Zero-Variance Principle for Monte Carlo Algorithms

By R. Assaraf, Michel Caffarel. 1999

We present a general approach to greatly increase at little cost the efficiency of Monte Carlo algorithms. To each observable to be computed we associate a renormalized observable (improved estimator) having the same average but a different variance. By writing down the zero-variance condition a fundamental equation determining the optimal choice for the renormalized observable is derived (zero-variance principle for each observable separately). We show, with several examples including classical and quantum Monte Carlo calculations, that the method can be very powerful.

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Citations in the corpus

Control functionals for Monte Carlo integration (2014, methodology): “Equation (1) appears in Stein’s classical test for approximate normality (Stein, 1970) and related to (but simpler than) control variates proposed by Assaraf and Caffarel (1999) and Mira et al. (2013). We make the following assumption (see for example equation (9) of Mira et al.”
Operator Variational Inference (2016, background): “There are many ways to construct such classes [1, 2].”
Control variates for estimation based on reversible Markov chain Monte Carlo samplers (2012, background): “In a different context, and partly motivated by considerations from statistical mechanics, Assaraf and Caffarel (1999) introduced a family of control variates that they called ‘zero-variance estimators’.”
Applications of quantum Monte Carlo methods in condensed systems (2010, unknown): “such that 〈#T|Â|#T〉 =〈 #T|Â|#T〉 while the fluctuations of AL are substantially reduced [44–48].”
The Controlled Thermodynamic Integral for Bayesian Model Comparison (2014, methodology): “In this paper we extend the zero-variance (ZV) control variate technique, introduced in the physics literature by Assaraf and Caffarel (1999), to estimators of model evidence that are based on MCMC and thermodynamic integration (TI; Frenkel and Smit, 2002).”
Zero variance Markov chain Monte Carlo for Bayesian estimators (2010, methodology): “Besides translating into statistical terms the paper by [4], the main effort of our work has been the discussion of unbiasedness and convergence of the ZV-MCMC estimator.”
Exploiting Multi-Core Architectures for Reduced-Variance Estimation with Intractable Likelihoods (2014, background): “(2013), following Assaraf and Caffarel (1999), since in several special cases the score has perfect canonical correlation with the target, generating an estimate that has zero sampling variance.”
Convergence rates for a class of estimators based on Stein’s method (2016, methodology): “This definition can be motivated in several ways, including via Schrödinger Hamiltonians [2] and via the generator method of Barbour applied to an overdamped Langevin diffusion [18].”
Zero Variance Differential Geometric Markov Chain Monte Carlo Algorithms (2014, background): “Assaraf and Caffarel (1999) suggest defining the auxiliary function g̃(θ) as g̃(θ) = g(θ) + H[ψ(θ)]√ π(θ) , (2) where H[ψ(θ)] denotes the Schrödinger Hamiltonian given by H[ψ(θ)] = −1 2 ∆θ[ψ(θ)] + ψ(θ) 2 √ π(θ) ∆θ[ √ π(θ)], (3) ∆θ := ∑nθ i=1 ∂ 2/∂θ2i is the Laplace operator and ψ(θ) is the…”