Nonexistence of a class of variate generation schemes

Henderson, Shane G.

doi:10.1016/s0167-6377(02)00217-1

Cited by 6 publications

(3 citation statements)

References 20 publications

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“…samples from, and a function ϕ : X → R such that µ(ϕ) > 0, one wishes to simulate a random variable with expectation µ(ϕ) that is almost surely non-negative. The non-existence of a general procedure for solving this problem without any assumptions on µ and ϕ has been shown in Jacob and Thiery (2013) following other non-existence results such as Keane and O'Brien (1994) and Henderson and Glynn (2003). We provide a positive result in the case where ϕ is a bounded function and µ(ϕ) ≥ δ > 0 for a known constant δ.…”

Section: B Lemmas For F and Fmentioning

confidence: 57%

Perfect simulation using atomic regeneration with application to Sequential Monte Carlo

Lee,

Doucet,

Łatuszyński

2014

Preprint

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Consider an irreducible, Harris recurrent Markov chain of transition kernel Π and invariant probability measure π. If Π satisfies a minorization condition, then the split chain allows the identification of regeneration times which may be exploited to obtain perfect samples from π. Unfortunately, many transition kernels associated with complex Markov chain Monte Carlo algorithms are analytically intractable, so establishing a minorization condition and simulating the split chain is challenging, if not impossible. For uniformly ergodic Markov chains with intractable transition kernels, we propose two efficient perfect simulation procedures of similar expected running time which are instances of the multigamma coupler and an imputation scheme. These algorithms overcome the intractability of the kernel by introducing an artificial atom and using a Bernoulli factory. We detail an application of these procedures when Π is the recently introduced iterated conditional Sequential Monte Carlo kernel. We additionally provide results on the general applicability of the methodology, and how Sequential Monte Carlo methods may be used to facilitate perfect simulation and/or unbiased estimation of expectations with respect to the stationary distribution of a non-uniformly ergodic Markov chain.

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Section: B Lemmas For F and Fmentioning

confidence: 57%

Perfect simulation using atomic regeneration with application to Sequential Monte Carlo

Lee,

Doucet,

Łatuszyński

2014

Preprint

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“…Since then, the problem has been generalized into finding an algorithm that given a p-coin -a coin that lands heads with unknown probability p -can produce an f (p)-coin for a given function f : D ⊆ (0, 1) → (0, 1). Keane and O'Brien [23] referred to this problem as the Bernoulli Factory and, motivated by problems in regenerative steady-state simulations [1,14], identified the class of functions f for which it is solvable. Since then, other studies have been carried out to provide ways of constructing and analysing the Bernoulli Factory algorithms [17,18,19,24,26,28,29] as well as extending it to quantum settings [6,31,40] and specialised multivariate scenarios [7,19].…”

Section: Introductionmentioning

confidence: 99%

From the Bernoulli Factory to a Dice Enterprise via Perfect Sampling of Markov Chains

Morina¹,

Łatuszyński²,

Nayar³

et al. 2019

Preprint

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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 [23,29]. 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; i.e., we consider rational functions f : ∆ m−1 → ∆ v−1 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.

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“…An ongoing research in Markov chain Monte Carlo and rejection sampling indicates that the Bernoulli factory problem is not only of theoretical interest (c.f. also [1], [8] and [2] Chapter 16). However using approximate algorithms in these applications perturbs simulations in a way difficult to quantify.…”

Section: Introductionmentioning

confidence: 99%

Simulating events of unknown probabilities via reverse time martingales

Łatuszyński

Kosmidis

Papaspiliopoulos

et al. 2011

Random Struct Algorithms

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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] 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.

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Nonexistence of a class of variate generation schemes

Cited by 6 publications

References 20 publications

Perfect simulation using atomic regeneration with application to Sequential Monte Carlo

Perfect simulation using atomic regeneration with application to Sequential Monte Carlo

From the Bernoulli Factory to a Dice Enterprise via Perfect Sampling of Markov Chains

Simulating events of unknown probabilities via reverse time martingales

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