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A non-asymptotic error analysis for parallel Monte Carlo estimation from many short Markov chains

Austin Brown

TL;DR

A novel many-short-chains Monte Carlo estimator is constructed by averaging over multiple independent sums from Markov chains of a guaranteed short length by a method to improve estimation properties in importance sampling by additionally simulating a Markov chain.

Abstract

Single-chain Markov chain Monte Carlo simulates realizations from a Markov chain to estimate expectations with the empirical average. The single-chain simulation is generally of considerable length and restricts many advantages of modern parallel computation. This paper constructs a novel many-short-chains Monte Carlo (MSC) estimator by averaging over multiple independent sums from Markov chains of a guaranteed short length. The computational advantage is the independent Markov chain simulations can be fast and may be run in parallel. The MSC estimator requires an importance sampling proposal and a drift condition on the Markov chain without requiring convergence analysis on the Markov chain. A non-asymptotic error analysis is developed for the MSC estimator under both geometric and multiplicative drift conditions. Empirical performance is illustrated on an autoregressive process and the Pólya-Gamma Gibbs sampler for Bayesian logistic regression to predict cardiovascular disease.

A non-asymptotic error analysis for parallel Monte Carlo estimation from many short Markov chains

TL;DR

A novel many-short-chains Monte Carlo estimator is constructed by averaging over multiple independent sums from Markov chains of a guaranteed short length by a method to improve estimation properties in importance sampling by additionally simulating a Markov chain.

Abstract

Single-chain Markov chain Monte Carlo simulates realizations from a Markov chain to estimate expectations with the empirical average. The single-chain simulation is generally of considerable length and restricts many advantages of modern parallel computation. This paper constructs a novel many-short-chains Monte Carlo (MSC) estimator by averaging over multiple independent sums from Markov chains of a guaranteed short length. The computational advantage is the independent Markov chain simulations can be fast and may be run in parallel. The MSC estimator requires an importance sampling proposal and a drift condition on the Markov chain without requiring convergence analysis on the Markov chain. A non-asymptotic error analysis is developed for the MSC estimator under both geometric and multiplicative drift conditions. Empirical performance is illustrated on an autoregressive process and the Pólya-Gamma Gibbs sampler for Bayesian logistic regression to predict cardiovascular disease.
Paper Structure (10 sections, 13 theorems, 98 equations, 3 figures, 1 algorithm)

This paper contains 10 sections, 13 theorems, 98 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The MSC estimator
  3. Mean squared error analysis under geometric drift conditions
  4. Concentration results under multiplicative drift conditions
  5. Toy example: Autoregressive process
  6. Cardiovascular disease prediction with the Pólya-Gamma Gibbs sampler
  7. Discussion and conclusions
  8. Supporting technical results for Section \ref{['section:mse_error_bound']}
  9. Supporting technical results for Section \ref{['section:concentration_mult']}
  10. Supporting technical results for Section \ref{['section:pg_application']}

Key Result

Proposition 1

Assume the drift condition assumption:drift holds, and the set $C$ is defined by eq:set_C. If $Y_i \sim Q$ for $i = 1, \ldots, N$ are independent and $\sup_{x \in C} V(x) < \infty$, where $\gamma_R = \gamma + K/R$ and $A_R = \sup_{x \in C} \left[ V(x) + (\gamma - 1) f(x) \right] + 2 K - 1.$

Figures (3)

  • Figure 1: Computation of the lower bound on the number of Markov chains $M$ and initialization samples $N$ for the MSC estimator using the autoregressive process with respect to increasing dimensions.
  • Figure 2: Figures \ref{['figure:ar_sim_a']} and Figure \ref{['figure:ar_sim_b']} plot the coordinates of the MSC mean within 2 estimated standard errors compared to standard MC. Figure \ref{['figure:ar_sim_b']} plots the length of the Markov sample paths over the independent runs of the Markov chains.
  • Figure 3: The mean within 2 estimated standard deviations of MSC the Pólya-Gamma sampler using compared to the single-chain Pólya-Gamma sampler and RWM algorithm.

Theorems & Definitions (26)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Proposition 3
  • Proposition 4
  • Theorem 2
  • Proposition 5
  • Proposition 6
  • Lemma 1
  • proof
  • ...and 16 more