Recent advancements in Bayesian modeling have allowed for likelihood-free posterior estimation. Such estimation techniques are crucial to the understanding of simulation-based models, whose likelihood functions may be difficult or even impossible to derive. However, current approaches are limited by their dependence on sufficient statistics and/or tolerance thresholds. In this article, we provide a new approach that requires no summary statistics, error terms, or thresholds and is generalizable to all models in psychology that can be simulated. We use our algorithm to fit a variety of cognitive models with known likelihood functions to ensure the accuracy of our approach. We then apply our method to two real-world examples to illustrate the types of complex problems our method solves. In the first example, we fit an error-correcting criterion model of signal detection, whose criterion dynamically adjusts after every trial. We then fit two models of choice response time to experimental data: the linear ballistic accumulator model, which has a known likelihood, and the leaky competing accumulator model, whose likelihood is intractable. The estimated posterior distributions of the two models allow for direct parameter interpretation and model comparison by means of conventional Bayesian statistics-a feat that was not previously possible.