(credit: Seaman Corey Hensley, U.S. Navy, via Wikimedia Commons) 


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 H based just 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 E based only on my old familiar background models B of how reality works. This term will be quite small if, for example, I see some evidence that at first I can’t quite believe is real because none of my old background knowledge B had prepared me for it.

These terms are not fractions in the normal sense. The forward slash is not working in its usual sense here. For example, the term Pr(H/B) 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 with my familiar, old set of background assumptions B about the world.

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 am starting to believe that the hypothesis H must be right now that I’ve seen this new bit of evidence, all the while assuming that the evidence E is as I saw it, not a trick or illusion of some kind, and that the rest of my old beliefs B 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 trying to understand seems more 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 my old models of reality. When I set the values of these terms, I will assume that the evidence E is as I saw it, not some mistake or trick or delusion, and that the rest of my background ideas B about reality are valid.

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 evidence that I keep encountering in this world and the less I can explain that evidence if I don’t accept the new hypothesis. So far, so good.