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trist comments on The Power of Noise - Less Wrong
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Let me try to express it more clearly here:
I agree that it is both reasonable and common for programs to require that their inputs satisfy certain properties (or in other words, for the inputs to lie within a certain set). But this is different than requiring that the inputs be drawn from a certain probability distribution (in other words, requiring 5% of the inputs to be 0000, 7% to be 0001, 6% to be 0010, etc.). This latter requirement makes the program very non-modular because invoking a method in one area of the program alters the ways in which I am allowed to invoke the method in other areas of the program (because I have to make sure that the total fraction of inputs that are 0000 remains 5% across the program as a whole).
Does this make more sense or is it still unclear? Thanks for the feedback.
So you're differentiating between properties where the probability of [0 1 2 3] is 1-ɛ while >3 is ɛ and probability distributions where the probability of 0 is 0.01, 1 is 0.003, etc? Got it. The only algorithms that I can think of that require the latter are those that require uniformly random input. I don't think those violate modularity though, as any are of the program that interfaces with that module must provide independently random input (which would be the straightforward way to meet that requirement with an arbitrary distribution).
There's a difference between requiring and being optimized for though, and there are lots of algorithms that are optimized for particular inputs. Sort algorithms are a excellent example, if most of your lists are almost already sorted, there algorithms that are cheaper on average, but might take a long time with a number of rare orderings.