(credit: Sigismund von Dobschütz, via Wikimedia Commons)

It’s important to point out here that the idea behind H&B, the set of the new hypothesis plus my background concepts, is more complex than the equation can capture. This part of the formula should be read: “If I integrate the hypothesis into my whole background concept set.” The formula can only attempt to capture in symbols something that is almost not capturable. This is so because the point of positing a hypothesis, H, is that it does not fit neatly into my background set of beliefs. It is built around a new way of seeing and comprehending reality, and thus it will only be integrated into my old background set of concepts and beliefs if some of those are removed, by careful, gradual tinkering, and then many other concepts also are adjusted.

Similarly, in the term Pr(H/E&B), the E&B is trying to capture something no math expression can capture. E&B is trying to say: “If I take both the evidence and my set of background beliefs to be 100 percent reliable.” 

But that way of stating the E&B part of the term merely highlights the issue with problematic old evidence. This evidence is problematic because I can’t make it consistent with my set of background concepts and beliefs, no matter how I tinker with them.

All the whole formula really does is try to capture the gist of human thinking and learning. It is a useful approximation, but we can’t become complacent about this formula for the Bayesian model of human thinking and learning any more than we can become complacent about any of our concepts. And that thought is consistent with the spirit of Bayesianism. It tells us not to become too blindly attached to any of our concepts; any one of them may have to be radically updated and revised at any time.

Thus, on closer examination, the criticism of Bayesianism which says the Bayesian model can’t explain why we find a fit between a hypothesis and some problematic old evidence so reassuring—turns out to be not a fatal criticism, but more of a useful tool, one that we may use to deepen and broaden our understanding of the Bayesian model of human thinking. 

We can hold onto the Bayesian model if we accept that all the concepts, thought patterns, and patterns of neuron firings in the brain—hypotheses, evidence, and assumed background concepts—are forming, reforming, aligning, realigning, and floating in and out of one another all the time, even concepts as basic as the ones we have about gravity, matter, space, and time. This whole view of this scary idea called “Bayesianism” arises naturally if we simply apply Bayesianism to itself. In short, we must learn and adjust until we die.