Control Variates for MCMC
This chapter describes several control variate methods for improving estimates of expectations from MCMC.
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arXiv subjects
Leah South
Publications and source records attributed to Leah South.
Adaptively switching between a particle marginal Metropolis-Hastings and a particle Gibbs kernel in SMC$^2$
Sequential Monte Carlo squared (SMC$^2$; Chopin et al., 2012) methods can be used to sample from the exact posterior distribution of intractable likelihood state space models. These methods are the SMC analogue to particle Markov chain Monte Carlo (MCMC; Andrieu et al., 2010) and rely on particle MCMC kernels to mutate the particles at each iteration. Two options for the particle MCMC kernels are particle marginal Metropolis-Hastings (PMMH) and particle Gibbs (PG). We introduce a method to adaptively select the particle MCMC kernel at each iteration of SMC$^2$, with a particular focus on switching between a PMMH and PG kernel. The resulting method can significantly improve the efficiency of SMC$^2$ compared to using a fixed particle MCMC kernel throughout the algorithm. Code for our methods is available at https://github.com/imkebotha/kernel_switching_smc2.
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Automatically adapting the number of state particles in SMC$^2$
Sequential Monte Carlo squared (SMC$^2$) methods can be used for parameter inference of intractable likelihood state-space models. These methods replace the likelihood with an unbiased particle filter estimator, similarly to particle Markov chain Monte Carlo (MCMC). As with particle MCMC, the efficiency of SMC$^2$ greatly depends on the variance of the likelihood estimator, and therefore on the number of state particles used within the particle filter. We introduce novel methods to adaptively select the number of state particles within SMC$^2$ using the expected squared jumping distance to trigger the adaptation, and modifying the exchange importance sampling method of \citet{Chopin2012a} to replace the current set of state particles with the new set of state particles. The resulting algorithm is fully automatic, and can significantly improve current methods. Code for our methods is available at https://github.com/imkebotha/adaptive-exact-approximate-smc.
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Nested sampling for physical scientists
We review Skilling's nested sampling (NS) algorithm for Bayesian inference and more broadly multi-dimensional integration. After recapitulating the principles of NS, we survey developments in implementing efficient NS algorithms in practice in high-dimensions, including methods for sampling from the so-called constrained prior. We outline the ways in which NS may be applied and describe the application of NS in three scientific fields in which the algorithm has proved to be useful: cosmology, gravitational-wave astronomy, and materials science. We close by making recommendations for best practice when using NS and by summarizing potential limitations and optimizations of NS.
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Efficient and Generalizable Tuning Strategies for Stochastic Gradient MCMC
Stochastic gradient Markov chain Monte Carlo (SGMCMC) is a popular class of algorithms for scalable Bayesian inference. However, these algorithms include hyperparameters such as step size or batch size that influence the accuracy of estimators based on the obtained posterior samples. As a result, these hyperparameters must be tuned by the practitioner and currently no principled and automated way to tune them exists. Standard MCMC tuning methods based on acceptance rates cannot be used for SGMCMC, thus requiring alternative tools and diagnostics. We propose a novel bandit-based algorithm that tunes the SGMCMC hyperparameters by minimizing the Stein discrepancy between the true posterior and its Monte Carlo approximation. We provide theoretical results supporting this approach and assess various Stein-based discrepancies. We support our results with experiments on both simulated and real datasets, and find that this method is practical for a wide range of applications.
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