A class of adaptive sampling methods is introduced for efficient posterior and predictive simulation. The proposed methods are robust in the sense that they can handle target distributions that exhibit non-elliptical shapes such as multimodality and skewness. The basic method makes use of sequences of importance weighted Expectation Maximization steps in order to efficiently construct a mixture of Student-t densities that approximates accurately the target distribution -typically a posterior distribution, of which we only require a kernel - in the sense that the Kullback-Leibler divergence between target and mixture is minimized. We label this approach Mixture of t by Importance Sampling and Expectation Maximization (MitISEM). We also introduce three extensions of the basic MitISEM approach. First, we propose a method for applying MitISEM in a sequential manner, so that the candidate distribution for posterior simulation is cleverly updated when new data become available. Our results show that the computational effort reduces enormously. This sequential approach can be combined with a tempering approach, which facilitates the simulation from densities with multiple modes that are far apart. Second, we introduce a permutation-augmented MitISEM approach, for importance sampling from posterior distributions in mixture models without the requirement of imposing identification restrictions on the model's mixture regimes' parameters. Third, we propose a partial MitISEM approach, which aims at approximating the marginal and conditional posterior distributions of subsets of model parameters, rather than the joint. This division can substantially reduce the dimension of the approximation problem.

Kullback-Leibler divergence, Metropolis-Hastings algorithm, expectation maximazation, importance sampling, mixture GARCH models, mixture of Student-t distributions, predictive likelihoods, value at risk
Bayesian Analysis (jel C11), Simulation Methods; Monte Carlo Methods; Bootstrap Methods (jel C15), Time-Series Models; Dynamic Quantile Regressions (jel C22)
Tinbergen Institute
hdl.handle.net/1765/22332
Tinbergen Institute Discussion Paper Series
Discussion paper / Tinbergen Institute
Tinbergen Institute

Hoogerheide, L.F, Opschoor, A, & van Dijk, H.K. (2011). A Class of Adaptive EM-based Importance Sampling Algorithms for Efficient and Robust Posterior and Predictive Simulation (No. TI 2011-004/4). Discussion paper / Tinbergen Institute. Tinbergen Institute. Retrieved from http://hdl.handle.net/1765/22332