Gibbs Sampling Qianji Zheng Oct. 5th, 2010.

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Presentation transcript:

Gibbs Sampling Qianji Zheng Oct. 5th, 2010

Outline Motivation & Basic Idea Algorithm Example Applications Why Gibbs Works

Gibbs Sampling: Motivation Gibbs sampling is a particular form of Markov chain Monte Carlo (MCMC) algorithm for approximating joint and marginal distribution by sampling from conditional distributions.

Gibbs Sampling: Basic Idea If the joint distribution is not known explicitly or is difficult to sample from directly, but the conditional distribution is known or easy to sample from. Even if the joint distribution is known, the computational burden needed to calculate it may be huge. Gibbs Sampling algorithm could generate a sequence of samples from conditional individual distributions, which constitutes a Markov chain, to approximate the joint distribution.

Characteristics of Gibbs Sampling Algorithm A particular Markov Chain Monte Carlo (MCMC) algorithm Sample from conditional distribution while other parameters are fixed Update a single parameter at a time

Gibbs Sampling Algorithm Let be the conditional distribution of the element given all the other parameters minus the , then Gibbs Sampling for an m-component variable is given by the transition from to generated as: Given an arbitrary initial value

Gibbs Sampling Algorithm Contd… Steps 1 through m can be iterated J times to get , j = 1, 2, … , J. 2. The joint and marginal distributions of generated converge at an exponential rate to joint and marginal distribution of , as . 3. Then the joint and marginal distributions of can be approximated by the empirical distributions of M simulated values (j=L+1,…, L+M). The mean of the marginal distribution of may be approximated by

Gibbs Sampling Algorithm In BN

Example

Example Example refer to Gibbs Sampling for Approximate Inference in Bayesian Networks http://www-users.cselabs.umn.edu/classes/Spring-2010/csci5512/notes/gibbs.pdf

Gibbs Sampling: Applications Gibbs Sampling algorithm has been widely used on a broad class of areas, e.g. , Bayesian networks, statistical inference, bioinformatics, econometrics. The power of Gibbs Sampling is: 1. Approximate joint and marginal distribution 2. Estimate unknown parameters 3. Compute an integral (e.g. mean, median, etc)

Why Gibbs Works The Gibbs sampling can simulate the target distribution by constructing a Gibbs sequence which converges to a stationary distribution that is independent of the starting value. The stationary distribution is the target distribution.

Online Resources Reference Gibbs Sampling for Approximate Inference in Bayesian Networks http://www-users.cselabs.umn.edu/classes/Spring-2010/csci5512/notes/gibbs.pdf Markov Chain Monte Carlo and Gibbs Sampling http://membres-timc.imag.fr/Olivier.Francois/mcmc_gibbs_sampling.pdf Markov Chains, the Gibbs Sampler and Data Augmentation http://athens.src.uchicago.edu/jenni/econ350/Salvador/h4.pdf Reference Kim, C. J. and Nelson, C. R. (1999), State-Space Models with Regime Switching, Cambridge, Massachusetts: MIT Press.