Pattern Recognition and Machine Learning

11.2. Markov Chain Monte Carlo 541 for some set of mixing coefficientsα 1 ,...,αKsatisfyingαk 0 and ∑ kαk=1. Alternatively, the ...

542 11. SAMPLING METHODS Figure 11.10 Schematic illustration of the use of an isotropic Gaussian proposal distribution (blue cir ...

11.3. Gibbs Sampling 543 in some particular order or by choosing the variable to be updated at each step at random from some dis ...

544 11. SAMPLING METHODS To show that this procedure samples from the required distribution, we first of all note that the distr ...

11.3. Gibbs Sampling 545 Figure 11.11 Illustration of Gibbs sampling by alter- nate updates of two variables whose distribution ...

546 11. SAMPLING METHODS Figure 11.12 The Gibbs sampling method requires samples to be drawn from the conditional distribution o ...

11.4. Slice Sampling 547 p ̃(z) z(τ) z u (a) p ̃(z) z(τ) z zmin u zmax (b) Figure 11.13 Illustration of slice sampling. (a) For ...

548 11. SAMPLING METHODS candidate point is drawn uniformly from this reduced region and so on, until a value ofzis found that l ...

11.5. The Hybrid Monte Carlo Algorithm 549 for each position variable there is a corresponding momentum variable, and the joint ...

550 11. SAMPLING METHODS During the evolution of this dynamical system, the value of the HamiltonianHis constant, as is easily s ...

11.5. The Hybrid Monte Carlo Algorithm 551 Section 11.3 we see that this also leaves the desired distribution invariant. Noting ...

552 11. SAMPLING METHODS 11.5.2 Hybrid Monte Carlo As we discussed in the previous section, for a nonzero step size, the discre ...

11.5. The Hybrid Monte Carlo Algorithm 553 ri zi r′i z′i Figure 11.14 Each step of the leapfrog algorithm (11.64)–(11.66) modifi ...

554 11. SAMPLING METHODS good approximation to the true continuous-time dynamics, it is necessary for the leapfrog integration s ...

11.6. Estimating the Partition Function 555 where{z(l)}are samples drawn from the distribution defined bypG(z). If the dis- trib ...

556 11. SAMPLING METHODS Exercises 11.1 ( ) www Show that the finite sample estimatorf̂defined by (11.2) has mean equal toE[f]an ...

Exercises 557 Figure 11.15 A probability distribution over two variablesz 1 andz 2 that is uniform over the shaded regions and t ...

558 11. SAMPLING METHODS 11.17 ( )www Verify that the two probabilities (11.68) and (11.69) are equal, and hence that detailed b ...

AppendixA InChapter9,wediscussedprobabilisticmodelshavingdiscretelatentvariables,such asthemixtureofGaussians.Wenowexploremodels ...

560 12.CONTINUOUSLATENTVARIABLES Figure12.1 A syntheticdataselobtainedbytakingoneoftheoff-linedigitimagesandcreatingmulti- pleco ...