But let’s get back to the so-called flaw in the formula for Bayesian decision making.

Suppose I am considering a new way of explaining how some part of the world around me works. The new way is usually called a hypothesis. Then suppose I decide to do some research and I come up with a new bit of evidence that definitely relates to the matter I’m researching. What kind of process is going on in my mind as I try to decide whether this new bit of evidence is making me more likely to believe the new hypothesis or less likely to do so? This thoughtful time of curiosity and investigation, for Bayesians, is at the core of how human knowledge forms and grows. 

Mathematically, the Bayesian situation can be represented if we set the following terms: let Pr(H/B) be the degree to which I trust the hypothesis just based on the background knowledge I had before I observed any bit of new evidence. If the hypothesis seems like a fairly radical one to me, then this term is going to be pretty small. Maybe less than 1%. This new hypothesis may sound pretty far-fetched to me.

Then let Pr(E/B) be the degree to which I expected to see this new evidence occur based only on my old familiar background models of how reality works. This term will be quite small if for example I’m seeing some evidence that at first I can’t quite believe is real because none of my background knowledge had prepared me for it.

These terms are not fractions in the normal sense. The slash is not a division sign. The term Pr(H/B), for example, is called my “prior expectation”. The term refers to my estimate of the probability (Pr) that the hypothesis (H) is correct if I base that estimate only on how well the hypothesis fits together with my old, already established, familiar set of background assumptions about the world (B).

The term Pr(E/H&B) means 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.

The most important part of the equation is Pr(H/E&B). It represents how much I now am inclined to believe that the hypothesis gives a correct picture of reality after I’ve seen this new bit of evidence, while assuming that the evidence is as I saw it and 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 be expressed in the following form:                                     

 Pr(H/E&B) =  Pr(E/H&B) x  Pr(H/B)

                        Pr(E/B)  

          

While this formula looks daunting, it actually says something fairly simple. A new hypothesis that I am thinking about and trying to understand seems to me increasingly likely to be correct the more I keep encountering new evidence that the hypothesis can explain and that I can’t explain using any of the models of reality 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 I saw (E) is 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.

Increasingly, then, I tend 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 increasingly 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 set of ways of understanding the world.