Zero variance Markov chain Monte Carlo for Bayesian estimators

By Antonietta Mira, Reza Solgi, D. Imparato. 2010

97 citations recorded by the source corpus.

Interest is in evaluating, by Markov chain Monte Carlo (MCMC) simulation, the expected value of a function with respect to a, possibly unnormalized, probability distribution. A general purpose variance reduction technique for the MCMC estimator, based on the zero-variance principle introduced in the physics literature, is proposed. Conditions for asymptotic unbiasedness of the zero-variance estimator are derived. A central limit theorem is also proved under regularity conditions. The potential of the idea is illustrated with real applications to probit, logit and GARCH Bayesian models. For all these models, a central limit theorem and unbiasedness for the zero-variance estimator are proved (see the supplementary material available on-line).

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

The Controlled Thermodynamic Integral for Bayesian Model Evidence Evaluation (2016, methodology): “Here Mira, Solgi, and Imparato (2013) proposed to use where (cid:7) θ = ∇ θ · ∇ θ is the Laplacian operator, the trial function Q ( θ ) is a polynomial in θ and is the score function.”
Convergence rates for a class of estimators based on Stein’s method (2016, background): “Control variates can alternatively be constructed based on gradient information on the sampling distribution [2,31,35,36].”
Zero Variance Differential Geometric Markov Chain Monte Carlo Algorithms (2014, methodology): “So ZV-RMHMC, for the linear and quadratic polynomials proposed by Mira et al. (2012), requires only to save z(θ) at each iteration of the Gibbs sampler.”
The Controlled Thermodynamic Integral for Bayesian Model Comparison (2014, methodology): “Following Mira et al. (2013), it is possible to rearrange the terms on the right hand side of Eqn.”
Exploiting Multi-Core Architectures for Reduced-Variance Estimation with Intractable Likelihoods (2014, methodology): “Recently Mira et al. (2013) proposed to use the score vector u ( θ | y ) := ∇ θ log p ( θ | y ) as the basis for a set of control variates, since this can be guaranteed to have zero expectation under mild boundary conditions (described below).”
Control variates for estimation based on reversible Markov chain Monte Carlo samplers (2012, methodology): “The Assaraf–Cafarel estimators have been adapted and applied to problems in statistics by Mira et al. (2003, 2010) and Dalla Valle and Leisen (2010).”