The Annals of Applied Probability

Approximation algorithms for the normalizing constant of Gibbs distributions

Mark Huber

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Consider a family of distributions $\{\pi_{\beta}\}$ where $X\sim\pi_{\beta}$ means that $\mathbb{P}(X=x)=\exp(-\beta H(x))/Z(\beta)$. Here $Z(\beta)$ is the proper normalizing constant, equal to $\sum_{x}\exp(-\beta H(x))$. Then $\{\pi_{\beta}\}$ is known as a Gibbs distribution, and $Z(\beta)$ is the partition function. This work presents a new method for approximating the partition function to a specified level of relative accuracy using only a number of samples, that is, $O(\ln(Z(\beta))\ln(\ln(Z(\beta))))$ when $Z(0)\geq1$. This is a sharp improvement over previous, similar approaches that used a much more complicated algorithm, requiring $O(\ln(Z(\beta))\ln(\ln(Z(\beta)))^{5})$ samples.

Article information

Ann. Appl. Probab., Volume 25, Number 2 (2015), 974-985.

First available in Project Euclid: 19 February 2015

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Zentralblatt MATH identifier

Primary: 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) [See also 68W20, 68W40] 65C60: Computational problems in statistics
Secondary: 65C05: Monte Carlo methods

Integration Monte Carlo methods cooling schedule self-reducible


Huber, Mark. Approximation algorithms for the normalizing constant of Gibbs distributions. Ann. Appl. Probab. 25 (2015), no. 2, 974--985. doi:10.1214/14-AAP1015.

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