Occam’s razor is the principle stating that, all else being equal, simpler explanations for a set of observations are preferred over more complex ones. This idea is central to multiple formal theories of statistical model selection and is posited to play a role in human perception and decision-making, but a general, quantitative account of the specific nature and impact of complexity on human decision-making is still missing. Here we use a preregistered, online study to show that, when faced with uncertain evidence, human subjects bias their decisions in favor of simpler explanations in a way that can be quantified precisely using the framework of Bayesian model selection. These biases reflect an aversion to complex explanations (statistical models of data) that depends on specific geometrical features of those models, namely their dimensionality, boundaries, volume, and curvature. Overall, our results imply that principled notions of statistical model complexity have direct, quantitative relevance to human decision-making and establish a new understanding of the computational foundations, and behavioral benefits, of our predilection for inferring simplicity in the latent properties of our complex world.