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army1987 comments on Hearsay, Double Hearsay, and Bayesian Updates - Less Wrong
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If your point is that there are a lot of people locked up for violating laws that are basically stupid, you're absolutely right.
But that issue is largely irrelevant to the subject of the primary post, which is the accuracy of courts. If the government bans pot, the purpose of evidence law is to determine whether people are guilty of that crime with accuracy.
In other words, your criticism of the normative value of the American legal system is spot-on; we imprison far more people than we should and we have a lot of stupid statutes. But since this context is a discussion of the accuracy of evidentiary rules and court procedure, your criticism is off-topic.
The original question was:
which I would interpret as ‘Is P(Conviction|Guilt) substantially larger than P(Conviction|Innocence)?’ Now, for some crimes such as copyright infringement, P(G) is very close to 1, so P(C|G) cannot be close to 1 simply because then there wouldn't be enough room in prisons to hold NP(C|G)P(G) people (N being the population -- times the mean sentence length, over the mean lifespan, and possibly some other factor of order unity I'm forgetting of), and since P(I) is small, in order for P(C|I) to be much less than P(C|G), P(C and I) = P(C|I)P(I) must be very small.
(Also, we want the system to be unbiased, i.e. P(C|G, brown skin) to be close to P(C|G, pink skin), P(C|G, penis) to be close to P(C|G, vagina), and so on, and so forth. The best way of achieving this would IMO be for all of these numbers to be close to 1, but that's impossible with the current definition of G and finite capacity of prisons.)
Are we restricting to cases that are prosecuted or doing this over all people?
The immediately obvious answer would be “over the prosecuted if you're only ‘testing’ the courtroom itself, over all the people if you're ‘testing’ the whole system”, but I'm not sure what the ‘ideal’ thing for a courtroom to do in terms of P(C|Prosecution, G) and P(C|P,I) if the police is ‘non-ideal’ so that P(P|G) is not close to or greater than P(P|I) to start with. Or even whether this question makes sense... I'll have to think more clearly about this when I'm not this tired.
It should be noted that the reasonable meaning of 'substantially' in such a case is relative not just the unadorned absolute difference in the ridiculously low probabilities. In comparisons of this nature a difference it is overwhelmingly obvious that "0.0001 - 0.000001" is much more significant than "0.6001 - 0.600001".
The above consideration is even more important when the substitution of "is significantly better than chance" with "‘Is P(Conviction|Guilt) substantially larger than P(Conviction|Innocence)?’" is yours and not the original question.
I'm not sure of this: I'd very much prefer a world where 0.01% of guilty people and 0.0001 of innocent people are convicted to one where 60.01% and 60.001% are. (If convicting a guilty person has utility A and convicting an innocent person has utility -B, what you want to maximise is AP(C|G) - BP(C|I), which also depends on the magnitudes of the probabilities, not only on their ratio.)
Huh? How else could the original question be interpreted? [ETA: Well, if better is interpreted instrumentally rather than epistemically, “the legal system is significantly better than chance” means “AP(C|G) - BP(C|I) is significantly greater than AP(C) - BP(C)”; and since A is orders of magnitude less than B (unless we're talking about serial killers or similarly serious stuff), that boils down to “P(C|I) is significantly less than P(C)”, where it's the magnitude of the difference that matters, rather than the ratio.]
So do I. This answers the question "do I consider false convictions worse than guilty parties who are not punished" but does not tell us much about the original question. The point of the original question was not "with trivial consideration of how much the conviction process is different to chance how high is the base rate of convictions for this crime?"
That is a reasonable interpretation - the point is that we must also interpret 'significance' in light of the original meaning. That original meaning does not contain an overwhelming emphasis on the base rate of convictions!