Large Sample Asymptotics of the Pseudo-Marginal Method

Schmon, Sebastian M.; Deligiannidis, George; Doucet, Arnaud; Pitt, M.

doi:10.48550/arxiv.1806.10060

Cited by 2 publications

(3 citation statements)

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“…Thus the algorithm proceeds exactly as a standard MH algorithm with proposal density q(θ |θ), with the difference that likelihood evaluations p(y|θ) are replaced by estimators p(y|θ, U ) with U ∼ m θ (•). The performance of the pseudo-marginal algorithm depends on the likelihood estimator: lower variance estimators typically yield ergodic averages with lower asymptotic variance, but the cost of producing lower variance estimators tends to be higher which leads to a trade-off analyzed in detail in , Schmon et al [2018].…”

Section: Pseudo-marginal Metropolis-hastingsmentioning

confidence: 99%

“…We investigate the distribution of meeting times as a function of T , with N scaling linearly with T . Such a scaling is motivated through the guarantee that the variance of the log-likelihood estimates obtained at each iteration are asymptotically constant [Bérard et al, 2014, Schmon et al, 2018. For the model as before, we consider a grid of T ∈ {100, ..., 1000}, using a single realisation of the data.…”

Section: Effect Of the Time Horizonmentioning

confidence: 99%

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Unbiased Markov chain Monte Carlo for intractable target distributions

Middleton¹,

Deligiannidis²,

Doucet³

et al. 2018

Preprint

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Performing numerical integration when the integrand itself cannot be evaluated point-wise is a challenging task that arises in statistical analysis, notably in Bayesian inference for models with intractable likelihood functions. Markov chain Monte Carlo (MCMC) algorithms have been proposed for this setting, such as the pseudo-marginal method for latent variable models and the exchange algorithm for a class of undirected graphical models. As with any MCMC algorithm, the resulting estimators are justified asymptotically in the limit of the number of iterations, but exhibit a bias for any fixed number of iterations due to the Markov chains starting outside of stationarity. This "burn-in" bias is known to complicate the use of parallel processors for MCMC computations. We show how to use coupling techniques to generate unbiased estimators in finite time, building on recent advances for generic MCMC algorithms. We establish the theoretical validity of some of these procedures, by extending existing results to cover the case of polynomially ergodic Markov chains. The efficiency of the proposed estimators is compared with that of standard MCMC estimators, with theoretical arguments and numerical experiments including state space models and Ising models.

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Section: Pseudo-marginal Metropolis-hastingsmentioning

confidence: 99%

Section: Effect Of the Time Horizonmentioning

confidence: 99%

Unbiased Markov chain Monte Carlo for intractable target distributions

Middleton¹,

Deligiannidis²,

Doucet³

et al. 2018

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The second term is proportional to the computing time for a single draw, a measure that is independent of the implementation. Starting with Pitt et al (2012), there is a large literature showing that CT is minimized by an m that targets a variance of the log of the likelihood estimator around 1 (Doucet et al, 2015;Sherlock et al, 2015;Tran et al, 2017;Deligiannidis et al, 2018;Schmon et al, 2018). Recent developments in pseudo-marginal methods induce dependence in the auxiliary variables u over the MCMC iterations such that the likelihood estimates over the iterations become dependent (Deligiannidis et al, 2018).…”

Section: Pseudo-marginal Mcmc Denote the Model Parameter Vector Bymentioning

confidence: 99%

Hamiltonian Monte Carlo with Energy Conserving Subsampling

Dang,

Quiroz,

Kohn

et al. 2017

Preprint

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Hamiltonian Monte Carlo (HMC) samples efficiently from high-dimensional posterior distributions with proposed parameter draws obtained by iterating on a discretized version of the Hamiltonian dynamics. The iterations make HMC computationally costly, especially in problems with large datasets, since it is necessary to compute posterior densities and their derivatives with respect to the parameters. Naively computing the Hamiltonian dynamics on a subset of the data causes HMC to lose its key ability to generate distant parameter proposals with high acceptance probability. The key insight in our article is that efficient subsampling HMC for the parameters is possible if both the dynamics and the acceptance probability are computed from the same data subsample in each complete HMC iteration. We show that this is possible to do in a principled way in a HMC-within-Gibbs framework where the subsample is updated using a pseudo marginal MH step and the parameters are then updated using an HMC step, based on the current subsample. We show that our subsampling methods are fast and compare favorably to two popular sampling algorithms that utilize gradient estimates from data subsampling. We also explore the current limitations of subsampling HMC algorithms by varying the quality of the variance reducing control variates used in the estimators of the posterior density and its gradients.

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Large Sample Asymptotics of the Pseudo-Marginal Method

Cited by 2 publications

References 0 publications

Unbiased Markov chain Monte Carlo for intractable target distributions

Unbiased Markov chain Monte Carlo for intractable target distributions

Hamiltonian Monte Carlo with Energy Conserving Subsampling

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