How do we make causal judgments? Many studies have demonstrated that people are capable causal reasoners, achieving success on tasks from reasoning to categorization to interventions. However, less is known about the mental processes people use to achieve such sophisticated judgments. We propose a new process model - the mutation sampler - that models causal judgments as based on a sample of possible states of the causal system generated using the Metropolis-Hastings sampling algorithm. Across a diverse array of tasks and conditions encompassing over 1,700 participants, we found that our model provided a consistently closer fit to participant judgments than standard causal graphical models. In particular, we found that the biases introduced by mutation sampling accounted for people’s consistent, predictable errors that the normative model by definition could not. Moreover, using a novel methodology, we found that those biases appeared in the samples that participants explicitly judged to be representative of a causal system. We advocate the mutation sampler as the heretofore missing process level account of the computations specified by the causal graphical model framework, and highlight opportunities for future research to identify not just what reasoners compute when drawing causal inferences, but also how they compute it.