Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator

By Arnaud Doucet, Michael K. Pitt, George Deligiannidis, Robert Kohn. 2012

322 citations recorded by the source corpus.

When an unbiased estimator of the likelihood is used within a Metropolis–Hastings chain, it is necessary to trade off the number of Monte Carlo samples used to construct this estimator against the asymptotic variances of the averages computed under this chain. Using many Monte Carlo samples will typically result in Metropolis–Hastings averages with lower asymptotic variances than the corresponding averages that use fewer samples; however, the computing time required to construct the likelihood estimator increases with the number of samples. Under the assumption that the distribution of the additive noise introduced by the loglikelihood estimator is Gaussian with variance inversely proportional to the number of samples and independent of the parameter value at which it is evaluated, we provide guidelines on the number of samples to select. We illustrate our results by considering a stochastic volatility model applied to stock index returns.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/db5fc1ffb7da5e9309bcd41d080d00baa3b9dd9a

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Citations in the corpus

Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers (2013, methodology): “While PMMH methods have been studied in a series of papers [4, 5, 16, 28, 11], a theoretical study of the PGibbs is still missing.”
Exploiting Multi-Core Architectures for Reduced-Variance Estimation with Intractable Likelihoods (2014, result): “These results are orthogonal to recent work by Doucet et al. (2012) and Sherlock et al. (2014) that deals with implementation of MCMC samplers themselves.”
The correlated pseudomarginal method (2015, methodology): “For the PM scheme, it is necessary to take N = 5000 samples to ensure that the variance of Z evaluated at a central value θ̂ is approximately one (Doucet et al., 2015).”
Bayesian Synthetic Likelihood (2017, methodology): “standard deviation of the log uSL is infinite generally, meaning that we cannot consider the guidance of Doucet et al. (2015) for pseudo-marginal methods.”
Bayesian State-Space Modelling on High-Performance Hardware Using LibBi (2013, background): “For PMMH and SMC2, a critical measure of performance is the variance of the estimate of the marginal likelihood delivered by SMC (see e.g., Pitt et al. 2012; Doucet et al. 2013; Murray et al. 2013).”
On Markov chain Monte Carlo methods for tall data (2015, background): “In this common scenario, Doucet et al. (2015) recommend keeping the variance around 1.5 as this is a value which ensures a small penalty in performance even in scenarios where 1.0 or 3.0 are actually optimal.”
Interacting Particle Markov Chain Monte Carlo (2016, background): “There exist theoretical and experimental results (Pitt et al., 2012; Bérard et al., 2014; Doucet et al., 2015) that show that the distributions of the normalisation constants are wellapproximated by their log-Normal limiting distributions.”
Particle gibbs with ancestor sampling (2014, result): “PMMH requires a larger number of particles to mix well—this is in agreement with previous analyzes and empirical studies (Doucet et al., 2014; Andrieu et al., 2010).”
Bayesian forecasting in economics and finance: A modern review (2022, methodology): “For the purpose of this review, the most pertinent development is the melding of particle filtering with MCMC in state space settings to produce a particle marginal MH (PMMH) algorithm (Andrieu et al., 2011; Flury and Shephard, 2011; Pitt et al., 2012; Doucet et al., 2015; Deligiannidis et al., 2018”
On the efficiency of pseudo-marginal random walk Metropolis algorithms (2013, methodology): “83, which corresponds (at least to 2 decimal places) with the value obtained in [14].”
Robust generalised Bayesian inference for intractable likelihoods (2021, unknown): “, 2018; Cherief-Abdellatif and Alquier, 2020; Frazier, 2020), (2) auxiliary variable MCMC (such as the exchange algorithm and pseudo-marginal MCMC; Møller et al., 2006; Murray et al., 2006; Andrieu and Roberts, 2009; Liang, 2010; Lyne et al., 2015; Doucet et al., 2015; Andrieu et al., 2020), and (3)”
Importance Sampling Squared for Bayesian Inference in Latent Variable Models (2013, background): “(2012) and Doucet et al. (2014) and make the following idealized assumption to make it possible to develop some theory.”