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
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
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:
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
- 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.
- 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
- 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.
- 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.
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.