ABSTRACT
Both Bayesians and frequentists recognize the importance of the parameters of probability distributions. In fact, we cannot compute probabilities of some of the most basic events using these distributions without knowledge of them. In fact, Bayesians almost always, while frequentists sometimes, refer to the probability distribution of the event x as [ ]| ,x θP or ( )| ,f x θ explicitly denoting the relationship between the probability distribution of x and the parameter on which use of that distribution is based. However, we commonly do not know the value of the relevant parameter, signified in this discussion by θ. Frequentists estimate θ as though it had one and only one value in the population. However, to the Bayesian, the parameter θ has intrinsic variability of its own. Bayesians capture this useful information about the location of θ in the prior distribution, denoted as [ ],θP or ( ).π θ
Identifying this information has two useful purposes. The first is that it permits a worker to identify the unconditional probability distribution of x, by carrying out a weighed sum of the values of the parameter of θ. This is what the Law of Total Probability permits, i.e.,
This equality allows us to compute the unconditional distribution of x from its conditional distribution, i.e., converting [ ]|x θP to [ ],xP a conversion known as compounding. However, useful as the compounding process is,* it is only an intermediate point in the Bayes computation. The Bayesian is not so interested in starting with
* Examples are provided in Chapters Two and Three.
