Not so. Priors/posteriors are only as good as the model they're based on. For instance, if you choose parental income as the feature it will can be a stronger signal than skin color, even though both may be good predictors. But the correlation between the income and skin color can account for the predictive power of one feature when the other features is the true cause.
So if Americans from race A commit ten times more violent crime than others and the police consequently accuse and manhandle vast numbers of innocent Americans of race A, there's nothing unjust about that? The vast majority of citizens of race A are innocent of all offenses but deserve constant suspicion and low level official humiliation and violence in a just world for no reason other than being the same color as some crooks.
accuse and manhandle vast numbers of innocent Americans of race A
One doesn't need to advocate accusing and manhandling to think the police should use statistically valid inferences in the name of justice. I myself am a member of a minority group—men—that is responsible for a vastly disproportionate share of crime, especially violent crime. You could mandate that cops ignore this reality and treat men and women with equal suspicion, but the result would be worse policing. For example, if you look at the statistics for New York's supposedly racist "stop-and-frisk" policy, you'll find that the disparity between whites and blacks is smaller than the disparity between men and women—indeed, smaller even than the disparity between white men and black women. Why have you never heard stop-and-frisk described as "sexist"?
This inconsistency is best explained politically: complaining about racial injustice against blacks is an effective route to power; complaining about gender injustice against men is not. It's the same reason you hear constant complaints about how white tech is, but not about how black sports are. Jesse Jackson can effectively shake down Apple and Intel [1], but there is no white equivalent shaking down the NFL. (Can you imagine if "increasing diversity in the NFL" meant "increasing the relative proportion of white players"? It would be a different world—not, incidentally, one I would particularly want to live in.)
Being male means people will infer based on a superficial assessment that I'm more likely to be a criminal than, say, my sister. But that inference is correct. Being a member of such a group is my lot in life, and complaining doesn't change what is.
