by Gene Callahan
In a seminar I’ve been attending at NYU this semester, David Chalmers contended that “resetting priors” is irrational in a Bayesian framework. (If you’re not familiar with Bayesian inference, the Wikipedia article just linked to does a good job of introducing the topic, so I will refer you to that, rather than lengthening an already long post with my own introduction to the subject.) This seemed wrong to me and seemed wrong to me long before Chalmers raised the issue for me again. But his remarks renewed my interest in the subject, and resulted in this post.
To get myself back in the “Bayesian spirit,” I began by re-reading two of my favorite papers on the subject, Wesley Salmon’s “Rationality and Objectivity in Science” and Clark Glymour’s “Why I am Not a Bayesian” (both available in the excellent survey work by Cover and Curd, Philosophy of Science: The Central Issues, 1998, New York: W. W. Norton & Company). I will quote from both of those papers here, but the conclusion about re-setting priors seems to be my own—at least a quick Google search turns up only Chalmers rejection of my central idea here!
Glymour’s paper begins by stating what he sees as the role of confirmation theory, of which he understands Bayesianism to be a variety: “The aim of confirmation theory is to provide a true account of the principles that guide scientific argument insofar as that argument is not, and does not purport to be, of a deductive kind” (584). He believes Bayesianism falls short of capturing important aspects of that process, and discusses several ways in which he finds that so.
However, he notes at the outset that he does not suggest these shortcomings render Bayesian inference useless—far from it!—a caveat that applies to my own criticism as well: “It is not that I think the Bayesian scheme or related probabilistic accounts capture nothing. On the contrary, they are clearly pertinent where the reasoning involved in explicitly statistical. Further, the accounts developed by Carnap, his predecessors, and his successors are impressive systematizations and generalizations, in a probabilistic framework, of certain principles of ordinary reasoning” (587-588).
Nevertheless, he continues: “What is controversial is that the general principles required for argument can best be understood as conditions restricting prior probabilities in a Bayesian framework. Sometimes they can, perhaps, but I think when arguments turn on relating evidence to theory, it is very difficult to explicate them in a plausible way within the Bayesian framework” (592)—for instance, many scientific discoveries, such as Newton’s dynamics and the General Theory of Relativity, were made much more convincing by how they handled old evidence—but, per Bayesian thinking, old evidence, being already known, should cause no change in the plausibility we assign to present theories at all.
However, it is Salmon, in his advocacy of a Bayesian model of theory acceptance, who points at the problem I see in Chalmers’ contention. He offers the following example:
“Let us look at a simple and non-controversial application of Bayes’s theorem. Consider a factor that produces can openers at the rate of 6,000 per day. This factory has two machines, a new one that produces 5,000 can openers per day, and an old one that produces 1,000 per day. Among the can openers produced by the new machine 1 percent are defective; among those by the old machine 3 percent are defective. We pick one can opener at random from today’s production and find it defective. What is the probability that it was produced by the new machine?” (554)
Salmon shows, by Bayesian inference, that the probability is 5/8ths. Now, this is all fine as far as it goes. But let’s say that we bring this analysis to the plant foreman, and he says, “But wait a second—the day that piece was produced, the new machine was offline most of the day, and only put out 1000 pieces!”
Now, in this situation, I contend, the right maneuver is not to update based on our initial priors, but to reset the priors and right away get the correct probability of 1/4—but this is the very procedure that, Chalmers contends, is irrational. And, interestingly, this example is very much like the one I recall Bayes himself presenting to illustrate his theory. As I remember (I haven’t been able to Google this example up), Bayes used an example of deciding where the middle of a pool table lay. The procedure went something like (again, this is all from memory at present): Pick a spot that looks like the middle to you. Now, have someone keep breaking and see how many balls wind up on each side of your “prior” middle, then update your prior based on the idea that, over the long run, half of all balls should wind up on each side of the middle.
Now, I believe that both of the examples show perfectly sound uses of Bayesian inference, and the reason is that the problems (determine where the bad can opener came from and determine the middle of the table) are set within a framework where we believe we understand everything relevant to our problem. And, once again with the latter example, I suggest Bayesian inference breaks down if it turns out we were wrong about the framework of the problem we are trying to answer. If, while calculating where the middle of the pool table is, we suddenly spy a dwarf with a magnet under the table, and then pick up a ball and note that it feels like it might have an iron core, the right move would be to say, “Wait a minute—all bets are off!” We’d then gauge the dwarf’s agility, the size of the balls’ iron core, the strength of the magnet, etc., and then reset our priors to handle the actual situation.
The force of these examples, if I am correct in my analysis of them, is that, I suggest, they capture what happens in a time of Kuhnian paradigm change. (And, to relate this post to economics, perhaps in Kirznerian entrepreneurship as well?) For instance, if we surveyed scientists in the 19th century about whether measuring rods could change their length simply by being moved about, they would have said, not “probably not” but “no way,” because they would have seen this not as an empirical issue, but an analytic one—it was not that steel rods didn’t change length because you put them on a fast train, but that anything that did so could not serve as a measuring rod. (And an analytical impossibility should get a prior probability of zero, so that no amount of new evidence will dislodge it—it might turn out that nothing can serve as an (ideal) measuring rod, but not that the analytical truth was false.) What Einstein achieved, with the Special Theory of Relativity, was to give science an entire, previously unimagined framework that showed that, given another constant (the speed of light) and an equation relating the rods changes in length to the percentage of the speed of light at which they were moving, this was not an analytical truth at all. And, I suggest, what scientists really did, confronted with this new theory, was much more akin to resetting their priors than it was to updating based on fixed priors. And that it was eminently rational!
Now, it might be proposed, to counter my argument, that we must envision an ideal Bayesian reasoner, who could already conceive every scientific theory when setting her priors. Such a construct might prove useful in some situations, but it totally disconnects Bayesian inference from any mooring to the actual, historical process by which science really advances. Essentially, we are being asked to envision science as already complete, which, of course, obviates any need to ever correct priors set based on the scientific knowledge and theories actually available at any given time!
What this all means is that Bayesian inference is an incomplete model of scientific reasoning. But after the work of Carroll, Wittgenstein, Gödel, and Polanyi, are we really surprised that formal models can’t capture everything that is reasonable about science?