How personal degrees of belief can be represented numerically by using imaginary gambles.
Chapters 1–10 were often deliberately ambiguous about different kinds of probability. That was because the basic ideas usually applied, across the board, to most kinds of probability.
Now we develop ideas that matter a lot for belief-type probabilities. They do not matter so much from the frequency point of view.
THE PROGRAM
There are three distinct steps in the argument, and each deserves a separate chapter.
▪ This chapter shows how you might use numbers to represent your degrees of belief.
▪ Chapter 14 shows why these numbers should satisfy the basic rules of probability. (And hence they should obey Bayes' Rule.)
▪ Chapter 15 shows how to use Bayes' Rule to revise or update personal probabilities in the light of new evidence. This is the fundamental motivation for the group of chapters, 13–15.
In these chapters we are concerned with a person's degrees of belief. We are talking about personal probabilities. But this approach can be used for other versions of belief-type probability, such as the logical perspective of Keynes and Carnap.
Because Bayes' Rule is so fundamental, this approach is often called Bayesian. “Belief dogmatists” are often simply called Bayesians because the use of Bayes' rule as a model of learning from experience plays such a large part in their philosophy. But notice that there are many varieties of Bayesian thinking. This perspective ranges from the personal to the logical.
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