Overview

We have seen in chapter 1 that the most general solution of an inverse problem provides a probability distribution over the model space. It is only when the probability distribution in the model space is very simple (for instance, when it has only one maximum) that analytic techniques can be used to characterize it.

For more general probability distributions, one needs to perform an extensive exploration of the model space. Except for problems with a very small number of dimensions, this exploration cannot be systematic (as the number of required points grows too rapidly with the dimension of the space). Well-designed random (or pseudorandom) explorations can solve many complex problems. These random methods were jokingly called Monte Carlo methods by the team at Los Alamos that was at the origin, among others, of the Metropolis sampling algorithm, and the name Monte Carlo has now become established.

2.1 Introduction

That Monte Carlo (i.e., random) methods can be used for computation has been known for centuries. For instance, one can use a Monte Carlo method to evaluate the number ? : on a regular floor, made of strips of equal width ?, one throws needles of length ?/2. The probability that a needle will intersect a groove in the floor equals...