Now, all of this may begin to seem intuitive, but once we have a formula set down it also is open to criticism and attack, and the critics of Bayesianism see a flaw in it that they consider fatal. The flaw they point to is usually called “the problem of old evidence.”

One of the ways a new hypothesis gets more respect among experts in the field the hypothesis covers is by its ability to explain old evidence that no other theories in the field have been able to explain. For example, physicists all over the world felt that the probability they assigned to Einstein’s theory of relativity took a huge jump upward when Einstein used the theory to account for the changes in the orbit of the planet Mercury—changes that were familiar to physicists but that had long defied explanation by the old familiar Newtonian model.

                                             Representation of the inner solar system

The constant, gradual shift in that planets’ orbit had baffled astronomers for decades since they had first acquired instruments that enabled them to detect that shift. This shift could not be explained by any pre-relativity models. But relativity theory could describe this gradual shift and make predictions about it that were extremely accurate. In other branches of science, instances of hypotheses that worked to explain old, anomalous phenomena could easily be listed. Kuhn, in his book, gives many of them.¹

What is wrong with Bayesianism, then, according to its critics, is that it cannot explain why we give more credence to a theory when we realize it can be used to explain pieces of old, anomalous evidence that had long defied explanation by the established theories in the field. When the formula given above is applied to this situation, critics say Pr(E/B) has to be considered equal to 100 percent, or absolute certainty, since the evidence (E) has been accepted as having been accurately observed for a long time.

For the same reasons, Pr(E/H&B) has to be thought of as equal to 100 percent because the evidence has been reliably observed and recorded many times since long before we ever had this new theory to consider adding to our stock of usable ideas. When these two quantities are put into the equation, according to the critics, it looks like this:

Pr(H/E&B) = Pr(H/B)

This new version of the formula emerges because Pr(E/B) and Pr(E/H&B) are now both equal to 100 percent, or a probability of 1.0, and thus they can be cancelled out of the equation. But that means that when I realize this new theory that I’m considering adding to my mental programming can be used to explain some old, nagging problems in my field, my overall confidence in the new theory is not raised at all. Or to put the matter another way, after seeing the new theory explain some troubling old evidence, I trust the theory not one jot more than I did before I realized it might explain that old evidence.

This is simply not what happens in real life. When we suddenly realize that a new theory or model can be used to solve some old problems that were previously not solvable, we are impressed and definitely more inclined to believe that this new theory or model of reality is a true one.

This indifferent reaction to a new theory’s handling of troubling old evidence is simply not what happens in real life. When we suddenly realize that a new theory or model can be used to solve some old problems that were previously not solvable, we are definitely impressed and definitely more inclined to believe that this new theory or model of reality is a true one. When physicists around the world realized that the Theory of Relativity could be used to explain the shift in the orbit of Mercury, their confidence that the theory just might be correct shot up.

Hence the critics suggest that Bayesianism, as a way of describing what goes on in human thinking, is obviously not adequate. It can’t account for some of the ways of thinking that we’re now certain we use. We do indeed test new theories against old, puzzling evidence all the time, and we do feel much more impressed with a new theory if it can fully account for that same evidence when all the old theories can’t.

The response in defense of Bayesianism is complex, but not that complex. What the critics seem not to grasp is the spirit of Bayesianism. In the deeply Bayesian way of seeing reality and our relationship to it, everything in the human mind is metamorphosing and floating. The Bayesian picture of the mind sees us as testing, doubting, reassessing, and restructuring all our mental models of reality all the time.

In the formula above, the term for my degree of confidence in the evidence, taking only my background assumptions as true and without letting the new hypothesis into my thinking—namely, the term Pr(E/B)—is never 100 percent. Not even for very familiar old evidence. Nor is the term for my degree of confidence in the evidence if I include the hypothesis in my set of mental assumptions—that is, the term Pr(E/H&B)—ever equal to 100 percent. I am never perfectly certain of anything, not of my background assumptions and not even any of the evidence I may have seen—sometimes repeatedly—with my own eyes.

To consider this crucial situation in which a hypothesis is used to try to explain old evidence, we need to examine closely the kinds of things that happen in the mind of the researcher in both the situation in which the new hypothesis successfully interprets the old evidence and the one in which it doesn’t.

When the hypothesis does successfully explain some old evidence, what the researcher is really considering and affirming to her satisfaction is that, in the term Pr(E/H&B), the evidence fits the hypothesis, the hypothesis fits the evidence, and the background set of assumptions can be integrated with the hypothesis in a consistent and comprehensive way. She is delighted that if she does commit to this hypothesis, it will mean she can be more confident that the old evidence really happened in the way she and her fellow researchers saw it, that they were observing the evidence in the right way, and that they were not prey to some kind of hallucination or mental lapse that might have caused them to misinterpret the old-evidence situations or even misperceive them altogether. In short, she and her colleagues can feel a bit more confident that they weren’t sloppy in recording the old evidence data, a source of error that scientists know plagues all research.

All of this becomes even more apparent when we consider what the researcher does when she finds that a hypothesis does not successfully account for the old evidence. Rarely in scientific research does a researcher in this situation simply drop the new hypothesis. Instead, she examines the hypothesis, the old evidence, and her background set of assumptions to see whether any or all of them may be adjusted, using new concepts or new calculations involving newly proposed and measured variables or different, closer observations of the old evidence, so that all of the elements in the Bayesian equation may be brought into harmony again.

When the old evidence is examined in light of the new hypothesis, if the hypothesis does successfully explain that old evidence, the scientist’s confidence in the hypothesis and her confidence in that old evidence both go up. Even if her prior confidence in that old evidence was really high, she can now feel more confident that she and her colleagues—even ones in the distant past—did observe that old evidence correctly and did record their observations accurately.

The value of this successful application of the new hypothesis to the old evidence may be small—perhaps it has raised the E value in the term Pr(E/H&B) only a fraction of 1 percent. But that is still a positive increase in the value of the whole term and therefore a kind of proof of the explicative value rather than the predictive value of the hypothesis being considered.

Meanwhile, the scientist’s degree of confidence in this new hypothesis—namely, the value of the term Pr(H/E&B)—as a result of the increase in her confidence in the evidence also goes up another notch. A scientist, like all of us, finds reassurance in the feeling of mental harmony when more of her perceptions, memories, and concepts about the world can be brought into cognitive consonance with each other.