Abstract:
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The pseudo-marginal Metropolis-Hastings (PMMH) approach is increasingly used for inference where the likelihood is intractable but is estimated unbiasedly. It's shown recently how the approach can be made much more efficient by correlating the underlying pseudo random numbers used to form the estimates of the likelihood, which greatly speeds up the standard algorithm. We present an alternative approach that divides the random numbers into blocks so that the likelihood estimates at the proposed and current states only differ by the random numbers in one block. Our approach has several advantages. First, it provides a direct way to control the correlation between the likelihood estimates. Second, the mathematical properties of the method are simplified and transparent. Third, the blocking strategy always takes a less CPU time in each iteration than the standard and correlated PMMH. Fourth, it offers a natural way to use quasi numbers to estimate the likelihood, where we show that the number of particles required is much smaller than that in correlated PMMH. We obtain theory for selecting the optimal number of particles, and document large speed-ups in a wide range of applications.
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