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thomblake comments on Maximizing Cost-effectiveness via Critical Inquiry - Less Wrong
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Comments (24)
This model doesn't seem to work well for extreme values. Most illustratively if gives zero for infinite outcomes. Zero is not a probability.
If you're going to have a probability distribution that covers continuous intervals, 0 has to be allowed as a probability.
That just looks like a proof you can't have probability distributions over continuous intervals.
0 shouldn't be assigned as a probability if you're going to do Bayesian updates. That doesn't interfere with the necessity of using 0 when assigning probabilities to continuous distributions, as any evidence you have in practice will be at a particular precision.
For example, say the time it takes to complete a task is x. You might assign a probability of 20% that the task is finished between 2.3 and 2.4 seconds, with an even distribution between. Then, the probability that it is exactly 2.35 seconds is 0; however, the measured time might be 2.3500 seconds to the precision of your timing device, whose prior probability would be .02%.
Edit: I need a linter for these comments. Where's the warning "x was declared but never used"?
I know that. But any possible interval must be non-zero.
Also, some exact numbers are exceptions, depending on how you measure things: for example, there is a possibility the "task" "takes" EXACTLY 0 seconds, because it was already done. For example, sorting something that was already in the right order. (In some contexts. In other contexts it might be a negative time, or how long it took to check that it really was already done, or something like that)
Infinite utility seems like it might be a similar case.