[I posted this to open thread a few days ago for review. I've only made some minor editorial changes since then, so no need to read it again if you've already read the draft.]
This is a belated reply to cousin_it's 2009 post Bayesian Flame, which claimed that frequentists can give calibrated estimates for unknown parameters without using priors:
And here's an ultra-short example of what frequentists can do: estimate 100 independent unknown parameters from 100 different sample data sets and have 90 of the estimates turn out to be true to fact afterward. Like, fo'real. Always 90% in the long run, truly, irrevocably and forever.
And indeed they can. Here's the simplest example that I can think of that illustrates the spirit of frequentism:
Suppose there is a machine that produces biased coins. You don't know how the machine works, except that each coin it produces is either biased towards heads (in which case each toss of the coin will land heads with probability .9 and tails with probability .1) or towards tails (in which case each toss of the coin will land tails with probability .9 and heads with probability .1). For each coin, you get to observe one toss, and then have to state whether you think it's biased towards heads or tails, and what is the probability that's the right answer.
Let's say that you decide to follow this rule: after observing heads, always answer "the coin is biased towards heads with probability .9" and after observing tails, always answer "the coin is biased towards tails with probability .9". Do this for a while, and it will turn out that 90% of the time you are right about which way the coin is biased, no matter how the machine actually works. The machine might always produce coins biased towards heads, or always towards tails, or decide based on the digits of pi, and it wouldn't matter—you'll still be right 90% of the time. (To verify this, notice that in the long run you will answer "heads" for 90% of the coins actually biased towards heads, and "tails" for 90% of the coins actually biased towards tails.) No priors needed! Magic!
What is going on here? There are a couple of things we could say. One was mentioned by Eliezer in a comment:
It's not perfectly reliable. They assume they have perfect information about experimental setups and likelihood ratios. (Where does this perfect knowledge come from? Can Bayesians get their priors from the same source?)
In this example, the "perfect information about experimental setups and likelihood ratios" is the information that a biased coin will land the way it's biased with probability .9. I think this is a valid criticism, but it's not complete. There are perhaps many situations where we have much better information about experimental setups and likelihood ratios than about the mechanism that determines the unknown parameter we're trying to estimate. This criticism leaves open the question of whether it would make sense to give up Bayesianism for frequentism in those situations.
The other thing we could say is that while the frequentist in this example appears to be perfectly calibrated, he or she is liable to pay a heavy cost for this in accuracy. For example, suppose the machine is actually set up to always produce head-biased coins. After observing the coin tosses for a while, a typical intelligent person, just applying common sense, would notice that 90% of the tosses come up heads, and infer that perhaps all the coins are biased towards heads. They would become more certain of this with time, and adjust their answers accordingly. But the frequentist would not (or isn't supposed to) notice this. He or she would answer "the coin is head-biased with probability .9" 90% of the time, and "the coin is tail-biased with probability .9" 10% of the time, and keep doing this, irrevocably and forever.
The frequentist magic turns out to be weaker than it first appeared. What about the Bayesian solution to this problem? Well, we know that it must involve a prior, so the only question is which one. The maximum entropy prior that is consistent with the information given in the problem statement is to assign each coin an independent probability of .5 of being head-biased, and .5 of being tail-biased. It turns out that a Bayesian using this prior will give the exact same answers as the frequentist, so this is also an example of a "matching prior". (To verify: P(biased heads | observed heads) = P(OH|BH)*P(BH)/P(OH) = .9*.5/.5 = .9)
But a Bayesian can do much better. A Bayesian can use a universal prior. (With a universal prior based on a universal Turing machine, the prior probability that the first 4 coins will be biased "heads, heads, tails, tails" is the probability that the UTM will produce 1100 as the first 4 bits of its output, when given a uniformly random input tape.) Using such a prior guarantees that no matter how the coin-producing machine works, as long as it doesn't involve some kind of uncomputable physics, in the long run your expected total Bayes score will be no worse than someone who knows exactly how the machine works, except by a constant (that's determined by the algorithmic complexity of the machine). And unless the machine actually settles into deciding the bias of each coin independently with 50/50 probabilities, your expected Bayes score will also be better than the frequentist (or a Bayesian using the matching prior) by an unbounded margin as time goes to infinity.
I consider this magic also, because I don't really understand why it works. Is our prior actually a universal prior, or is the universal prior just a handy approximation that we can substitute in place of the real prior? Why does the universe that we live in look like a giant computer? What about uncomputable physics? Just what are priors, anyway? These are some of the questions that I'm still confused about.
But as long as we're choosing between different magics, why not pick the stronger one?