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Simulating events of unknown probabilities via reverse time martingales

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Łatuszyński, Krzysztof, Kosmidis, Ioannis, Papaspiliopoulos, Omiros and Roberts, Gareth O. (2009) Simulating events of unknown probabilities via reverse time martingales. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. Working papers, Volume 2009 (Number 30).

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Abstract

Assume that one aims to simulate an event of unknown probability
s ∈ (0, 1) which is uniquely determined, however only its approximations
can be obtained using a finite computational effort. Such settings are often
encountered in statistical simulations. We consider two specific examples.
First, the exact simulation of non-linear diffusions ([3]). Second, the celebrated Bernoulli factory problem ([10], [16], [13], [12], [9], and also [1] and
[8]) of generating an f(p)-coin given a sequence X1,X2, ... of independent
tosses of a p-coin (with known f and unknown p). We describe a general
framework and provide algorithms where this kind of problems can be
fitted and solved. The algorithms are straightforward to implement and
thus allow for effective simulation of desired events of probability s: In the
case of diffusions, we obtain the algorithm of [3] as a specific instance of
the generic framework developed here. In the case of the Bernoulli factory,
our work offers a statistical understanding of the Nacu-Peres algorithm
for f(p) = min{2p; 1 - 2ε} (which is central to the general question, c.f.
[13]) and allows for its immediate implementation that avoids algorithmic
difficulties of the original version. In the general case we link our results
to existence and construction of unbiased estimators. In particular we
show how to construct unbiased estimators given sequences of under- and
overestimating reverse time super- and submartingales.

Item Type: Working or Discussion Paper (Working Paper)
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Statistics
Library of Congress Subject Headings (LCSH): Martingales (Mathematics), Probabilities
Series Name: Working papers
Publisher: University of Warwick. Centre for Research in Statistical Methodology
Place of Publication: Coventry
Official Date: 2009
Dates:
DateEvent
2009Published
Volume: Volume 2009
Number: Number 30
Number of Pages: 11
Institution: University of Warwick
Status: Not Peer Reviewed
Access rights to Published version: Open Access (Creative Commons)
Date of first compliant deposit: 1 August 2016
Date of first compliant Open Access: 1 August 2016
Funder: Spain
Grant number: MTM2008-06660 (Spain)
Version or Related Resource: Published as: Łatuszyński, K., et al. (2011). Simulating events of unknown probabilities via reverse time martingales. Random Structures & Algorithms, 38(4), pp. 441-452. http://wrap.warwick.ac.uk/id/eprint/41524
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