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**Unformatted text preview: **Machine Learning Srihari 1 Monte Carlo Sampling Methods Sargur Srihari [email protected] Machine Learning Srihari 2 Topics 1. Markov Chain Monte Carlo 2. Basic Metropolis Algorithm 3. Markov Chains 4. Metropolis-Hastings Algorithm 5. Gibbs Sampling 6. Slice Sampling Machine Learning Srihari 3 1. Markov Chain Monte Carlo (MCMC) • Rejection sampling and importance sampling for evaluating expectations of functions – suffer from severe limitations, particularly with high dimensionality • MCMC is a very general and powerful framework – Markov refers to sequence of samples rather than the model being Markovian • Allows sampling from large class of distributions • Scales well with dimensionality • MCMC origin is in statistical physics (Metropolis 1949) Machine Learning Srihari 4 2. Basic Metropolis Algorithm • As with rejection and importance sampling use a proposal distribution (simpler distribution) • Maintain a record of current state z ( τ ) • Proposal distribution q( z | z ( τ ) ) depends on current state • Thus sequence of samples z (1) , z (2) … forms a Markov chain • Write p( z ) =p ~ ( z )/Z p where p ~ ( z ) is readily evaluated • At each cycle generate candidate z* and test for acceptance Machine Learning Srihari 5 Metropolis Algorithm • Assumes simple proposal distribution that is symmetric q( z A | z B ) = q( z B | z A ) for all z A , z B • Generated sample z* is accepted with probability – Done by choosing u ~ U(0,1) and accepting if A( z *, z ( τ ) )>u • If accepted then z ( τ +1) = z * • Otherwise: – z * is discarded, – z ( τ +1) is set to z ( τ ) and – another candidate drawn from q( z | z ( τ +1) ) Accepted steps in green Rejected steps in red Gaussian whose one-standard deviation contour is shown Machine Learning Srihari 6 Inefficiency of Random Walk • Consider simple random walk • State space z consisting of...

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