by Gene Callahan
Dan Klein’s Knowledge and Coordination has something interesting to say about Bayesian inference, although he never explicitly addresses that topic. Consider the following:
Here, we have the distinction between responding to the realization of events within a framework of recognized variables and relationships and the discovery of a fresh opportunity to embrace a new and better framework or interpretation. This element of epiphany, of finding fortune by interpreting the world differently, is the subtle and vital element in human decision making. Yet, it is absent from equilibrium model building. In equilibrium stories, agents never have a “light bulb” moment… (p. 13)
Kirzner’s alertness is the individual’s re-interpretation of that world [of a world of already-interpreted "facts"]. (p. 14)
“Equilibrium” is meaningful only in reference to a specified model… (p. 28)
Bayesian inference, similar to equilibrium theorizing, works within a fixed frame of interpretation: it “is meaningful only in reference to a specified model.” It cannot extend across instances when a new interpretive framework takes the place of the old. Consider Bayes’s original paper introducing his calculus:
Postulate. 1. Suppose the square table or plane ABCD to be so made and levelled, that if either of the balls o or W be thrown upon it, there shall be the same probability that it rests upon any one equal part of the plane as another, and that it must necessarily rest somewhere upon it.
2. I suppose that the ball W shall be 1st thrown, and through the point where it rests a line os shall be drawn parallel to AD, and meeting CD and AB in s and o; and that afterwards the ball O shall be thrown p + q or n times, and that its resting between AD and os after a single throw be called the happening of the event M in a single trial. These things supposed…
The point here is the Bayes only sets his calculus going within a very definite framework of interpretation. If, for instance, it was found that our interpretive framework was all wrong — perhaps, say, that the balls contained an iron core, and there was a man under the table manipulating them with a magnet, postulate 1 would not hold. We would no longer be working in this interpretive framework. The right thing to do in this case is not proceed with Bayesian updating, but throw out one’s priors and start all over again setting up new postulates.
I would suggest that in science and in practical life, both types of situations occur. Einstein’s theory of relativity was an instance of the creating an entirely new interpretive framework. Until he did so, it would have been quite reasonable for a scientists to have had a prior as near zero as one wishes for the idea that one’s speed of travel would make time flow differently for one. But, given Einstein’s new interpretive framework, the idea suddenly became quite plausible. It was time, not to proceed with Bayesian inference as usual, but to set new priors and start over. In fact, we can see here a close parallel between the situations that require only Bayesian updating and those that require new priors to Kuhn’s distinction between “normal science” and paradigm shifts.