Bayesian epistemology
Richard Bradley argues (Philosophy of Science, 72 (April 2005)), against Bayesian analysis, or classical conditioning, as a universal tool for changes in belief. I only disagree in part, and here I intend a partial response and an outline my own thoughts.

My general thesis is that for any question that we can adequately specify, and given a set of data we wish to apply, Bayesian analysis gives us the correct and unique answer in terms of the probability of the hypotheses.

First I would note that the question we ask will define an unbiased prior. If we ask “Will the next marble pulled from the sack be red?” the question defines two possibilities, “red” and “not red”. The unbiased prior in this case would be a 50% chance of each possibility.

Secondly, Bradley raises the issue of “sensuously given data propositions”. In the above example, we might ask “How do we know we have a red marble?” “Are we sure it is red?” “What if it is maroon?” Here I would respond that if we don’t have any idea what a “red marble” is, we could not have asked the question in the first place. And for that matter we have to know what a sack is, and what procedure is involved in pulling a marble.

If the question is inadequately specified then we will have to ask other questions in terms that are better specified, until we reach terms we all agree on. At this point we have reached the level of verifiable data inputs. These are not absolute facts. They are verifiable facts, and these occur in cases where we universally agree that our sensory inputs and the definitions of our terms are in 1 to 1 relation. If we universally agree that “red marble” describes the sensory inputs we receive from the object in question, then we have reached the level of verifiable fact, and we do not need to go any deeper for any practical application.

I would also argue that starting only with verifiable facts is a defining characteristic of science. Scientific inputs must be verifiable and/or repeatable. If we can not reduce our question to questions involving verifiable facts, then I would argue it is not a scientific question. Thus, what to do with uncertain data inputs may be more important for a general epistemology than for the philosophy of science. Here, the anti-foundationalist might ask “But what happens if there are no questions at all that exist where our terms are adequately defined”? To which I would respond, “That question is a self-referential paradox”.

Finally, Bradley brings up cases where he argues that a change in our partial believe may cause us to change our conditional degree of belief, given the truth of the prospect. In his final example, he sets up an experiment where there are three possible paths for grain. Grain may go down tube A into bin X, or down tube B, into bin X, or down tube B and into bin Y.

He tells us that we are very sure that 50% of the grain ends up in bin X. He then tells us we do a measurement of the flow into pipe B, and revise our probability there. If X|B remained the same, then Jeffrey conditioning would tell us to update the probability of X. But, he points out, this would be incorrect, since we know that the probability of X is 50%. We should, he argues, update X|B instead. Thus knowledge that updates B, also updates X|B.

I would say this is another version of the type of criticism that has been leveled against Bayesian analysis from very early on. A Bayesian problem is set-up, including a set specific set of information, and it is then shown to give a counter-factual result, when compared to information that was not included in the original problem. Here the information that is excluded, and then included, is information about the logical relation of the pipe system.

Bradley says (p. 345), “It is possible that A, together with other beliefs will justify a change to one’s conditional beliefs given A”. I would respond that if those other beliefs were relevant, they should be incorporated into the structure of the Bayesian question that is posed.

If we were unaware that BX + AX = X, then measuring the flow B would not change our assessment of X|B, we would simply increase our probability of X by Jeffrey conditioning

If however we are fully aware of the logical relationships, then we would set up the Bayesian problem with 3 hypotheses. AX, BX, and BY. Given that we know BY is 50%, and given that we measure the flow BX+BY, we would revise our probabilities of AX and BX accordingly. Thus a change in our partial belief does not cause us to change our conditional degree of believe in a correctly specified problem that includes all of the information given.

Suppose we were not aware of the logical relationships at all. Then we would have 3 separate Bayesian problems. We would have X vs. Y, and A vs. B, and BX vs. BY. We would revise each of these independently. The results could be logically inconsistent, because we are not yet aware that a logical relationship should exist. This could be seen as a phase parallel to Kuhn’s “normal science”.

What will happen when the logical contradiction is realized? There will probably be some initial instability in our beliefs, as we shift rapidly back and forth between paradigms. Once our mental model as been revised, our beliefs will stabilize again. This could be seen as equivalent to a Kuhnian revolution.

We will have changed the questions we are asking. Before hand we would have asked 3 separate questions, with two hypotheses each. After the restructuring, we will ask one question, with 3 possible outcomes.

So, in effect, I would argue that classical conditioning is the correct procedure for “normal science”. As long as our questions remain well defined, and we encounter no logical contradictions, we can update our probabilities with new verifiable facts.

If, however, we find that our terms have become ill-defined, or that we have arrived at some apparent logical contradiction, we will have to ask new questions, and restructure our knowledge (and probabilities) accordingly.

Then, of course, Bayesian analysis can be used to choose between competing complex models. It naturally takes into account both simplicity of the model (a generalization of Occam’s razor), and the explanatory power of the model, and yields the model that will probably have the most predictive power. If we crate a model that has greater predictive power than our previous model, we again have a case where we will have to restructure our knowledge, again involving changing the questions we are asking.

It seems that, assuming something like the “certainty” condition holds, then we should always be able to construct a question that we can answer with Bayesian analysis where the “rigidity” condition also holds. However, if we were to say that the “certainty” condition not holds, and if we take as a given that in fact our inputs can not be reduced to verifiable inputs in a given problem, then it seems that “rigidity” need not hold.

My example would be that an art expert evaluates a painting. She simultaneously updates he probably that the average gallery patron will find the painting beautiful (upward), and the probability that they will pay the asking price given that they find in beautiful (downward). (Because of other intangible characteristics of the piece).

I don’t think this works in the grain example however. The measurement of the grain flow at B, can almost certainly be reduced to verifiable data inputs. Effectively the “certainty” condition is met, and therefore it is possible to frame the question in such a way that the “rigidity” condition is also met.

In conclusion, I would restate that as long as we can adequately specify the question, and agree on the data to be used, Bayesian analysis gives the correct, unique, answer in terms of the probabilities of the hypotheses. However, nothing guarantees that we have asked the right questions.
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