# Lost Relatives of the Gumbel Trick

## TL;DR

This paper introduces a family of methods related to the Gumbel trick for sampling and estimating partition functions in discrete distributions, demonstrating improved properties and bounds with minimal extra computation.

## Contribution

It derives new methods related to the Gumbel trick, providing better bounds and sampling techniques for discrete graphical models.

## Key findings

- New upper and lower bounds on the log partition function.
- Development of sequential samplers for the Gibbs distribution.
- Enhanced methods with superior properties over the original Gumbel trick.

## Abstract

The Gumbel trick is a method to sample from a discrete probability distribution, or to estimate its normalizing partition function. The method relies on repeatedly applying a random perturbation to the distribution in a particular way, each time solving for the most likely configuration. We derive an entire family of related methods, of which the Gumbel trick is one member, and show that the new methods have superior properties in several settings with minimal additional computational cost. In particular, for the Gumbel trick to yield computational benefits for discrete graphical models, Gumbel perturbations on all configurations are typically replaced with so-called low-rank perturbations. We show how a subfamily of our new methods adapts to this setting, proving new upper and lower bounds on the log partition function and deriving a family of sequential samplers for the Gibbs distribution. Finally, we balance the discussion by showing how the simpler analytical form of the Gumbel trick enables additional theoretical results.