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″.
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.)
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.
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.]
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.
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?”
Huh? How else could the original question be interpreted?
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!
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!