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Just in case it's not clear from the above: there are uncountably many degrees of freedom to an arbitrary complex function on the real line, since you can specify its value at each point independently.
A continuous function, however, has only countably many degrees of freedom: it is uniquely dete...(read more)
Eliezer: poetic and informative. I like it.
Felix: Yes, for example see http://en.wikipedia.org/wiki/NC_%28complexity%29
Eliezer: "You could see someone else's engine operating materially, through material chains of cause and effect, to compute by "pure thought" that 1 + 1 = 2. How is observing this pattern in someone else's brain any different, as a way of knowing, from observing your own brain doing the same thing?...(read more)
One reason is Cox's theorem, which shows any quantitative measure of plausibility must obey the axioms of probability theory. Then this result, conservation of expected evidence, is a theorem.
What is the "confidence level"? Why is 50% special here?
Perhaps this formulation is nice:
0 = (P(H|E)-P(H))*P(E) + (P(H|~E)-P(H))*P(~E)
The expected change in probability is zero (for if you expected change you would have already changed).
Since P(E) and P(~E) are both positive, to maintain balance if P(H|E)-P(H) < 0 then P(H|~E)-P(H) > 0. If P(E) is...(read more)
It seems the point of the exercise is to think of non-obvious cognitive strategies, ways of thinking, for improving things. The chronophone translation is both a tool both for finding these strategies by induction, and a rationality test to see if the strategies are sufficiently unbiased and meta.