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Comment author: Eliezer_Yudkowsky 12 March 2008 04:57:15PM 8 points [-]

Constant: The competent frequentist would presumably not be befuddled by these supposed paradoxes.

Not the last two paradoxes, no. But the first case given, the biased coin whose bias is not known, is indeed a classic example of the difference between Bayesians and frequentists. The frequentist says:

"The coin's bias is not a random variable! It's a fixed fact! If you repeat the experiment, it won't come out to a 0.5 long-run frequency of heads!" (Likewise when the fact to be determined is the speed of light, or whatever.) "If you flip the coin 10 times, I can make a statement about the probability that the observed ratio will be within some given distance of the inherent propensity, but to say that the coin has a 50% probability of turning up heads on the first occasion is nonsense - that's just not the real probability, which is unknown."

According to the frequentist, apparently there is no rational way to manage your uncertainty about a single flip of a coin of unknown bias, since whatever you do, someone else will be able to criticize your belief as "subjective" - such a devastating criticism that you may as well, um, flip a coin. Or consult a magic 8-ball.

Sudeep: If quantum mechanics is true, then ignorance/uncertainty is a part of nature and not just something that agents have.

A common misconception - Jaynes railed against that idea too, and he wasn't even equipped with the modern understanding of decoherence. In quantum mechanics, it's an objective fact that the blobs of amplitude making up reality sometimes split in two, and you can't predict what "you" will see, when that happens, because it is an objective fact that different versions of you will see different things. But all this is completely mechanical, causal, and deterministic - the splitting of observers just introduces an element of anthropic pseudo-uncertainty, if you happen to be one of those observers. The splitting is not inherently related to the act of measurement by a conscious agent, or any kind of agent; it happens just as much when a system is "measured" by a photon bouncing off and interacting with a rock.

There are other interpretations of quantum mechanics, but they don't make any sense. Making this fully clear will require more prerequisite posts first, though.

Comment author: radfordd 18 May 2011 03:44:34PM *  7 points [-]

Eliezer:

"The coin's bias is not a random variable! It's a fixed fact! If you repeat the experiment, it won't come out to a 0.5 long-run frequency of heads!"

You're repeating the wrong experiment.

The correct experiment for a frequentist to repeat is one where a coin is chosen from a pool of biased coins, and tossed once. By repeating that experiment, you learn something about the average bias in the pool of coins. For a symmetrically biased pool, the frequency of heads would approach 0.5.

So your original premise is wrong. A frequentist approach requires a series of trials of the correct experiment. Neither the frequentist nor the Bayesian can rationally evaluate unknown probabilities. A better way to say that might be, "In my view, it's okay for both frequentists and Bayesians to say "I don't know.""