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On perfect sampling: ROCFTP with Metropolis-multishift coupler

Majid Nabipoor

TL;DR

The paper tackles exact sampling for complex Bayesian posteriors, including those with unbounded state spaces, where standard MCMC diagnostics are insufficient. It introduces ROCFTP with a Metropolis-multishift coupler to produce i.i.d. samples without constructing a target-specific Markov chain, and it uses the Most Interest Range (MIR) to bound starting points and control hitting unlikely regions. The authors establish convergence properties, derive an exponential decay bound for coupling probabilities, and demonstrate robust performance on unimodal and multimodal targets, supported by simulations and QQ analyses. An R package ROCFTP.MMS is provided to enable practical adoption, underscoring the method’s potential for efficient, exact sampling in challenging Bayesian settings.

Abstract

ROCFTP is a perfect sampling algorithm that employs various random operations, and requiring a specific Markov chain construction for each target. To overcome this requirement, the Metropolis algorithm is incorporated as a random operation within ROCFTP. While the Metropolis sampler functions as a random operation, it isn't a coupler. However, by employing normal multishift coupler as a symmetric proposal for Metropolis, we obtain ROCFTP with Metropolis-multishift. Initially designed for bounded state spaces, ROCFTP's applicability to targets with unbounded state spaces is extended through the introduction of the Most Interest Range (MIR) for practical use. It was demonstrated that selecting MIR decreases the likelihood of ROCFTP hitting $MIR^C$ by a factor of (1 - ε), which is beneficial for practical implementation. The algorithm exhibits a convergence rate characterized by exponential decay. Its performance is rigorously evaluated across various targets, and tests ensure its goodness of fit. Lastly, an R package is provided for generating exact samples using ROCFTP Metropolis-multishift.

On perfect sampling: ROCFTP with Metropolis-multishift coupler

TL;DR

The paper tackles exact sampling for complex Bayesian posteriors, including those with unbounded state spaces, where standard MCMC diagnostics are insufficient. It introduces ROCFTP with a Metropolis-multishift coupler to produce i.i.d. samples without constructing a target-specific Markov chain, and it uses the Most Interest Range (MIR) to bound starting points and control hitting unlikely regions. The authors establish convergence properties, derive an exponential decay bound for coupling probabilities, and demonstrate robust performance on unimodal and multimodal targets, supported by simulations and QQ analyses. An R package ROCFTP.MMS is provided to enable practical adoption, underscoring the method’s potential for efficient, exact sampling in challenging Bayesian settings.

Abstract

ROCFTP is a perfect sampling algorithm that employs various random operations, and requiring a specific Markov chain construction for each target. To overcome this requirement, the Metropolis algorithm is incorporated as a random operation within ROCFTP. While the Metropolis sampler functions as a random operation, it isn't a coupler. However, by employing normal multishift coupler as a symmetric proposal for Metropolis, we obtain ROCFTP with Metropolis-multishift. Initially designed for bounded state spaces, ROCFTP's applicability to targets with unbounded state spaces is extended through the introduction of the Most Interest Range (MIR) for practical use. It was demonstrated that selecting MIR decreases the likelihood of ROCFTP hitting by a factor of (1 - ε), which is beneficial for practical implementation. The algorithm exhibits a convergence rate characterized by exponential decay. Its performance is rigorously evaluated across various targets, and tests ensure its goodness of fit. Lastly, an R package is provided for generating exact samples using ROCFTP Metropolis-multishift.

Paper Structure

This paper contains 19 sections, 2 theorems, 14 equations, 23 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Literature review
  3. Preliminaries
  4. Random operations
  5. Normal multishift coupler
  6. CFTP
  7. ROCFTP
  8. Metropolis-multishift coupler
  9. Metropolis sampler
  10. Metropolis-multishift coupler
  11. Convergence properties
  12. The most interest range
  13. How many starting paths are needed?
  14. ROCFTP with Metropolis-multishift coupler
  15. Discussion
  16. ...and 4 more sections

Key Result

Proposition 1

Suppose $Q^t$ represents the joint distribution of $t$ iterations of the Markov chain $\{ X_t \}$ governed by a $\pi$-irreducible Metropolis-multishift coupler, and $T^*$ is the coupling time of the two extreme auxiliary chains $\{ \hat{0}_t \}$ and $\{ \hat{1}_t \}$ with the primary chain $\{ X_t \

Figures (23)

  • Figure 1: Illustration of normal multishift coupler.
  • Figure 2: Illustration of CFTP in monotone case with normal multishift coupler.
  • Figure 3: Illustration of ROCFTP: output is $X_{mT}$.
  • Figure 4: Applying Metropolis simultaneously on $(-10,10)$ with target $N(0,1)$ and Markov chain $X_{t+1}= X_t + N(0,1)$.
  • Figure 5: Applying Metropolis-multishift simultaneously on $(-30,30)$ with target $0.2N(-5,1)+0.2N(5,1)+0.6N(15,1)$ and proposal multishift $N(0,3.5^2)$.
  • ...and 18 more figures

Theorems & Definitions (4)

  • Definition 1
  • Proposition 1
  • Definition 2
  • Proposition 2