## The Annals of Applied Probability

### Approximation algorithms for the normalizing constant of Gibbs distributions

Mark Huber

#### Abstract

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

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

Dates
First available in Project Euclid: 19 February 2015

https://projecteuclid.org/euclid.aoap/1424355135

Digital Object Identifier
doi:10.1214/14-AAP1015

Mathematical Reviews number (MathSciNet)
MR3313760

Zentralblatt MATH identifier
1328.65011

#### Citation

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. https://projecteuclid.org/euclid.aoap/1424355135

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