However, existing methods for both classes of algorithms require either perfect samplers or sequential samplers for complex models, which are often either not available, or suffer from poor mixing, especially in high dimensions. We develop a method that does not require perfect or sequential sampling, and can be applied to both classes of methods: exact and approximate MCMC. The key to our approach is to utilize the tractable independence model underlying the intractable probabilistic graphical model for the purpose of constructing a finite sample unbiased Monte Carlo (and not MCMC) estimate of the Metropolis--Hastings ratio. This innovation turns out to be crucial for scalability in high dimensions. The method is demonstrated on the Ising model. Gradient-based alternatives to construct a proposal, such as Langevin and Hamiltonian Monte Carlo approaches, also arise as a natural corollary to our general procedure, and are demonstrated as well.

Paper Structure

This paper contains 29 sections, 5 theorems, 43 equations, 6 figures, 10 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. Related Works in Intractable Models
  3. Key Intuition Behind the Current Work
  4. Summary of Our Contributions
  5. BACKGROUND
  6. Pairwise Exponential Family Graphical Models
  7. Pseudo-marginal MCMC
  8. The Exchange Algorithm
  9. EXACT MCMC USING AN UNBIASED ESTIMATE OF THE LIKELIHOOD
  10. Variance of $T$
  11. Choosing $\nu$
  12. The Pseudo-marginal Sampler
  13. THE NOISY SAMPLER
  14. NUMERICAL EXPERIMENTS
  15. Calibration of $\nu$
  16. ...and 14 more sections

Key Result

Proposition 1

Suppose $\nu$ is such that ${\mathbb{E} }|1 - \nu \widetilde{T}|< 1$. Then ${\mathbb{E} }(|T|)$ is finite.

Figures (6)

  • Figure 1: Log posterior trace plots for $p = 20$.
  • Figure 2: Posterior mean estimates of the parameter matrix $\theta$ for $p = 20$. (a) PMRW (b) NRW (c) EXRW. The true non-zero elements in $\theta_0$ have value $-3$, with their locations indicated by black dots.
  • Figure S.1: Log posterior trace plots for the PM, N, EX samplers with Langevin proposal for the movie data.
  • Figure S.2: PM(L)-based Ising ($\theta^{50\times50}$) Model Movie Network.Thicker edges indicate higher absolute values of posterior mean estimates, while larger nodes represent higher degrees, and red versus gray distinguishes between shared and contrasting preferences.
  • Figure S.3: EX(L)-based Ising ($\theta^{50\times50}$) Model Movie Network.Thicker edges indicate higher absolute values of posterior mean estimates, while larger nodes represent higher degrees, and red versus gray distinguishes between shared and contrasting preferences.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Theorem 2
  • Corollary 1