Related to: Beautiful Probability, Probability is in the Mind

Arnold Kling ponders probability:

How one thinks about probability affects how one thinks about economics. Consider the use of the word "probability" in each of the following sentences:

1. What is the probability that when a fair coin is flipped it will come up heads?
2. What is the probability that exactly two number-one seeds will make it to the final four in the March Madness basketball tournament?
3. What is the probability that New York City will rank higher relative to other cities five years from now in terms of college graduates?

We would answer the first question by saying that the probability is 50 percent, based on the very definition of a fair coin. This is an axiomatic interpretation of probability. The axiomatic view treats probability as a matter of pure logic, with statements that do not require any empirical testing.

We would answer the second question by looking up historical records for the NCAA basketball tournament. This is the frequentist account of probability, which treats probability as counting outcomes from repeated trials. A frequentist would claim that the only way we can know that a coin has a 50 percent probability of coming up heads is by actually flipping a coin enough times to verify this empirically.

The third question cannot be answered on the basis of axioms or observed frequencies. The probability estimate is purely subjective. The subjective account of probability is that it reflects an individual belief that cannot be proven either logically or empirically.

In the tradition of Reddit, and a little inspired by Robin, this is a simple link to an interesting page somewhere else - I leave comment and discussion to the very awesome Less Wrong community.

Edit: Eliezer has in the past been uncomplimentary of the "accursèd frequentists". In at least Beautiful Probability and Probability is in the Mind, he has characterized (for at least some problems) the "frequentist" approach as being wrong, and the "Bayesian" approach as being right. Kling suggests different problems for which different approaches are approrpriate.


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Here Arnold just repeats a very standard distinction in probability. Not sure what there is to talk about unless we want to repeat standard debates about which concept is better for what.

unless we want to repeat standard debates about which concept is better for what.

Well, yes. Eliezer has suggested that "The ancient war between the Bayesians and the accursèd frequentists stretches back through decades", but Kilng's post describes a class of problems for which he presents the frequentist approach as being the reasonable one. I had rather hoped for a discussion, even if it started "standard".

The usual Bayesian position is that the Bayesian picture subsumes the frequentist one. In cases where there's a clear single frequentist answer to the question "what's P(X)?" the answer is typically the same for any Bayesian who doesn't have a crazy prior. Likewise for instances where you can get the answer "axiomatically".

So, with Kling's first example, everyone agrees: modulo quibbling about the usual idealizations, the probability is 1/2. An accursed frequentist might say that the Bayesian apparatus is unnecessary or confusing when asking this question; a Bayesian might say that saying simply "the probability is 1/2" (without any mention of what you're conditioning on) is setting oneself up for confusion when subtleties arise later; but working out the probability that a fair coin, fairly tossed, comes up heads is not an instance where anyone thinks Bayesians get the wrong answer.

With Kling's second example, I think even the most hardened (and accursed) frequentist would agree that "what is the probability that X happens?" and "what fraction of the time has X happened in the past?" are not the same question. Frequentism isn't the idea that you always answer probability questions by looking at historical data. It's the idea that what statements about probability mean are about long-term averages -- historical and future. Here, again, I don't think Kling has given an example where the frequentist approach is "the reasonable one"; I'm not sure he's even given an example where the frequentist approach is a reasonable one.

It seems to me that to call that third probability "purely subjective" is unhelpful. Compare a statement to which applying that label is more obviously reasonable: "Beethoven's third symphony is better than his fifth". It's not clear that there's even any way to state that without using evaluative terms whose primary meaning is subjective. Whereas in principle, at least, one could estimate the probability Kling asks for in a way that even a frequentist might accept: do some sort of vast computer model of the world and run a big Monte Carlo simulation. Totally impractical, of course, but any question you could in principle answer that way surely isn't purely subjective.

Quite possibly Kling does in fact have a decent grasp of this stuff, but I don't see a great deal of insight in that page.