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Predictive modeling of criminal recidivism, or whether people will re-offend in the future, has a long and contentious history. Modern causal inference methods allow us to move beyond prediction and target the "treatment effect" of a specific intervention on an outcome in an observational dataset. In this paper, we look specifically at the effect of incarceration (prison time) on recidivism, using a well-known dataset from North Carolina. Two popular causal methods for addressing confounding bias are explained and demonstrated: directed acyclic graph (DAG) adjustment and double machine learning (DML), including a sensitivity analysis for unobserved confounders. We find that incarceration has a detrimental effect on recidivism, i.e., longer prison sentences make it more likely that individuals will re-offend after release, although this conclusion should not be generalized beyond the scope of our data. We hope that this case study can inform future applications of causal inference to criminal justice analysis.