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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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Official URL: http://www2.warwick.ac.uk/fac/sci/statistics/crism...
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 nonlinear 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 pcoin (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 NacuPeres 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 > 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 
Volume:  Volume 2009 
Number:  Number 30 
Number of Pages:  11 
Status:  Not Peer Reviewed 
Access rights to Published version:  Open Access 
Funder:  Spain 
Grant number:  MTM200806660 (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. 441452. http://wrap.warwick.ac.uk/id/eprint/41524 
Related URLs:  
References:  [1] S. Assmussen, P. W. Glynn and H. Thorisson. Stationarity Detection in the 
URI:  http://wrap.warwick.ac.uk/id/eprint/35218 
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