These two lectures will review basic Monte Carlo methodology. The starting point is the ability to generate an i.i.d. uniform sequence. Lecture 1 is a review of the transformation and accept-reject methods for generating from distributions, and their application to Monte Carlo integration. Lecture 2 reviews importance sampling, Markov chain methods, and the assessment of Monte Carlo error. Statistical examples are used to illustrate the various techniques.

A perfect sampler is an algorithm that allows one to use a Markov chain with stationary density pi to make exact (or perfect) draws from pi. A simple, three-state Markov chain is used to explain the perfect sampling algorithm called coupling from the past (CFTP) (Propp & Wilson 1996). Extending CFTP to Markov chains with uncountable state spaces has proved difficult. One success story is Murdoch & Green's (1998) multigamma coupler, which is based on the fact that a minorization condition can be used to represent the Markov transition density as a two-component mixture. The multigamma coupler is illustrated using a Markov chain from Diaconis & Freedman (1999). Our main result is a representation of pi as an infinite mixture that is based on a minorization condition. When the minorization condition is of a certain type, it is possible to make exact draws from this mixture and hence from pi. The resulting algorithm turns out to be equivalent to the multigamma coupler. Finally, when it is not possible to draw from the mixture, it is still possible to construct a statistical estimate of pi from which it is possible to sample. (This is joint work with C.P. Robert - Universite Paris Dauphine.)