Introduction

We have described various probability distributions, on the assumption that the parameters are known. We have dealt with classical assignments of probability, in which events were judged as being equally likely, and their close relation, the maximum-entropy distributions of Chapter 6. We have considered the Central Limit Theorem – the model of sums – and often this can be used to justify the use of the normal distribution. We might have an excellent candidate distribution for a particular application. The next step in our analysis is to estimate the model parameters, for example the mean µ and variance σ2 for a normal distribution. We often wish to use measured data to estimate these parameters. The subject of the present chapter, ‘inference’, refers to the use of data to estimate the parameter(s) of a probability distribution. This is key to the estimation of the probability of a future event or quantity of a similar kind. The fitting of distributions to data is outlined in Chapter 11.

References of interest for this chapter are de Finetti (1937, 1974), Raiffa and Schlaifer (1961), Lindley (1965, Part 2), Heath and Sudderth (1976), O'Hagan (1994) and Bernardo and Smith (1994).