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pragmatist comments on Stupid Questions Open Thread Round 3 - Less Wrong
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Yeah, but if your observation does not have a probability of 1 then Bayesian conditionalization is the wrong update rule. I take it this was Alex's point. If you updated on a 0.7 probability observation using Bayesian conditionalization, you would be vulnerable to a Dutch book. The correct update rule in this circumstance is Jeffrey conditionalization. If P1 is your distribution prior to the observation and P2 is the distribution after the observation, the update rule for a hypothesis H given evidence E is:
P2(H) = P1(H | E) * P2(E) + P1(H | ~E) * P2(~E)
If P2(E) is sufficiently close to 1, the contribution of the second term in the sum is negligible and Bayesian conditionalization is a fine approximation.
This is a strange distinction, Jeffrey conditionalization. A little google searching shows that someone got their name added to conditioning on E and ~E. To me that's just a straight application of probability theory. It's not like I just fell off the turnip truck, but I've never heard anyone give this a name before.
To get a marginal, you condition on what you know, and sum across the other things you don't. I dislike the endless multiplication of terms for special cases where the general form is clear enough.
I don't know. i like having names for things. Makes it easier to refer to them. And to be fair to Jeffrey, while the update rule itself is a trivial consequence of probability theory (assuming the conditional probabilities are invariant), his reason for explicitly advocating it was the important epistemological point that absolute certainty (probability 1) is a sort of degenerate epistemic state. Think of his name being attached to the rule as recognition not of some new piece of math but of an insight into the nature of knowledge and learning.