Friday - Saturday, 13-14 October 2006
In the Temple of the Reverend Bayes
People seem to like workshops more than any other academic event. They typically last a day or a day and a bit and draw together a small group of scholars all working in the same area. The emphasis is on free and leisurely discussion. A speaker presents material and then expects the participants to enter into a searching examination and discussion. As the participants rotate through the various roles, the audience member asking the searching questions in one session may be their target after presenting in another session. In the best workshops, a sense of community and camaraderie develops from the formal proceedings and the informal interactions over coffee breaks and dinners. It is often a chance to put faces and personalities with what were just names in a bibliography of readings.
The room began to fill for James Joyces' Annual Lecture series on Friday October 13. Jim's talk in the series was the gateway event opening a workshop on Bayesianism that will continue the following morning and fill the day. The attendance was very good. No ill-fortune coming here from Friday the 13th. And what curious bedfellows the topic had brought: an audience essentially equally divided among philosophers of science and statisticians.
The reason for this curious intersection was one of the oldest problems in philosophy of science: how is it that evidence can supply support to science? That evidential relation is almost universally inductive. The evidence of red shifts in distant galaxies does not prove with deductive certainty that the universe is expanding. Yet it provides such strong support that dissent, while logically possible, is unreasonable.
What is the logic that governs induction? Over the last few decades, a new orthodoxy has grown in philosophy of science. Jim's talk and a major part of his research program is designed to shore up this new orthodoxy. The probability calculus is the logic of induction and it is used in the following way. At any moment, a scientist has a distribution of belief that is represented by the assigning of probabilities to the different possibilities. When new evidence is learned, these probabilities are adjusted by a simple formula known as Bayes' theorem that was introduced for this purpose by the Reverend Thomas Bayes in the 18th century. As new evidence comes in, the probabilities are continually adjusted so as to reflect better and better what the evidence is bringing.
Imagine that we trace the process in reverse. We wind back to look at the probabilities prior to the application of evidence. What ought these "prior probabilities" to be? That simple question has created enduring problems that will dominate the workshop. Any probability distribution says something, yet we would like the original priors to say nothing at all. All the real content of our beliefs ought to come from the evidence we learn.
Jim Joyce belongs to the "objectivist" tradition. Even in the absence of evidence, there ought to be just one right way to ascribe prior probabilities. He reviews the various ways people have tried to find these prior probabilities. The review is a catalog of uncomfortable failures that culminates in a proposal that priors should be assessed for accuracy against the truth by scoring rules. With just the right set of demands placed on these scoring rules, all sorts of good properties follow. The development becomes dense and technical. Just these technicalities are picked up with energy in question time as the benefits and need for this or that condition are weighed and measured.
The following morning we assembled for a 9am, breakfast of fruit and pastries in the small Center lounge.
The seminar room had been filled to capacity on Friday afternoon, so that we had worried that we may be mobbed the following day when people could come to hear not just one speaker but five. Now there seemed to be far fewer people there. We had underestimated the lure of a warm bed on a cold Saturday morning. Those who were there, however, were dedicated. The Center has developed a policy of encouraging graduate students from other places to come to these events by offering a small amount of financial support. I was pleased to see three accepting: one from North Carolina, one from India on tour in the US and one from the London School of Economics. In the course of the day, the numbers swelled.
The podium was now to be handed over to the statisticians. Jayanta Ghosh and Jay Kadane, both senior members of the statistics community, would speak in the morning. I was immediately struck by the way both began their talks. They each recalled their original training and background in traditional statistical methods and then described their moment of conversion to the Bayesian approach to statistics, which is built around the notions of prior and posterior probabilities and Reverend Bayes' formula. They were sinners who had found the right way and proclaimed it with joy.
That they needed to do this reminded me of the curious fact that, while Bayesians form a majority in the relevant group in philosophy of science, they are a vocal minority amount statisticians.These professions of faith were statements of reassurance and solidarity for members of a besieged minority. In this venue they were needed since the talks that followed were about to highlight their differences.
Ghosh, like Joyce, is an objectivist and he described in careful detail the various ingenious methods that can be used to pick out the right prior probability. Ghosh was careful to emphasize that these methods can only be used when one already has a very well defined problem with lots of data.
Kadane, however, believes none of these methods work reliably when one moves from such circumstances. He displayed polished examples of familiar ways in which they fail. When you just don't know which outcome is right, there are many ways to distribute your indifference and they give different probability distributions. For example, there are two marbles in a jar and each can be red or black. Should we let indifference tell us that no red, one red and two red are the equally probable cases? Or should the equally probable cases be both red, the first only red, the second only red or none red? That's my example. Kadane's were more elegant, but more complicated.
Kadane is a subjectivist and his solution to the surfeit of priors is that there is no right prior probability. Everyone picks their own. While I am drawn to the objectivist's idea of a unique probability, the problems Kadane recalled are so profound that his subjectivism seems unavoidable. In question time I press him on the obvious worry. "If the probability calculus is the logic of induction, ought it not to supply a unique import of evidence?" "No," he assured me, "the import of evidence is always relative to the person." That is not an idea I find reassuring. "Perhaps that means that the probability calculus is not the right logic of induction. Perhaps we should seek another logic." Kadane was momentarily startled by this apostasy but saw no reason to condemn the thought. "Good luck in finding another! Go for it!"
We broke for lunch and drifted off in small groups to find lunch somewhere on the streets of Oakland. We could have had sandwiches brought it but we thought it better to let everyone stretch their legs, lest the confinement to the one room for the day make us feel like Cardinals locked down until they elect a Pope.
In the subjectivist tradition, probabilities differ from person to person with no one being the right distribution. So the emphasis shifts to methods of eliciting these probabilities from the confusion of each person's thoughts. One ingenious device for doing this is to see what combinations of bets they will accept. What is needed in this eliciting is to know the value--the utility--assigned to each outcome.
After lunch we assembled to hear another subjectivist, Teddy Seidenfeld. He had unalloyed bad news for that aspect of the subjectivist tradition. What he proceeded to show through simple examples was that the nice apparatus of elicitation fails if we allow that utilities can be infinite. Someone's preferences ought to be set fully by the specification of the probabilities and utilities of outcomes. But, in Teddy's example, that demand could not be reconciled with the principle of dominance. That simple idea just says that if you prefer A to B in each possible state of the world, then you prefer A to B overall.
I could hear murmurs of dismay as the results were developed and, after the talk, the room fragmented into small groups trying to assess just how seriously this new threat should be taken.
Teddy is half statistician and half philosopher. So his talk made the transition to the philosophers speaking in the afternoon. Branden Fitelson's talk commenced with tales of the founding fathers of modern philosophy of science: Hempel, Carnap, Popper. He included portraits of each. The eternally kind Hempel smiled warmly with his tie disarming askew, just as I remembered him from the few years we overlapped in Pittsburgh. Carnap and Popper posed majestically, as befits someone whose thoughts are truly deep.
The talk moved on to an old study which seemed to establish that people are just very bad at reasoning probabilistically. When surveyed, people tend to affirm that a suitably described "Linda" is more probably a bank teller than a bank teller and a feminist. Since the reverse is a theorem in probability, people seem badly confused about elementary probability. It is hard to resist the sense that this is too simple. Branden proceeded to show that sense was quite right. He elaborated different senses of confirmatory support that could be recovered from probabilistic relations and showed how in their terms people come off as not quite so confused.
No one really wants to give the last talk of the day. People in the audience are tired; their concentration is flagging; and so I wasn't surprised that we needed extra efforts to shepherd them out of our lounge with the coffee and cookies and more. I happened to have the misfortune of being that last speaker. Since I was the organizer, I could not in clear conscience assign the time to anyone else.
As the event had progressed, I found myself getting more anxious about the talk. For their all differences and worries, what united the group was complete acceptance of the core doctrine: beliefs are probabilities and their dynamics is governed by Bayes' theorem. Woe to anyone who would challenge that. Woe to me, for that was the purpose of my talk.
The subjectivists, I thought, were quite correct to display the failure of the objectivists' methods at picking a unique prior probability. However I had become convinced that the failure was no failure of their methods. The methods depended upon platitudes of evidence. If, for example, you have no grounds for picking between outcomes, your belief in each should be the same. The problems arose, I argued, from the assumption that, in cases of ignorance, belief distributions are still probability distributions. I took the case of complete ignorance and urged that the platitudes of evidence pick out a single, non-probabilistic belief distribution.
I was not relishing what may happen in question time. So I was quite relieved to find that I was treated with tolerance and kindness, rather as an insightful parent might deal with a teenager who has adopted some outrageous behavior specifically designed to annoy the elders. They could easily afford that indulgence since I was in a minority of one and, perhaps, some of them recalled that I would need to sign off on dinner. Teddy Seidenfeld, whose critical dressing downs are legendary both for their sharpness and incisiveness, had left early with apologies to attend a family wedding.
As the room cleared, I assembled the small group I would take to dinner. Branden Fitelson had become fully distracted by a calculation and a small group assembled around him as he hammered on his laptop keyboard. Could I print out just one page for him? No, never mind, it's OK. Not needed. I noticed as the evening unwound that he carried a small notepad from the hotel with him. Whenever he had a quiet moment, he would stare at it and make small marks on it with his pen. That, I thought with some satisfaction, is the mark of a good workshop. We leave with more than we brought.
John D. Norton
::: Rational Belief and Reasonable Belief, A Ramseyian Distinction
James Joyce, University of Michigan, Philosophy
Friday, 13 October 2006, 3:30 pm
817R Cathedral of Learning
Annual Lecture Series
Saturday, 14 October 2006