Calculating Bayesian Prior Distributions by Assuming Gamma-Distributed Data Sources and K-Factor Uncertainty Measurements

摘要

Bayesian updating is a standard and widely-accepted approach to failure rate quantification in the probabilistic risk assessment world. The Bayesian approach combines a prior distribution-representing our beliefs about the probability distribution on the failure rate before a particular set of evidence is observed-with a likelihood function representing observed evidence (expressed as failures in time or per demand) to produce the posterior distribution. The Bayesian prior is typically calculated as a weighted average of a number of data points taken from similar components operating in similar environments, with weights generally chosen by subject matter experts as a subjective measure of the applicability or relevance of each data source. In this paper, we develop a novel method for combining data sources in which no subjective weights for data sources need be estimated; instead, implicit weights are derived from uncertainty surrounding the multiplicative factors necessary to convert data observed in one context or environment into another (K-factors, called π-factors in MIL-HDBK-217). The method also applies an equivalence between evidence expressed as hours and failures and Gamma-distributed probability distributions to superimpose multiple data sources in a statistically defensible manner, avoiding some common pitfalls of a combination based on weighted averages of means and variances.