I Think I'm a Radical Baysean

To learn something that isn't math, I've been picking up the readings from some of my friends' classes if they look interesting. This last week I read a pair of articles by Kahneman and Tversky about flaws in heuristic reasoning and an objection by the frequentist Gigerenzer.

K&T illustrate one of the flaws of heuristic reasoning by asking people to evaluate the probability that a particular woman is a librarian or a librarian and feminist, given a description of her that fits the stereotype of a feminist. Many people say the latter is more probable, when in fact the probability of any subset (that someone is both A and B) must be less than the probability of the superset (that someone is A). They call this the "representativeness" error -- that is, that if a particular event is very representative of a particular class, then people overestimate its likelihood. An example of the same effect is that people consider the sequence of flips of a fair coin "H-T-H-H-T" more likely than the sequence "H-H-H-H-H" because the first evokes randomness more, even though we know that any particular sequence must have equal probability.

Gigerenzer objects to this in a couple of ways. One is that people, when asked to evaluate two possibilities, A or A and B, they naturally assume that the asker actually means "A and not B" or "A and B", since otherwise it's kind of a stupid question. This makes sense to me. His second objection is the one I really want to talk about, though, and that is that it doesn't make sense to ascribe probability to a single, non-repeatable event.

In case you don't know, like I didn't before I came here, there are two different schools of thought concerning probability: the Frequentists and the Bayseans. The former believe, like Gigerenzer, that in order for an event to have meaningful probability, it must be repeatable, and that the probability is interpreted as the limit of the frequency of occurances as the number of samples goes to infinity. Bayseans allow one to speak about the probability even of events that occur only once, such as a particular team winning the twentieth Superbowl, and interpret probability as a measure of certainty concerning a particular outcome.

I take great objection to the Frequentist point of view, not only in cases where we are evaluating the probability of a single event, but also when the measured outcome is "repeatable". For example, take the flipping of a coin. The fact that when I flip a coin, we reasonably describe the probability of it coming up heads has nothing to do with the fact that many coins have been flipped before, or the possibility of flipping the coin many more times. It has to do with the symmetry between the two sides, the information loss introduced by the flipping action (negating the asymmetry of the coin starting either heads up or heads down), and a reasonable physical model of the world that suggests that symmetric outcomes should happen with equal likelihood. Were this theory in doubt, a coin has added benefit that we can flip it many times and get a more rigorous estimate of the likelihood of it coming up heads through the reasonable assumptions of statistical inference, but this possibility can't reasonably be said to inform our estimate of the probability of heads. The first coin that was ever flipped, long before statistics textbooks, had just as much of a fifty-fifty chance of heads and tails as any coin today. (Ignoring recent improvements in the manufacture of symmetric coins, I suppose.)

In fact, the moment the coin has left your thumb, the outcome is as determined as the profession of an unknown person. Depending on what your ideas about free will are, you could say it is determined even before that. We describe its outcome in probabilistic terms because of a lack of information, not because of its similarity to possible future situations. Looking at it this way, there's no basic philosophical difference between a coin flip and evaluating the probability that a single person has a particular profession -- we are just able, through simplifying physical assumptions and statistical inference, to make a much better estimate of the probability of the latter knowing very little about the particular event.

Statistics and probability are primarily concerned with trying to make inferences about situations about which there is insufficient information. The notion of "probability" is simply a convenient way of thinking about deterministic and determined events that we cannot observe completely. When we say an event is "repeatable", we simply mean that a series of events in time are similar enough with respect to the variables we can observe and think are salient to its outcome. This allows us to use convenient mathematical models and make accurate predictions about the future. Random events have a continuum of repeatability, but they all have one thing in common, whether they happen once or many times: we don't know enough about them to predict their outcomes, and this is what allows us to speak of their probability.

I think this makes me a radical Baysean, a label that I think sounds pretty cool. (Ever since my friend Fred described himself as a "radical agronomist" this summer, I think I've wanted to be a radical something.) Of course, my understanding of this issue is still pretty shallow, and probably biased by the fact that I've now read heavyweight Bayseans (Kahneman won the Nobel prize in economics) and a less prestigious Frequentist. I'm sure you're all sitting on the edge of your seats, wondering how this will all turn out.