Unless you believe that young black men in US are the most criminal group in history of the world, most of them who are in prisons must be innocent by pure statistics.

Or there is a significant selection bias in the quasi-random selection process known as "arresting people."

By the way, have you read Bernard Harcourt's research, which suggests that the "imprisonment" rate in the United States has been constant over time, provided that you count commitment for mental illness as imprisonment. Thus, the recent growth in prison population in the United States reflects the shrinkage in long-term involuntary commitment of the mentally ill. In other words, a lot of the restrictions on the extremely mentally ill that used to be "provided" by dedicated institutions (i.e. mental hospitals) are now "provided" by jails and prisons.