Improving Approximate Bayesian Computation via Quasi-Monte Carlo

Buchholz, Alexander; Chopin, Nicolás

doi:10.1080/10618600.2018.1497511

Cited by 19 publications

(14 citation statements)

References 37 publications

(48 reference statements)

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“…The (in some cases) optimal convergence rates, as well as sound statistical properties, of QMC have recently led to interest within statistics (e.g. Gerber and Chopin, 2015;Buchholz and Chopin, 2017). A related method with non-uniform weights was explored in Stein (1995a,b).…”

Section: Monte Carlo Methodsmentioning

confidence: 99%

Probabilistic Integration: A Role in Statistical Computation?

Briol¹,

Oates²,

Girolami³

et al. 2019

Statist. Sci.

153

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A research frontier has emerged in scientific computation, wherein discretisation error is regarded as a source of epistemic uncertainty that can be modelled. This raises several statistical challenges, including the design of statistical methods that enable the coherent propagation of probabilities through a (possibly deterministic) computational work-flow, in order to assess the impact of discretisation error on the computer output. This paper examines the case for probabilistic numerical methods in routine statistical computation. Our focus is on numerical integration, where a probabilistic integrator is equipped with a full distribution over its output that reflects the fact that the integrand has been discretised. Our main technical contribution is to establish, for the first time, rates of posterior contraction for one such method. Several substantial applications are provided for illustration and critical evaluation, including examples from statistical modelling, computer graphics and a computer model for an oil reservoir. arXiv:1512.00933v6 [stat.ML] 18 Oct 2017 2 computer code to reproduce experiments reported in this paper can be downloaded from http://www. warwick.ac.uk/fxbriol/probabilistic_integration.

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Section: Monte Carlo Methodsmentioning

confidence: 99%

Probabilistic Integration: A Role in Statistical Computation?

Briol¹,

Oates²,

Girolami³

et al. 2019

Statist. Sci.

153

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show abstract

“…, S}, the function E[ψ(y s i (θ), x i )|ũ s i = u] does not depend on x so that Theorem 1 can be applied to this conditional expectation, assuming it has finite variance. This insight was used to derive CLTs for moments based on hybrid sequences which combine MC draws with qMC sequences in Ökten et al (2006) and Buchholz and Chopin (2017) for bounded ψ. The results in Proposition 2 extend these results to unbounded empirical processes over θ ∈ Θ, allowing ψs n to be non-smooth in θ.…”

Section: Algorithm 1 Simulated Methods Of Moments For Static Modelsmentioning

confidence: 99%

“…These sequences were initially designed to compute integrals of iid sequences and can achieve faster than √ nrate convergence. More details are given in Section 2. qMC integration has been extended to non-linear state-space filtering (Gerber andChopin, 2015, 2017), MCMC sampling (Owen and Tribble, 2005) and importance sampling for ABC estimation (Buchholz and Chopin, 2017). A key takeaway from these papers is that a lot of care is required in implementing qMC integration in non iid settings (MCMC or filtering) where 'naive' implementations may be inconsistent.…”

Section: Related Literaturesmentioning

confidence: 99%

A Scrambled Method of Moments

Forneron

2019

Preprint

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Quasi-Monte Carlo (qMC) methods are a powerful alternative to classical Monte-Carlo (MC) integration. Under certain conditions, they can approximate the desired integral at a faster rate than the usual Central Limit Theorem, resulting in more accurate estimates. This paper explores these methods in a simulation-based estimation setting with an emphasis on the scramble of Owen (1995). For cross-sections and short-panels, the resulting Scrambled Method of Moments simply replaces the random number generator with the scramble (available in most softwares) to reduce simulation noise. Scrambled Indirect Inference estimation is also considered. For time series, qMC may not apply directly because of a curse of dimensionality on the time dimension. A simple algorithm and a class of moments which circumvent this issue are described. Asymptotic results are given for each algorithm. Monte-Carlo examples illustrate these results in finite samples, including an income process with "lots of heterogeneity."

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“…As a result, this strategy helps to reduce the variance in the outer sampling. Buchholz and Chopin [3] applied this strategy in ABC. They found that the resulting ABC estimate has a lower variance than the MC counter-part.…”

Section: (B)mentioning

confidence: 99%

Unbiased MLMC-based variational Bayes for likelihood-free inference

He¹,

Xu²,

Wang³

2021

Preprint

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Variational Bayes (VB) is a popular tool for Bayesian inference in statistical modeling. Recently, some VB algorithms are proposed to handle intractable likelihoods with applications such as approximate Bayesian computation. In this paper, we propose several unbiased estimators based on multilevel Monte Carlo (MLMC) for the gradient of Kullback-Leibler divergence between the posterior distribution and the variational distribution when the likelihood is intractable, but can be estimated unbiasedly. The new VB algorithm differs from the VB algorithms in the literature which usually render biased gradient estimators. Moreover, we incorporate randomized quasi-Monte Carlo (RQMC) sampling within the MLMC-based gradient estimators, which was known to provide a favorable rate of convergence in numerical integration. Theoretical guarantees for RQMC are provided in this new setting. Numerical experiments show that using RQMC in MLMC greatly speeds up the VB algorithm, and finds a better parameter value than some existing competitors do.

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Improving Approximate Bayesian Computation via Quasi-Monte Carlo

Cited by 19 publications

References 37 publications

Probabilistic Integration: A Role in Statistical Computation?

Probabilistic Integration: A Role in Statistical Computation?

A Scrambled Method of Moments

Unbiased MLMC-based variational Bayes for likelihood-free inference

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