# Simple perfect samplers using monotone birth-and-death processes

## TL;DR

This paper introduces simple perfect sampling algorithms based on monotone birth-and-death processes that efficiently generate exact samples from arbitrary finite discrete distributions, with bounds on their expected running times.

## Contribution

It constructs monotone birth-and-death processes matching the target distribution and derives bounds for the coalescence and running times of two perfect samplers, including a memory-efficient method.

## Key findings

- Derived upper bounds for coalescence times of the processes.
- Established bounds for the running times of Doubling and Read-once CFTP algorithms.
- Demonstrated the effectiveness of the proposed sampler for unnormalized distributions.

## Abstract

This paper proposes simple perfect samplers using monotone birth-and-death processes (BD-processes), which draw samples from an arbitrary finite discrete target distribution. We first construct a monotone BD-process whose stationary distribution is equal to the target distribution. We then derive upper bounds for the expected coalescence time of the copies of the monotone BD-process. We also establish upper bounds for the expected values and tail probabilities of the running times of two perfect samplers, which are Doubling CFTP and Read-once CFTP using our monotone BD-process. The latter sampler can draw samples exactly from unnormalized target distributions with little memory consumption.