Investigation of the Effects of Probability Density Function Kurtosis on Evaluated Data Results

Abstract

In two previous investigations that are documented in this IAEA report series, we examined the effects of non-Gaussian, non-symmetric probability density functions (PDFs) on the outcomes of data evaluations. Most of this earlier work involved considering just two independent input data values and their respective uncertainties. They were used to generate one evaluated data point. The input data are referred to, respectively, as the mean value and standard deviation pair (y0,s0) for a prior PDF p0(y) and a second mean value and standard deviation pair (ye,se) for a likelihood PDF pe(y). Conceptually, these input data could be viewed as resulting from theory (subscript “0”) and experiment (subscript “e”). In accordance with Bayes Theorem, the evaluated mean value and standard deviation pair (ysol,ssol) corresponds to the posterior PDF p(y) which is related to p0(y) and pe(y) by p(y) = Cp0(y)pe(y). The prior and likelihood PDFs are both assumed to be normalized so that they integrate to unity for all y ≥ 0. Negative values of y are viewed as non-physical so they are not permitted. The product function p0(y)pe(y) is not normalized, so a positive multiplicative constant C is required to normalize p(y). In the earlier work, both normal (Gaussian) and lognormal functions were considered for the prior PDF. The likelihood functions were all taken to be Gaussians. Gaussians are symmetric, with zero skewness, and they always possess a fixed kurtosis of 3. Lognormal functions are inherently skewed, with widely varying values of skewness and kurtosis that depend on the function parameters. In order to explore the effects of kurtosis, distinct from skewness, the present work constrains the likelihood function to be Gaussian, and it considers three distinct, inherently symmetric prior PDF types: Gaussian (kurtosis = 3), Continuous Uniform (kurtosis = 1.8), and Laplace (kurtosis = 6). A product of two Gaussians produces a Gaussian even if ye ≠ y0. The product of a Gaussian PDF and a Uniform PDF, or a Laplace PDF, yields a symmetric PDF with zero skewness only when ye = y0. A pure test of the effect of kurtosis on an evaluation is provided by considering combinations of s0 and se with ye = y0. The present work also investigates the extent to which p(y) exhibits skewness when ye ≠ y0, again by considering various values for s0 and se. The Bayesian results from numerous numerical examples have been compared with corresponding least-squares solutions in order to arrive at some general conclusions regarding how the evaluated result (ysol,ssol) depends on various combinations of the input data y0, s0, ye, and se as well as on prior-likelihood PDF combinations: Gaussian-Gaussian, Uniform-Gaussian, and Laplace-Gaussian.