Three Methods for Combining Probability Distributions and an Alternative to Random-Effects Meta-Analysis

Many fields or disciplines (e.g. uncertainty analysis in measurement science) require a combination of probability distributions. This paper examines three methods for combining probability distributions: weighted linear pooling, geometric pooling, and the law of combination of distributions (LCD). It provides insights into these three methods under the normality assumption. It shows that the weighted linear pooling method preserves all the variability (including heterogeneity) information in the original distributions; neither the geometric pooling method nor the LCD method preserves all the variability information, leading to information loss. We propose an index for measuring the information loss of a method with respect to the weighted linear pooling method. This paper also shows that the weighted linear pooling method can be used as an alternative to random-effects meta-analysis. Three examples are presented: the combination of two normal distributions, the combination of three discrete distributions, and the determination of the Newtonian constant of gravitation.