Nick writes:
“I thought the whole point was to consider all the branches.” That’s a good goal, if it can be done. But it’s like saying, “Why don’t we just kill the terrorists?” Obviously, because there are no maps and schedules of their activities. It’s hard to find all the branches short of making guesses. Glen write: “If there is a way to induce an analytically formed property (like maximum likelihood or spread), then fantastic, the mathematician has arrived. But if there's not, MC still provides a way to place the system into any region of its entire behavior space.” An “analytically formed property” is a perhaps a bit generous for what is often done in practice for maximizing likelihood. Little more than counting prediction mistakes until a model doesn’t make them. It’s not even necessary to have a carefully constructed statement of probability for that, just something that reliably distinguishes between better and worse. But this is one explanation why both MC and MH are so appealing, especially with big computers available. One can look at not just thousands of possibilities, but trillions. In simple cases, that may be almost total exploration. Like the proof by computer of the four color theorem. In other cases, like if each likelihood evaluation takes an hour to calculate, it is probably worth thinking harder about the problem. Marcus
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