Information Theory, Inference, and Learning ... - MAELabs UCSD

Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981
You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.
About Chapter 29
The last couple of chapters have assumed that a Gaussian approximation to
the probability distribution we are interested in is adequate. What if it is not?
We have already seen an example – clustering – where the likelihood function
is multimodal, and has nasty unboundedly-high spikes in certain locations in
the parameter space; so maximizing the posterior probability and fitting a
Gaussian is not always going to work. This difficulty with Laplace’s method is
one motivation for being interested in Monte Carlo methods. In fact, Monte
Carlo methods provide a general-purpose set of tools with applications in
Bayesian data modelling and many other fields.
This chapter describes a sequence of methods: importance sampling, rejection
sampling, the Metropolis method, Gibbs sampling and slice sampling.
For each method, we discuss whether the method is expected to be useful for
high-dimensional problems such as arise in inference with graphical models.
[A graphical model is a probabilistic model in which dependencies and independencies
of variables are represented by edges in a graph whose nodes are
the variables.] Along the way, the terminology of Markov chain Monte Carlo
methods is presented. The subsequent chapter discusses advanced methods
for reducing random walk behaviour.
For details of Monte Carlo methods, theorems and proofs and a full list
of references, the reader is directed to Neal (1993b), Gilks et al. (1996), and
Tanner (1996).
In this chapter I will use the word ‘sample’ in the following sense: a sample
from a distribution P (x) is a single realization x whose probability distribution
is P (x). This contrasts with the alternative usage in statistics, where ‘sample’
refers to a collection of realizations {x}.
When we discuss transition probability matrices, I will use a right-multiplication
convention: I like my matrices to act to the right, preferring
to
u = Mv (29.1)
u T = v T M T . (29.2)
A transition probability matrix Tij or T i|j specifies the probability, given the
current state is j, of making the transition from j to i. The columns of T are
probability vectors. If we write down a transition probability density, we use
the same convention for the order of its arguments: T (x ′ ; x) is a transition
probability density from x to x ′ . This unfortunately means that you have
to get used to reading from right to left – the sequence xyz has probability
T (z; y)T (y; x)π(x).
356