Only non-informative Bayesian prior distributions agree with the GUM Type A evaluations of input quantities

Published

June 21, 2011

Author(s)

Raghu N. Kacker

Abstract

The Guide to the Expression of Uncertainty in Measurement (GUM) is self-consistent when Bayesian statistics is used for the Type A evaluations and the standard deviation of the posterior state-of-knowledge distribution is used as the Bayesian standard uncertainty. Bayesian statistics yields posterior state-of-knowledge probability distributions which have the same probabilistic interpretation as the Type B state-of-knowledge probability distributions. Thus the Type A and the Type B evaluations can be combined logically. This much is well known. We show that there are limitations on the kind of Bayesian statistics that can be used together with the GUM. The GUM recommends that the (central) measured value should be an unbiased estimate of the corresponding (true) quantity value. Also, the GUM uses the expected value of state-of-knowledge probability distributions as the measured value for both the Type A and the Type B evaluations. When Bayesian prior distributions are proper probability density functions (pdfs), the expected value of the Bayesian posterior state-of-knowledge distribution can never be an unbiased estimate. Thus a measured value can be unbiased only when non-informative prior distributions (which are not proper pdfs) are used. Therefore only non-informative improper prior distributions agree with the GUM. This note is relevant because wide availability of computational software for doing Bayesian analysis numerically has stimulated great interest in the use of Bayesian statistics for the evaluation of uncertainty in measurement. The Bayesian computational software requires that the prior distributions to be proper pdfs.

Proceedings Title

Proceedings of Advanced Mathematical and Computational Tools in Metrology and Testing (AMCTM 2011)

Conference Dates

June 19-22, 2011

Conference Location

Gotenborg

Pub Type

Conferences

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Keywords

Bayesian statistics, Unbiased estimate, Uncertainty in measurement

Citation

Kacker, R. (2011), Only non-informative Bayesian prior distributions agree with the GUM Type A evaluations of input quantities, Proceedings of Advanced Mathematical and Computational Tools in Metrology and Testing (AMCTM 2011), Gotenborg, -1, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=907999 (Accessed October 9, 2025)