Your thought experiment here is incorrect, given that the analysis compares COMPAS results to actual recidivism rates and shows over- and under-prediction in comparison to them.
The thought experiment is a mathematical proof that the two concepts are causally unrelated, nothing more. I really suggest you brush up on your basic math - you seem to not be following along.
Your claims about the emirical means of recidivism rates do not prove what you think they prove. Different races might be misclassified at different rates for a variety of reasons - e.g., one race might be affected more by some high-variance predictor, or there could be composition effects (e.g. the pdf of blacks|high score might be different than whites|high score).
The way to factor out whether they scores are biased is to do the cox survival analysis with interaction terms. Which they did. You just don't like the result.
Could you clearly lay out the statistical argument that you believe implies that E[\hat{\theta} - \theta] > 0?
> Could you clearly lay out the statistical argument that you believe implies that E[\hat{\theta} - \theta] > 0?
Again: I don't care about your domain-specific definition of "bias". I care about whether the end result of this secret algorithm is unequal and inaccurate treatment of different demographic groups.
inaccurate treatment of different demographic groups.
This is exactly what the standard mathematical definition of bias (restricted to a given group) addresses. The authors of this article ran exactly that analysis - see lines [36] and [46].
I know that you are trying to retreat from statistics, since the stats don't support your mood affiliation, but don't retreat to "accuracy". Retreat to something vague and undefined instead. It'll work better.
As for "unequal", I don't know what you mean. Do you consider disparate impact to be "unequal"? If so, then I'm sorry to tell you that reality is imposing an unfortunate choice on you: equal or accurate, you can't have both. (According to ProPublica this algorithm chooses accurate.)
Criticizing an algorithm for revealing this unfortunate fact is like blaming telescopes for Saturn having rings.
By "equal and accurate", I mean a result that corresponds to actual recidivism rates for as many demographic groups as possible (males, females, different races, different ages, combinations of the above, etc). The analysis shows that it is substantially likely (certainly well-beyond the oft-vaunted "reasonable doubt" standard, nevermind the specifics of p-values being slightly above 0.5) that the algorithm being used doesn't provide such a result.
Your claims about the emirical means of recidivism rates do not prove what you think they prove. Different races might be misclassified at different rates for a variety of reasons - e.g., one race might be affected more by some high-variance predictor, or there could be composition effects (e.g. the pdf of blacks|high score might be different than whites|high score).
The way to factor out whether they scores are biased is to do the cox survival analysis with interaction terms. Which they did. You just don't like the result.
Could you clearly lay out the statistical argument that you believe implies that E[\hat{\theta} - \theta] > 0?