Pattern Recognition and Machine Learning

Exercises 521 10.29 ( ) www Show that the functionf(x)=ln(x)is concave for 0 <x<∞ by computing its second derivative. Dete ...

522 10. APPROXIMATE INFERENCE whereZjis the normalization constant defined by (10.197). By applying this result recursively, and ...

11 Sampling Methods For most probabilistic models of practical interest, exact inference is intractable, and so we have to resor ...

524 11. SAMPLING METHODS Figure 11.1 Schematic illustration of a function f(z) whose expectation is to be evaluated with respect ...

11. SAMPLING METHODS 525 straightforward to sample from the joint distribution (assuming that it is possible to sample from the ...

526 11. SAMPLING METHODS methods for statistical inference. Diagnostic tests for convergence of Markov chain Monte Carlo algorit ...

11.1. Basic Sampling Algorithms 527 Figure 11.2 Geometrical interpretation of the trans- formation method for generating nonuni- ...

528 11. SAMPLING METHODS y 1 = z 1 ( −2lnz 1 r^2 ) 1 / 2 (11.10) y 2 = z 2 ( −2lnz 2 r^2 ) 1 / 2 (11.11) Exercise 11.4 wherer^2 ...

11.1. Basic Sampling Algorithms 529 Figure 11.4 In the rejection sampling method, samples are drawn from a sim- ple distribution ...

530 11. SAMPLING METHODS Figure 11.5 Plot showing the gamma distribu- tion given by (11.15) as the green curve, with a scaled Ca ...

11.1. Basic Sampling Algorithms 531 Figure 11.7 Illustrative example of rejection sampling involving sampling from a Gaussian di ...

532 11. SAMPLING METHODS Figure 11.8 Importance sampling addresses the prob- lem of evaluating the expectation of a func- tionf( ...

11.1. Basic Sampling Algorithms 533 samples{z(l)}drawn fromq(z) E[f]= ∫ f(z)p(z)dz = ∫ f(z) p(z) q(z) q(z)dz  1 L ∑L l=1 p(z(l) ...

534 11. SAMPLING METHODS distributionp(z). If, as is often the case,p(z)f(z)is strongly varying and has a sig- nificant proporti ...

11.1. Basic Sampling Algorithms 535 value forkin that any value that is sufficiently large to guarantee a bound on the desired d ...

536 11. SAMPLING METHODS evaluated directly using the original samples together with the weights, because E[f(z)] = ∫ f(z)p(z)dz ...

11.2. Markov Chain Monte Carlo 537 Now suppose we move from a maximum likelihood approach to a full Bayesian treatment in which ...

538 11. SAMPLING METHODS and which scales well with the dimensionality of the sample space. Markov chain Monte Carlo methods hav ...

11.2. Markov Chain Monte Carlo 539 Figure 11.9 A simple illustration using Metropo- lis algorithm to sample from a Gaussian dist ...

540 11. SAMPLING METHODS conditional probabilities for subsequent variables in the form oftransition probabil- itiesTm(z(m),z(m+ ...