Normally, testing a new hypothesis involves performing an experiment that will generate new evidence. If the experiment delivers new evidence that was predicted by the hypothesis but not by our background set of concepts, then the hypothesis, as a way of explaining the real world, seems more likely or probable to us. The new evidence confirms the hypothesis.

But I may also decide to try to use a hypothesis and the theory or model it is based on to explain some problematic old evidence. I will be looking at whether the hypothesis and its predictions did in fact fit the old evidence situations. If I find that the hypothesis and the theory it is based on do successfully explain that problematic old evidence, what I’m actually confirming is not just the hypothesis and theory but also the consistency between the evidence, the hypothesis, and my background set of concepts.


   

                                                                            (credit: Wikipedia) 

                                            

And no, it is not obvious that evidence seen with my own eyes is 100 percent reliable, not even if I’ve seen a particular phenomenon repeated many times. Neither my longest-held, most familiar background concepts nor the sense data I see in everyday experiences are trusted that much. If they were, then I and anyone who trusts gravity and light and human anatomy would be unable to watch a good magic show without having a nervous breakdown. Elephants disappear, men float, and women get sawn in half. By pure logic, if my most basic concepts were believed at the 100 percent level, then either I would have to gouge my eyes out or go mad. But I know it’s all a trick of some kind. And I choose, for just the duration of the show, to suspend my desire to connect all my sense data with my set of background concepts. It is supposed to be a performance of fun and wonder. If I did figure out how the trick was done, I would ruin my grandkids’ fun … and my own.

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 then many other concepts also 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 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 last 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, this 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.