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Register a new userIntroducing Monte Carlo Methods with R Solutions to Odd-Numbered Exercises
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Added by
Christian P. Robert
Publication date
2010
fields
Mathematical Statistics
and research's language is
English
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This is the solution manual to the odd-numbered exercises in our book Introducing Monte Carlo Methods with R, published by Springer Verlag on December 10, 2009, and made freely available to everyone.
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In this paper, we study the asymptotic variance of sample path averages for inhomogeneous Markov chains that evolve alternatingly according to two different $pi$-reversible Markov transition kernels $P$ and $Q$. More specifically, our main result allows us to compare directly the asymptotic variances of two inhomogeneous Markov chains associated with different kernels $P_i$ and $Q_i$, $iin{0,1}$, as soon as the kernels of each pair $(P_0,P_1)$ and $(Q_0,Q_1)$ can be ordered in the sense of lag-one autocovariance. As an important application, we use this result for comparing different data-augmentation-type Metropolis-Hastings algorithms. In particular, we compare some pseudo-marginal algorithms and propose a novel exact algorithm, referred to as the random refreshment algorithm, which is more efficient, in terms of asymptotic variance, than the Grouped Independence Metropolis-Hastings algorithm and has a computational complexity that does not exceed that of the Monte Carlo Within Metropolis algorithm.
We propose approaches for testing implementations of Markov Chain Monte Carlo methods as well as of general Monte Carlo methods. Based on statistical hypothesis tests, these approaches can be used in a unit testing framework to, for example, check if individual steps in a Gibbs sampler or a reversible jump MCMC have the desired invariant distribution. Two exact tests for assessing whether a given Markov chain has a specified invariant distribution are discussed. These and other tests of Monte Carlo methods can be embedded into a sequential method that allows low expected effort if the simulation shows the desired behavior and high power if it does not. Moreover, the false rejection probability can be kept arbitrarily low. For general Monte Carlo methods, this allows testing, for example, if a sampler has a specified distribution or if a sampler produces samples with the desired mean. The methods have been implemented in the R-package MCUnit.
Monte Carlo methods are widely used for approximating complicated, multidimensional integrals for Bayesian inference. Population Monte Carlo (PMC) is an important class of Monte Carlo methods, which utilizes a population of proposals to generate weighted samples that approximate the target distribution. The generic PMC framework iterates over three steps: samples are simulated from a set of proposals, weights are assigned to such samples to correct for mismatch between the proposal and target distributions, and the proposals are then adapted via resampling from the weighted samples. When the target distribution is expensive to evaluate, the PMC has its computational limitation since the convergence rate is $mathcal{O}(N^{-1/2})$. To address this, we propose in this paper a new Population Quasi-Monte Carlo (PQMC) framework, which integrates Quasi-Monte Carlo ideas within the sampling and adaptation steps of PMC. A key novelty in PQMC is the idea of importance support points resampling, a deterministic method for finding an optimal subsample from the weighted proposal samples. Moreover, within the PQMC framework, we develop an efficient covariance adaptation strategy for multivariate normal proposals. Lastly, a new set of correction weights is introduced for the weighted PMC estimator to improve the efficiency from the standard PMC estimator. We demonstrate the improved empirical convergence of PQMC over PMC in extensive numerical simulations and a friction drilling application.
Hamiltonian Monte Carlo (HMC) is a popular sampling method in Bayesian inference. Recently, Heng & Jacob (2019) studied Metropolis HMC with couplings for unbiased Monte Carlo estimation, establishing a generic parallelizable scheme for HMC. However, in practice a different HMC method, multinomial HMC, is considered as the go-to method, e.g. as part of the no-U-turn sampler. In multinomial HMC, proposed states are not limited to end-points as in Metropolis HMC; instead points along the entire trajectory can be proposed. In this paper, we establish couplings for multinomial HMC, based on optimal transport for multinomial sampling in its transition. We prove an upper bound for the meeting time - the time it takes for the coupled chains to meet - based on the notion of local contractivity. We evaluate our methods using three targets: 1,000 dimensional Gaussians, logistic regression and log-Gaussian Cox point processes. Compared to Heng & Jacob (2019), coupled multinomial HMC generally attains a smaller meeting time, and is more robust to choices of step sizes and trajectory lengths, which allows re-use of existing adaptation methods for HMC. These improvements together paves the way for a wider and more practical use of coupled HMC methods.
We propose Adaptive Incremental Mixture Markov chain Monte Carlo (AIMM), a novel approach to sample from challenging probability distributions defined on a general state-space. While adaptive MCMC methods usually update a parametric proposal kernel with a global rule, AIMM locally adapts a semiparametric kernel. AIMM is based on an independent Metropolis-Hastings proposal distribution which takes the form of a finite mixture of Gaussian distributions. Central to this approach is the idea that the proposal distribution adapts to the target by locally adding a mixture component when the discrepancy between the proposal mixture and the target is deemed to be too large. As a result, the number of components in the mixture proposal is not fixed in advance. Theoretically, we prove that there exists a process that can be made arbitrarily close to AIMM and that converges to the correct target distribution. We also illustrate that it performs well in practice in a variety of challenging situations, including high-dimensional and multimodal target distributions.
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