Instead of holding criminal defendants on cash bail, courts around the United States are increasingly using algorithmic risk-assessment tools to help judges decide if a defendant should be jailed or go free while awaiting trial.

One such model now used by New Jersey and other state and regional courts is the Public Safety Assessment, developed by the Houston-based Laura and John Arnold Foundation.

Supporters of such assessments say they limit prejudicial judgments, while critics say they can turn judging into a rote exercise — one that might perpetuate hidden biases within the criminal justice...