Statistical uncertainty associated with histograms in the Earth sciences

[1] Two types of quantitative information can be distinguished in the Earth sciences: categorical data (e.g., mineral type, fossil name) and continuous data (e.g., apparent age, strike, dip). Many branches of the Earth sciences study populations of such data by collecting a random sample and binning it into a histogram. Histograms of categorical data follow multinomial distributions. All possible outcomes of a multinomial distribution with M categories must plot on a (M − 1) simplex ΔM−1 because they are subject to a constant sum constraint. Confidence regions for such multinomial distributions can be computed using Bayesian statistics. The conjugate prior/posterior to the multinomial distribution is the Dirichlet distribution. A 100(1-α)% confidence interval for the unknown multinomial population given an observed sample histogram is a polygon on ΔM−1 containing 100(1-α)% of its Dirichlet posterior. The projection of this polygon onto the sides of the simplex yields M confidence intervals for the M bin counts. These confidence intervals are “simultaneous” in the sense that they form a band completely containing the 100(1-α)% most likely multinomial populations. As opposed to categorical variables, adjacent bins of histograms containing continuous variables are not mutually independent. If this “smoothness” of the unknown population is not taken into account, the Bayesian confidence bands described above will be overly conservative. This problem can be solved by introducing an ad hoc prior of “smoothing weights” w = e−sr, where r is the integrated squared second derivative of the histogram and s is a “smoothing parameter.”