Occam’s razor is the principle stating that, all else being equal, simpler explanations for a set of observations are to be preferred to more complex ones. This idea can be made precise in the context of statistical inference, where a geometrical characterization of statistical model complexity emerges naturally from an approach based on Bayesian model selection. The broad applicability of this formulation suggests a normative reference point for decision making under uncertainty. However, little is known about if and how humans intuitively quantify the complexity of competing interpretations of noisy data. In this work, we: 1) extend an existing geometrical characterization of model complexity to apply to models with bounded parameters; 2) measure the sensitivity of naive human subjects to statistical model complexity; and 3) introduce a deep neural network architecture for statistical model selection. Our data shows that human subjects bias their decisions in favor of simple explanations based not only on the dimensionality of the alternatives (number of model parameters), but also on finer-grained aspects of their geometry, such as volume, curvature, and presence of prominent boundaries. We conclude by studying the behavior and learned representations of our deep network architecture when trained on the same task as the human subjects. Overall, our results imply that principled notions of statistical model complexity have direct quantitative relevance to human decision making.