10,340 research outputs found
Likelihood-Free Parallel Tempering
Approximate Bayesian Computational (ABC) methods (or likelihood-free methods)
have appeared in the past fifteen years as useful methods to perform Bayesian
analyses when the likelihood is analytically or computationally intractable.
Several ABC methods have been proposed: Monte Carlo Markov Chains (MCMC)
methods have been developped by Marjoramet al. (2003) and by Bortotet al.
(2007) for instance, and sequential methods have been proposed among others by
Sissonet al. (2007), Beaumont et al. (2009) and Del Moral et al. (2009). Until
now, while ABC-MCMC methods remain the reference, sequential ABC methods have
appeared to outperforms them (see for example McKinley et al. (2009) or Sisson
et al. (2007)). In this paper a new algorithm combining population-based MCMC
methods with ABC requirements is proposed, using an analogy with the Parallel
Tempering algorithm (Geyer, 1991). Performances are compared with existing ABC
algorithms on simulations and on a real example
Computation of Gaussian orthant probabilities in high dimension
We study the computation of Gaussian orthant probabilities, i.e. the
probability that a Gaussian falls inside a quadrant. The
Geweke-Hajivassiliou-Keane (GHK) algorithm [Genz, 1992; Geweke, 1991;
Hajivassiliou et al., 1996; Keane, 1993], is currently used for integrals of
dimension greater than 10. In this paper we show that for Markovian covariances
GHK can be interpreted as the estimator of the normalizing constant of a state
space model using sequential importance sampling (SIS). We show for an AR(1)
the variance of the GHK, properly normalized, diverges exponentially fast with
the dimension. As an improvement we propose using a particle filter (PF). We
then generalize this idea to arbitrary covariance matrices using Sequential
Monte Carlo (SMC) with properly tailored MCMC moves. We show empirically that
this can lead to drastic improvements on currently used algorithms. We also
extend the framework to orthants of mixture of Gaussians (Student, Cauchy
etc.), and to the simulation of truncated Gaussians
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