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Introduction to sampling

Introduction to sampling

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Introduction to sampling

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  1. Introduction to sampling Discussion on An Introduction to MCMC for Machine Learning, Andrieu et al., 2001

  2. Sampling • What is sampling? • Useful for? • Bayesian inference and learning • Normalization • Marginalization • Expectation • Optimization • Model selection

  3. Sampling • Monte Carlo principle (pg. 5) • Law of large numbers • Central limit theorem

  4. Rejection sampling • Rejection • Drawbacks?

  5. Importance sampling • Importance • Drawbacks?

  6. Importance sampling • {ui, wi }: Sampled representation off(u) • Expectation under f(u)

  7. Markov chains • Homogeneous: • T is time-invariant • Represented using a transition matrix Series of samples such that

  8. Markov chains • Stationary distribution • Conditions for stationary distribution • Irreducible? • Aperiodic? • Detailed balance • Sufficient condition for stationarity of p

  9. MCMC • Markov Chain Monte Carlo • Markov Chain • Monte Carlo • Metropolis Hastings • Special cases • Independent sampler • Metropolis algorithm

  10. Metropolis-Hastings • Target distribution: p(x) • Set up a Markov chain with stationary p(x) • Resulting chain has the desired stationary • Detailed balance Propose (Easy to sample from q) with probability otherwise

  11. Metropolis-Hastings • “Mixing”

  12. Gibbs sampler • Idea • Proposals • Acceptance probability • Always possible?