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# Importance sampling & Markov chain Monte Carlo (MCMC) - Nando de Freitas March, 2013 University of British Columbia

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We propose a new class of learning algorithms that combines variational approximation and Markov chain Monte Carlo (MCMC) simulation. Naive algorithms that use the variational approximation as proposal distribution can perform poorly because this approximation tends to underestimate the true variance and other features of the data. We solve this problem by introducing more sophisticated MCMC algorithms. One of these algorithms is a mixture of two MCMC kernels: a random walk Metropolis kernel and a block Metropolis-Hastings (MH) kernel with a variational approximation as proposal distribution. The MH kernel allows one to locate regions of high probability efficiently. The Metropolis kernel allows us to explore the vicinity of these regions. This algorithm outperforms variational approximations because it yields slightly better estimates of the mean and considerably better estimates of higher moments, such as covariances. It also outperforms standard MCMC algorithms because it locates the regions of high probability quickly, thus speeding up convergence. We also present an adaptive MCMC algorithm that iterates between improving the variational approximation and improving the MCMC approximation. We demonstrate the algorithms on the problem of Bayesian parameter estimation for logistic (sigmoid) belief networks.

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### Importance sampling & Markov chain Monte Carlo (MCMC) - Nando de Freitas March, 2013 University of British Columbia

1. 1. CPSC540 Nando de Freitas March, 2013 University of British Columbia Importance sampling & Markov chain Monte Carlo (MCMC)
2. 2. Outline of the lecture This is about Monte Carlo methods. We will revise importance sampling. Revise how Google works (Markov chains). Introduce Markov chain Monte Carlo (MCMC)
3. 3. Bayesian logistic regression The logistic regression model speciﬁes the probability of a binary output yi ∈ {0, 1} given the input xi as follows: p(y|X, θ) = n i=1 Ber(yi|sigm(xiθ)) = n i=1 1 1 + e−xiθ yi 1 − 1 1 + e−xiθ 1−yi i=1 − −
4. 4. Importance sampling
5. 5. Importance sampling
6. 6. Importance sampling
7. 7. Example: Logistic Regression
8. 8. Un-normalized importance sampling
9. 9. Markov Chain Monte Carlo
10. 10. Markov Chain Monte Carlo
11. 11. Markov Chain Monte Carlo
12. 12. Markov Chain Monte Carlo
13. 13. Markov Chain Monte Carlo
14. 14. Markov Chain Monte Carlo
15. 15. Markov Chain Monte Carlo
16. 16. Markov Chain Monte Carlo
17. 17. Metropolis-Hastings for logistic regression
18. 18. MCMC: Metropolis-Hastings
19. 19. MCMC: Metropolis-Hastings
20. 20. MCMC: Choosing the Right Proposal
21. 21. MCMC: Theory
22. 22. MCMC: Theory
23. 23. Variations of Metropolis-Hastings
24. 24. Extending MH to directed probabilistic graphical models MINVOLSET PCWP CO HRBP HREKG HRSAT ERRCAUTERHRHISTORY CATECHOL SAO2 EXPCO2 ARTCO2 VENTALV VENTLUNG VENITUBE DISCONNECTVENTMACHKINKEDTUBEINTUBATIONPULMEMBOLUS PAP SHUNT ANAPHYLAXIS MINOVL PVSAT FIO2 PRESS INSUFFANESTHTPR LVFAILURE ERRBLOWOUTPUTSTROEVOLUMELVEDVOLUME HYPOVOLEMIA CVP BP
25. 25. Bayesian graphical models and Gibbs
26. 26. Gibbs Sampling
27. 27. Gibbs Sampling
28. 28. Gibbs Sampling For Graphical models
29. 29. Auxiliary Variable Samplers
30. 30. Hybrid (Hamiltonian) Monte Carlo
31. 31. Hybrid Monte Carlo
32. 32. Next lecture In the next lecture, we look at constrained optimization and sparse methods in learning.