CPSC540

Nando de FreitasMarch, 2013University of British Columbia

Importance sampling &

Markov chain Monte Carlo (MCMC)

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)

Bayesian logistic regressionThe logistic regression model specifies the probability of a binary outputyi ∈ {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

[

−

] [

−

]

Importance sampling

Importance sampling

Importance sampling

Example: Logistic Regression

Un-normalized importance sampling

Markov Chain Monte Carlo

Markov Chain Monte Carlo

Markov Chain Monte Carlo

Markov Chain Monte Carlo

Markov Chain Monte Carlo

Markov Chain Monte Carlo

Markov Chain Monte Carlo

Markov Chain Monte Carlo

Metropolis-Hastings for logistic regression

MCMC: Metropolis-Hastings

MCMC: Metropolis-Hastings

MCMC: Choosing the Right Proposal

MCMC: Theory

MCMC: Theory

Variations of Metropolis-Hastings

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

FIO2PRESS

INSUFFANESTHTPR

LVFAILURE

ERRBLOWOUTPUTSTROEVOLUMELVEDVOLUME

HYPOVOLEMIA

CVP

BP

Bayesian graphical models and Gibbs

Gibbs Sampling

Gibbs Sampling

Gibbs Sampling For Graphical models

Auxiliary Variable Samplers

Hybrid (Hamiltonian) Monte Carlo

Hybrid Monte Carlo

Next lecture

In the next lecture, we look at constrained optimization and sparse methods in learning.