It‘s important to point out here that the idea behind H&B 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 be integrated into my old background set of concepts and beliefs only if some of those are removed by careful, gradual tinkering and if many other concepts are adjusted.

Similarly, in the term Pr(H/E&B), the E&B is trying to capture something that 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.

Thus, all the whole formula really does is try to capture the general 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 of them may have to be radically updated and revised at any time.

In short, 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.

And what of the spirit of Bayesianism? Bayesian thinking requires us to be willing to float all of our concepts, even our most deeply held ones. Some are more central, and we can stand on them more often and with more confidence. A few we may believe almost, but not quite, absolutely. But in the end, none of our concepts is irreplaceable.

For our species, the mind is our means of surviving. If it has to, it will adapt to almost anything. Most of us choose to gamble most heavily on the concepts we use to organize most of our sense data and memories most of the time.

I use my concepts to organize both the memories already stored in my brain and the new sense data that are flooding into my brain all the time. I keep trying to acquire more concepts, including concepts for organizing other concepts that will enable me to utilize my memories more efficiently to make faster and better decisions and to act increasingly effectively. In this constant, restless, searching mental life of mine, I never trust anything absolutely. If I did, a simple magic show would mesmerize and paralyze me. Or reduce me to catatonia.

When I see elephants disappear, women get sawn in half, and men defy gravity, and all come through their ordeals in fine shape, some of my most basic and trusted concepts are obviously being violated. But I choose to stand by my concepts in almost every such case, not because I am certain they are perfect but because they have been tested and found effective over so many trials and for so long that I’m willing to keep gambling on them. I don’t know whether they are “sure things,” but they do seem like the most promising of the options available to me.

                        

                                        Harry Houdini with his “disappearing” elephant, Jennie

Life is constantly making demands on me to move and keep moving. I have to gamble on some things; I go with my best horses. And sometimes, I change my mind.

This mental flexibility on my part means that the critics of Bayesianism simply haven’t grasped its spirit. Bayesianism is telling us pretty much what Thomas Kuhn said in his influential book The Structure of Scientific Revolutions. We are constantly adjusting our mental constructs to try to make our ways of dealing with reality more effective.

And when a researcher begins to grasp a new hypothesis and the model or theory it is based on, the resulting experience is like a philosophical or religious awakening—profound, all-encompassing, and even life altering. Everything changes when we accept a new model or theory—because we change. In order to “get it,” we have to change. We have to eliminate some of the old beliefs from our familiar background set.

And what of the shifting nature of our view of reality and the gambling spirit that is implicit in the Bayesian model? The general tone of all our experiences tells us that this overall view of our world and ourselves, though it may seem scary or, perhaps for more confident individuals, challenging—it’s just life.

We have now arrived at a point where we can feel confident that Bayesianism gives us a solid base on which to build further reasoning. It can answer its critics decisively—both those who attack it with real-world counterexamples and those who attack it with pure logic.

For now, then, let us be content to summarize our points so far in a new chapter devoted solely to that summing up.

Notes

1. Thomas Kuhn, The Structure of Scientific Revolutions (Chicago: The University of Chicago Press, 3rd ed., 1996).