As I keep mentioning ad nauseam, I think probability’s really important in understanding all sorts of things about science. Here’s a really basic question that’s easy to ask but maybe not so easy to answer: What do probabilities mean anyway?
Not surprisingly, this is a question that philosophers have taken a lot of interest in. Here’s a nice article reviewing a bunch of interpretations of probability. The article lists a bunch of different interpretations, but ultimately I think the most important distinction to draw is between objective and subjective interpretations: Are probabilities statements about the world, or are they statements about our knowledge and belief about the world?
The poster child for the objective interpretation is frequentism, and the main buzzword associated with the subjective interpretation is Bayesianism. If you’re a frequentist, you think that probabilities are descriptions of the frequency with which a certain outcome will occur in many repeated trials of an experiment. If you want to talk about thing that don’t have many repeated trials (e.g., a weatherman who wants to say something about the chance of rain tomorrow, or a bookie who wants to set odds for the next Super Bowl), then you have to be willing to imagine many hypothetical repeated trials.
The frequentist point of view strikes me as so utterly bizarre that I can’t understand why anyone takes it seriously. Suppose that I say that the Red Sox have a 20% chance of winning the World Series this year. Can anyone really believe that I’m making a statement about a large (in principle infinite) number of imaginary worlds, in some of which the Red Sox win and in others of which they lose? And that if Daisuke Matsuzaka breaks his arm tomorrow, something happens to a bunch of those hypothetical worlds, changing the relative numbers of winning and losing worlds? These sound utterly crazy to me, but as far as I can tell, frequentists really believe that that’s what probabilities are all about.
It seems completely obvious to me that, when I say that the Red Sox have a 20% chance of winning the World Series, I’m describing my state of mind (my beliefs, knowledge, etc.), not something about a whole bunch of hypothetical worlds. That makes me a hardcore Bayesian.
So here’s what I’m wondering. Lots of smart people seem to believe in a frequentist idea of probability. Yet to me the whole idea seems not just wrong, but self-evidently absurd. This leads me to think maybe I’m missing something. (Then again, John Baez agrees with me on this, which reassures me.) So here’s a question for anyone who might have read this far: Do you know of anything I can read that would help me understand why frequentism is an idea that even deserves a serious hearing?