# Sythesis of MCMC and Belief Propagation

## Abstract

Markov Chain Monte Carlo (MCMC) and Belief Propagation (BP) are the most popular algorithms for computational inference in Graphical Models (GM). In principle, MCMC is an exact probabilistic method which, however, often suffers from exponentially slow mixing. In contrast, BP is a deterministic method, which is typically fast, empirically very successful, however in general lacking control of accuracy over loopy graphs. In this paper, we introduce MCMC algorithms correcting the approximation error of BP, i.e., we provide a way to compensate for BP errors via a consecutive BP-aware MCMC. Our framework is based on the Loop Calculus (LC) approach which allows to express the BP error as a sum of weighted generalized loops. Although the full series is computationally intractable, it is known that a truncated series, summing up all 2-regular loops, is computable in polynomial-time for planar pair-wise binary GMs and it also provides a highly accurate approximation empirically. Motivated by this, we first propose a polynomial-time approximation MCMC scheme for the truncated series of general (non-planar) pair-wise binary models. Our main idea here is to use the Worm algorithm, known to provide fast mixing in other (related) problems, and then design an appropriate rejection scheme to sample 2-regularmore »

- Authors:

- Korea Advanced Institute of Science and Technology, Daejeon (South Korea)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Korea Advanced Institute of Science and Technology, Daejeon (South Korea)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE Office of Electricity Delivery and Energy Reliability (OE)

- OSTI Identifier:
- 1254988

- Report Number(s):
- LA-UR-16-23749

- DOE Contract Number:
- AC52-06NA25396

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Computer Science

### Citation Formats

```
Ahn, Sungsoo, Chertkov, Michael, and Shin, Jinwoo.
```*Sythesis of MCMC and Belief Propagation*. United States: N. p., 2016.
Web. doi:10.2172/1254988.

```
Ahn, Sungsoo, Chertkov, Michael, & Shin, Jinwoo.
```*Sythesis of MCMC and Belief Propagation*. United States. doi:10.2172/1254988.

```
Ahn, Sungsoo, Chertkov, Michael, and Shin, Jinwoo. Fri .
"Sythesis of MCMC and Belief Propagation". United States.
doi:10.2172/1254988. https://www.osti.gov/servlets/purl/1254988.
```

```
@article{osti_1254988,
```

title = {Sythesis of MCMC and Belief Propagation},

author = {Ahn, Sungsoo and Chertkov, Michael and Shin, Jinwoo},

abstractNote = {Markov Chain Monte Carlo (MCMC) and Belief Propagation (BP) are the most popular algorithms for computational inference in Graphical Models (GM). In principle, MCMC is an exact probabilistic method which, however, often suffers from exponentially slow mixing. In contrast, BP is a deterministic method, which is typically fast, empirically very successful, however in general lacking control of accuracy over loopy graphs. In this paper, we introduce MCMC algorithms correcting the approximation error of BP, i.e., we provide a way to compensate for BP errors via a consecutive BP-aware MCMC. Our framework is based on the Loop Calculus (LC) approach which allows to express the BP error as a sum of weighted generalized loops. Although the full series is computationally intractable, it is known that a truncated series, summing up all 2-regular loops, is computable in polynomial-time for planar pair-wise binary GMs and it also provides a highly accurate approximation empirically. Motivated by this, we first propose a polynomial-time approximation MCMC scheme for the truncated series of general (non-planar) pair-wise binary models. Our main idea here is to use the Worm algorithm, known to provide fast mixing in other (related) problems, and then design an appropriate rejection scheme to sample 2-regular loops. Furthermore, we also design an efficient rejection-free MCMC scheme for approximating the full series. The main novelty underlying our design is in utilizing the concept of cycle basis, which provides an efficient decomposition of the generalized loops. In essence, the proposed MCMC schemes run on transformed GM built upon the non-trivial BP solution, and our experiments show that this synthesis of BP and MCMC outperforms both direct MCMC and bare BP schemes.},

doi = {10.2172/1254988},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Fri May 27 00:00:00 EDT 2016},

month = {Fri May 27 00:00:00 EDT 2016}

}