Optimal Data-Based Binning for Histograms

Histograms are convenient non-parametric density estimators, which continue to be used ubiquitously. Summary quantities estimated from histogram-based probability density models depend on the choice of the number of bins. In this paper we introduce a straightforward data-based method of determining the optimal number of bins in a uniform bin-width histogram. Using the Bayesian framework, we derive the posterior probability for the number of bins in the density model given the data. The most probable solution is determined naturally by a balance between the likelihood function, which increases with increasing number of bins, and the prior probability of the model, which decreases with increasing number of bins. We demonstrate how these results outperform several well-accepted rules for choosing bin sizes even in the integrated square error sense. Last, we demonstrate that these results can be applied directly to multi-dimensional histograms.