16.11.2011, Patrik Huber

One of our goals: Evaluation of the posterior p(Z|X)

Exact inference In practice: often infeasible to evaluate the

posterior distribution, or to compute expectations with respect to this distribution▪ Dimensionality of latent space too high▪ Posterior distribution has highly complex form,

expectations not analytically tractable▪ Integrations may not have analytical solutions

Approximate inference Deterministic approximation: Variational

algorithms (last week) Stochastic approximation: Monte Carlo methods

(today)

Pick a number uniformly at random. What’s the probability of hitting the red area?

What’s the probability of a dart thrown uniformly at random hitting the red area?

Really took off in 1940’s Motivation was nuclear

power, simulate samples (=neutrons), exploring the behavior of neutron chain reactions in nuclear devices

Stan Ulam / Von Neumann: Inspired by the idea of doing sampling using the newly developed electronic computing techniques (ENIAC)

50’s: Metropolis-Sampling

a P(m)

t .70

f .001

P(e) = 0.002

a P(j)

t .90

f .05

b e P(a)

t t .95

t f .94

f t .29

f f .001

Conditional probability tables:

P(b) = 0.001

When (inverse) CDF is known:

Matlab example

Sampling algorithms Can generate exact results if given

infinite computational resource (in contrast to variational inference)

Can be computationally demanding Difficult to know whether a sampling

scheme is generating independent samples from the required distribution