# 5.1 Bayesian Metrology: New Methods in Physical Measurement

Keith Eberhardt, David L. Banks, Charles Hagwood, Raghu Kacker, Mark Levenson, Hung-kung Liu, Mark Vangel, James Yen, Nien Fan Zhang
Statistical Engineering Division, ITL

Christoph Witzgall
Mathematical and Computational Sciences Division, ITL

There are two approaches to statistics: frequentist and Bayesian. They use different definitions of probability (entailing philosophical disagreements), but in many simple cases give answers that are similar. Frequentist statistics has long been the dominant view, partly for computational reasons. In the last decade, attention has shifted towards Bayesian statistics, which has advantages in complex problems and better reflects the way scientists think about evidence.

Our project develops Bayesian methods appropriate to the metrological problems faced by NIST scientists. There are four specific areas under consideration:

1.
Traceability and Expression of Uncertainty. This has a natural representation as a Bayesian hierarchical model, which reflects the uncertainty at each stage in a chain of measurements as a probability distribution. A Bayesian would express this chain as a sequence of probability distributions, each having (hyper)parameters that are modeled by the preceding distribution.
2.
Interlaboratory Comparisons. Bayes' Theorem is a blueprint for how information (even inconsistent information) should be combined; it enables divergent or unreliable data to be downweighted based upon current and past discrepancies. Most applications depend upon the Bayesian hierarchical model.
3.
Inspection. In part inspection problems, the mathematically correct procedure is given by decision theory, which is entirely formulated in terms of the Bayes risk. This approach enables one to balance the cost of passing a bad part against the loss from failing a good part.
4.
Calibration. Finding a calibration curve is an ill-posed problem, since an uncountable number of curves can agree equally well with the data. All solutions inject some kind of structure; the Bayesian formulation uses prior information. This prior information can penalize roughness in the fitted curve, and enforce monotonicity.
The technical work needed on these topics is different, but the common challenge is to find computable ways in which prior domain information can be used in the analysis. Recent work, including the hierarchical model, probability elicitation, and especially Markov chain Monte Carlo are the key tools needed for success.

Figure 29: This shows a profile likelihood surface used as part of a Bayesian analysis of combined data from many laboratories. Note that the a single laboratory is extremely influential; removing it changes the surface dramatically.

Date created: 7/20/2001
Last updated: 7/20/2001