Chapter 7.                           Part B 

Mathematically, the Bayesian decision situation can be represented if we let Pr(H/B) be the degree to which we trusted the hypothesis before we observed a bit of new evidence,  Pr(E/H&B)  be the degree to which we expected the evidence if, for the sake of argument, we briefly assumed that the hypothesis was true, and Pr(E/B) be the degree to which we expected this evidence to happen based on what we knew before we ever met this hypothesis (using only our old, familiar background models, and not the new hypothesis, in other words).

Note that these terms are not fractions in the normal algebraic sense at all. The term Pr(H/B) is called my “prior expectation” and should be read “my estimate of the probability that the hypothesis is a correct one if I base my estimate just on how well the hypothesis fits together with my whole familiar set of background assumptions about the world.” 

The term Pr(E/H&B) should be read “my estimate of the probability that the evidence will happen if I assume just for the sake of this term that my background assumptions and this new hypothesis are both true”. Finally, the term Pr(E/B) can be read “my estimate of the probability that the evidence (the event that the hypothesis predicts) will occur if I base my estimate only on my ordinary set of background assumptions and do not use the new hypothesis at all”.

The really important symbol in the equation comes now, and it is Pr(H/E&B). It stands for how much I now am inclined to believe that the hypothesis gives a correct picture of reality after I have seen this new bit of evidence, while taking as a given that the evidence is as I saw it - not a trick or illusion of some kind - and that the rest of my background beliefs are still in place.

Thus, the whole probability formula that describes this relationship can now be expressed in the following form:

Pr(H/E&B) =  Pr(E/H&B) x  Pr(H/B)

                      Pr(E/B)  

       Now this formula looks daunting, but it actually says something fairly simple. A new hypothesis that I am thinking about, and trying to understand, seems more and more likely to be correct the more that I keep encountering new evidence that the hypothesis can explain and that I can’t explain using any of the models of reality that I already have in my background stock of ideas. When I set the values of these terms, I will assume, at least for the time being, that the evidence that I saw (E) was as I saw it, not some mistake or trick or delusion, and that the rest of my background ideas/beliefs about reality (B) are valid.

               

        I tend more and more, then, to believe that a hypothesis is a true one the bigger Pr(E/H&B) gets and the smaller Pr(E/B) gets. 

        In other words, I more and more tend to believe that a new way of explaining the world is true, the more it can be used to explain the evidence that I keep encountering in this world, and the less I can explain that evidence if I don’t accept this new hypothesis into my total set of ways of explaining and understanding the world.