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Introduction of Markov Chain Monte Carlo Jeongkyun Lee

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Usage Why MCMC is called MCMC MCMC methods Appendix Reference Contents 2

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Goal : 1) Estimate an unknown target distribution (or posterior) for a complex function, or 2) draw samples from the distribution. 1. Simulation Draw samples from a probability governed by a system. 2. Integration / computing Integrate or compute a high dimensional function 3. Optimization / Bayesian inference Ex. Simulated annealing, MCMC-based particle filter 4. Learning MLE learning, unsupervised learning Usage 3

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Why MCMC is called MCMC 4

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3. Markov Chain Monte Carlo Construct a Markov Chain representing a target distribution. http://www.kev-smith.com/tutorial/flash/markov_chain.swf http://www.kev-smith.com/tutorial/flash/markov_chain.swf Why MCMC is called MCMC 6 …

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MCMC Methods 7 MetropolisMetropolis-Hastings

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MCMC Methods 8

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3. Reversible Jump(or trans-dimensional) MCMC When the dimension of the state is changed, Additionally consider a move type. MCMC Methods 10

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1. Markov Chain property Stationary distribution (or detailed balance) Irreducible (all pi > 0) Aperiodic Appendix 11

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2. MH sampling as a Markov Chain The transition probability kernel in the MH algorithm Thus, if the MH kernel satisfies then the stationary distribution from this kernel corresponds to draws from the target distribution. Appendix 12

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2. MH sampling as a Markov Chain Appendix 13

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http://vcla.stat.ucla.edu/old/MCMC/MCMC_tutorial.htm http://vcla.stat.ucla.edu/old/MCMC/MCMC_tutorial.htm http://www.kev-smith.com/tutorial/rjmcmc.php http://www.kev-smith.com/tutorial/rjmcmc.php http://www.cs.bris.ac.uk/~damen/MCMCTutorial.htm http://www.cs.bris.ac.uk/~damen/MCMCTutorial.htm B. Walsh, “Markov Chain Monte Carlo and Gibbs Sampling”, Lecture Notes, MIT, 2004 Reference 14

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