The Library
From the Bernoulli factory to a dice enterprise via perfect sampling of Markov chains
Tools
Tools
Tools
Morina, Giulio, Latuszynski, Krzysztof, Nayar, Piotr and Wendland, Alex (2022) From the Bernoulli factory to a dice enterprise via perfect sampling of Markov chains. Annals of Applied Probability, 32 (1). pp. 327-359. doi:10.1214/21-AAP1679 ISSN 1050-5164.
|
PDF
WRAP-from-Bernoulli-factory-dice-enterprise-via-perfect-sampling-Markov-chains-Latuszynski-2021.pdf - Accepted Version - Requires a PDF viewer. Download (585Kb) | Preview |
Official URL: https://doi.org/10.1214/21-AAP1679
Abstract
Given a p-coin that lands heads with unknown probability p, we wish to produce an f(p)-coin for a given function f:(0,1)→(0,1). This problem is commonly known as the Bernoulli factory and results on its solvability and complexity have been obtained in (ACM Trans. Model. Comput. Simul. 4 (1994) 213–219; Ann. Appl. Probab. 15 (2005) 93–115). Nevertheless, generic ways to design a practical Bernoulli factory for a given function f exist only in a few special cases. We present a constructive way to build an efficient Bernoulli factory when f(p) is a rational function with coefficients in R. Moreover, we extend the Bernoulli factory problem to a more general setting where we have access to an m-sided die and we wish to roll a v-sided one; that is, we consider rational functions between open probability simplices. Our construction consists of rephrasing the original problem as simulating from the stationary distribution of a certain class of Markov chains—a task that we show can be achieved using perfect simulation techniques with the original m-sided die as the only source of randomness. In the Bernoulli factory case, the number of tosses needed by the algorithm has exponential tails and its expected value can be bounded uniformly in p. En route to optimizing the algorithm we show a fact of independent interest: every finite, integer valued, random variable will eventually become log-concave after convolving with enough Bernoulli trials.
| Item Type: | Journal Article | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics Faculty of Science, Engineering and Medicine > Science > Statistics |
||||||||
| Library of Congress Subject Headings (LCSH): | Bernoulli numbers, Markov processes, Perfect simulation (Statistics) | ||||||||
| Journal or Publication Title: | Annals of Applied Probability | ||||||||
| Publisher: | Institute of Mathematical Statistics | ||||||||
| ISSN: | 1050-5164 | ||||||||
| Official Date: | February 2022 | ||||||||
| Dates: |
|
||||||||
| Volume: | 32 | ||||||||
| Number: | 1 | ||||||||
| Page Range: | pp. 327-359 | ||||||||
| DOI: | 10.1214/21-AAP1679 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Reuse Statement (publisher, data, author rights): | |||||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||||
| Date of first compliant deposit: | 12 March 2021 | ||||||||
| Date of first compliant Open Access: | 24 March 2022 | ||||||||
| Related URLs: | |||||||||
| Open Access Version: |
Request changes or add full text files to a record
Repository staff actions (login required)
 |
View Item |
Downloads
Downloads per month over past year
