MCMC Convergence Slides

# MCMC Convergence Slides - Convergence of MCMC Algorithms in...

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Convergence of MCMC Algorithms in Finite Samples Anna Kormilitsina and Denis Nekipelov SMU and UC Berkeley September 2009 Kormilitsina, Nekipelov Divergence of MCMC September 2009

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Introduction Motivation MCMC widely used Bayesian method in frequentist context Due to simplicity of coding and promise to converge to global extremum, popular in structural estimation In general requires verification of set of “regularity conditions”: practitioners rarely consider These assumptions can be violated in very common structural models Violation can lead to divergence of algorithm We use example of macro DSGE model: erroneous inference can lead to misinterpretation of policy parameters Kormilitsina, Nekipelov Divergence of MCMC September 2009
Introduction Our approach MCMC chain: complex dynamic system; in general, stability in such systems can be an issue We use continuous-time approximation for Markov chain; allows us to use results on Lyapunov stability Lyapunov in -stability implies divergence of MCMC chain (with probability 1) We formulate requirements for objective function to guarantee stability If stability is local, convergence will not occur from some regions of parameter space Kormilitsina, Nekipelov Divergence of MCMC September 2009

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Introduction Our results: preview MCMC can diverge even when structural model is identified Create test for stability of chain initialized in particular subset of parameter space: based on Lyapunov stability of its continuous-time approximation Test creates potential for “automatic” choice of support of prior distribution Results are illustrated using commonly used model of (Christiano, Eichenbaum, Evans, 2005) Find that even in simple case MCMC does not have global convergence Kormilitsina, Nekipelov Divergence of MCMC September 2009
Theory Our definition of “MCMC” MCMC - very large class of algorithms We analyze narrow class of quasi-Bayesian procedures in (Chernozhukov, Hong, 2003) Based on using objective for M-estimation to form a quasi-density (Laplace quasi-posterior) Idea: convergence of statistics of quasi-posterior to extremum estimates (Bernstein-von Mises theorem) leads to convergence of quasi-posterior moment to the M-estimator Study this problem in the context of sampling based on Metropolis-Hastings Kormilitsina, Nekipelov Divergence of MCMC September 2009

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Theory Characterization of Markov chains Create sample of parameter draws from quasi-posterior Procedure can be treated as dynamic system Elements: proposal density, objective function + tuning parameters; output { θ t } Usually have large samples, proposals can be chosen normal/truncated normal Sequence of draws can be approximated by diffusion-based stochastic process Kormilitsina, Nekipelov Divergence of MCMC September 2009
Theory Characterization of Markov chains Result from theory of SDE: form stochastic differential equation for Langevin diffusion process L t dL t = 1 2 log f ( L t ) dt + dW t where W t standard Brownian motion f will be the stationary distribution of the solution to Langevin equation

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