1. 16.11.2011, Patrik Huber

2.  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)

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

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

5.  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

6. Earthqua ke Burglary Alarm Mary calls  auf Whiteboard Joint distributionPrior distributions (Bishop p.525) Bsp Sample: [false, true, true, true, true ] Sampling-probability for this: 0.998 * 0.001 * 0.94 * 0.7 * 0.9 = 0.06%. In large N, we expect 0.06% of the samples to be of this event If we generate 10’000 samples from the california network, and 12 of them have burglary = true, then the estimated probability of a burglary P(burglary=true) is 0.12%. John calls

7. aP(m) t.70 f.001 P(e) = 0.002 aP(j) t.90 f.05 beP(a) tt.95 tf.94 ft.29 ff.001 Conditional probability tables: P(b) = 0.001

8.  When (inverse) CDF is known:

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10.  Matlab example

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12.  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