Information Theory, Inference, and Learning ... - Inference Group

Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.396 30 — Efficient Monte Carlo MethodsI’ll use R to denote the number of vectors in the population. We aim tohave P ∗ ({x (r) } R 1 ) = ∏ P ∗ (x (r) ). A genetic algorithm involves moves of two orthree types.First, individual moves in which one state vector is perturbed, x (r) → x (r)′ ,which could be performed using any of the Monte Carlo methods we havementioned so far.Second, we allow crossover moves of the form x, y → x ′ , y ′ ; in a typicalcrossover move, the progeny x ′ receives half his state vector from one parent,x, and half from the other, y; the secret of success in a genetic algorithm isthat the parameter x must be encoded in such a way that the crossover oftwo independent states x and y, both of which have good fitness P ∗ , shouldhave a reasonably good chance of producing progeny who are equally fit. Thisconstraint is a hard one to satisfy in many problems, which is why geneticalgorithms are mainly talked about and hyped up, and rarely used by seriousexperts. Having introduced a crossover move x, y → x ′ , y ′ , we need to choosean acceptance rule. One easy way to obtain a valid algorithm is to accept orreject the crossover proposal using the Metropolis rule with P ∗ ({x (r) } R 1 ) asthe target density – this involves comparing the fitnesses before and after thecrossover using the ratioP ∗ (x ′ )P ∗ (y ′ )P ∗ (x)P ∗ (y) . (30.15)If the crossover operator is reversible then we have an easy proof that thisprocedure satisfies detailed balance and so is a valid component in a chainconverging to P ∗ ({x (r) } R 1 ).⊲ Exercise 30.9. [3 ] Discuss whether the above two operators, individual variationand crossover with the Metropolis acceptance rule, will give a moreefficient Monte Carlo method than a standard method with only onestate vector and no crossover.The reason why the sexual community could acquire information faster thanthe asexual community in Chapter 19 was because the crossover operationproduced diversity with standard deviation √ G, then the Blind Watchmakerwas able to convey lots of information about the fitness function by killingoff the less fit offspring. The above two operators do not offer a speed-up of√G compared with standard Monte Carlo methods because there is no killing.What’s required, in order to obtain a speed-up, is two things: multiplicationand death; and at least one of these must operate selectively. Either we mustkill off the less-fit state vectors, or we must allow the more-fit state vectors togive rise to more offspring. While it’s easy to sketch these ideas, it is hard todefine a valid method for doing it.Exercise 30.10. [5 ] Design a birth rule and a death rule such that the chainconverges to P ∗ ({x (r) } R 1 ).I believe this is still an open research problem.Particle filtersParticle filters, which are particularly popular in inference problems involvingtemporal tracking, are multistate methods that mix the ideas of importancesampling and Markov chain Monte Carlo. See Isard and Blake (1996), Isardand Blake (1998), Berzuini et al. (1997), Berzuini and Gilks (2001), Doucetet al. (2001).