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Multiple Jump MCMC: A Scalable Algorithm for Bayesian Inference on Binary Model Spaces

Lucas Vogels, Reza Mohammadi, Marit Schoonhoven, Sinan Yildirim, Ilker Birbil

Abstract

This article considers Bayesian model inference on binary model spaces. Binary model spaces are used by a large class of models, including graphical models, variable selection, mixture distributions, and decision trees. Traditional strategies in this field, such as reversible jump or birth-death MCMC algorithms, are still popular, despite suffering from a slow exploration of the model space. In this article, we propose an alternative: the Multiple Jump MCMC algorithm. The algorithm is simple, rejection-free, and remarkably fast. When applied to undirected Gaussian graphical models, it is 100 to 200 times faster than the state-of-the-art, solving models with $500,000$ parameters in less than a minute. We provide theorems showing how accurately our algorithm targets the posterior, and we show how to apply our framework to Gaussian graphical models, Ising models, and variable selection, but note that it applies to most Bayesian posterior inference on binary model spaces.

Multiple Jump MCMC: A Scalable Algorithm for Bayesian Inference on Binary Model Spaces

Abstract

This article considers Bayesian model inference on binary model spaces. Binary model spaces are used by a large class of models, including graphical models, variable selection, mixture distributions, and decision trees. Traditional strategies in this field, such as reversible jump or birth-death MCMC algorithms, are still popular, despite suffering from a slow exploration of the model space. In this article, we propose an alternative: the Multiple Jump MCMC algorithm. The algorithm is simple, rejection-free, and remarkably fast. When applied to undirected Gaussian graphical models, it is 100 to 200 times faster than the state-of-the-art, solving models with parameters in less than a minute. We provide theorems showing how accurately our algorithm targets the posterior, and we show how to apply our framework to Gaussian graphical models, Ising models, and variable selection, but note that it applies to most Bayesian posterior inference on binary model spaces.
Paper Structure (24 sections, 5 theorems, 20 equations, 4 figures, 1 algorithm)

This paper contains 24 sections, 5 theorems, 20 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Binary model spaces:
  4. Bayesian model selection:
  5. Discrete-time Markov chains (DTMC):
  6. Continuous-time Markov chains (CTMC):
  7. Bayes factor:
  8. Proposed Approach
  9. Our Algorithm
  10. Theoretical Support
  11. Homogeneous case:
  12. Inhomogeneous case:
  13. Additional discussion and results
  14. A notable quirk of Algorithm \ref{['alg:DTMC']}:
  15. An exact (and slower) algorithm:
  16. ...and 9 more sections

Key Result

Lemma 2.1

Let $\mathcal{M} = \{0,1\}^k$ be a binary model space. A CTMC with rates where has the posterior $p(m \mid y)$ as its invariant distribution.

Figures (4)

  • Figure 1: The decay during the first $500$ iterations for a quickly (red) and slowly (black) decaying sequence.
  • Figure 2: AUC-PR scores over running time for the BD-MPL algorithm as well as for our approach with homogeneous and inhomogeneous transition probabilities. Zoomed out on the left, zoomed in on the right. Results are the average of eight replications on instances with $p=1000$ variables, $n=400$ observations, and edge density of $0.2\%$.
  • Figure 3: Scatterplot comparing the edge inclusion probabilities of the BD-MPL algorithm with our MJ-MCMC algorithm with $\varepsilon=0.3$.
  • Figure 4: The running time (left) and required MCMC iterations (right) of the BD-MPL algorithm compared to Algorithm \ref{['alg:DTMC']} for different values of $\varepsilon_s$, on the mice gene dataset with $p = 623$.

Theorems & Definitions (7)

  • Lemma 2.1
  • proof
  • Theorem 3.1: Homogeneous DTMC
  • Theorem 3.2: Inhomogeneous DTMC
  • Lemma 3.3
  • Lemma 3.4
  • proof