Overview: Bayesian Metrology

Bayesian Metrology: Overview Topics
What is Statistical Metrology?	Metrology is the science of weights and measures as well as the study of measuring devices. The ultimate goal for metrology is to outline ways in which metrological constants can be measured to acceptable accuracies. The word constant encompasses all engineering, physical science, information technology, biological, etc constants sought via instrumentation. The three major tasks needed to accomplish this goal are: a precise definition of the metrological constant, a measuring device to determine the unknown constant in terms of an accepted unit of measurement, and the assignment of a realistic uncertainty to the results. Generally, basic equations of physics are used to precisely define the unknown constant. For example, mass metrology is based on Newton's Second Law . The measuring device is an ideal balance and the uncertainty is determined by how accurately one can determine all the significant forces that make up the left hand side of the above equation. An acceptable metrology system requires traceability of measurements to an international standard and world class precision. Traceability is the means by which measurements are standardized. Traceability provides a common means by which measurements are comparable and ensures the uniformity of manufactured goods and industrial processes. Although measurements are traceable to a very accurately measured international standard, along the traceability chain precision can differ from manufacturer to manufacturer. Of course, the goal is to produce the best quality product, that is, the one with the lowest total uncertainty or with the highest precision. Although, the precision of products may differ from manufacturer to manufacturer, an accurate statement of uncertainty of the measurand is often a requirement in manufacturing. Statistical metrology was initially coined to describe the use of electrical measurements to deduce physical structure. The emphasis on uncertainty as opposed to the mean or other parameters of central tendency makes statistical metrology quite unique from other statistical fields. Although precise instrumentation can drastically reduce uncertainty, uncertainty cannot be completely removed. So, statistical methods must be applied in order to account for all the measurement uncertainty. The factors affecting uncertainty are set partly by experience, partly by discussions with experts, and partly by analysis.

What is Bayesian Metrology?	Because experience is a part of the accuracy determination, Bayesian statistics has a natural part to play in statistical metrology. By experience, we mean knowledge of the measurement process that is independent of the measurements being taken. The statistical way to characterize these apriori knowledge events is with a set of probabilites describing the probability or plausibility of them occurring. This set is called the prior probability distribution of the measurand. The mechanism for combining prior information with physical measurements is Bayes Theorem. The outcome is the posterior distribution, where its variance or a credible region is taken as the uncertainty of the measurand.

Why is Bayesian Metrology important?	In the last two decades, great advances in technology and industry have produced various high throughput measurement instruments. Large scale measurements have been collected in different areas in industry and technology, in the form of data as curves (e.g. thermal diffusivity measurements), data array and images (e.g. DNA microarray experiments), and spatial observations of space-time systems (e.g., satellite images), etc. Efficient use of these great resources raises a host of new methodological and experimental design issues in statistics, such as: how to assess and calibrate the instruments? how to combine data from heterogeneous sources? how to do the error propagation in high dimensional and nonlinear input-output systems? how to combine model uncertainty and prior information (e.g., type B uncertainty) with experimental results? how to design multi-stage experiments used in automated testing and other areas? The Bayesian approach offers a systematic and flexible approach to these problems. By adopting an objective or noninformative prior, the Bayesian approach produces estimates and uncertainty measures comparable to the classical approach. However, in most of the problems considered here, it is either imperative to take into account prior information or physical knowledge, or there are underlying latent and unobservable processes. Thus, the Bayesian approach offers the only viable and rigorous solution, though there is also the added benefit of providing much-needed uncertainty and probability assessments in nonlinear and non-Gaussian situations in a valid and rigorous way. The Bayesian metrology project will spend significant efforts in developing and implementing fast and reliable computational algorithms, most of which are based on recent developments in Markov Chain Monte Carlo (MCMC) in statistics and physics, and will pay particular attention to making the developed products accessible to scientists within NIST and other NMIs, as well as customers in industry. This will represent a drastic upgrade over existing statistical tools which have existed since the 1970s and a major methodological development for a number of new problems and areas not covered by the Guide to Uncertainty in Measurement (GUM) specification. The research directions identified in the Bayesian metrology project will have far-reaching consequences for the twenty first century statistical metrology.

What are the Benefits of Bayesian Metrology?	The benefits of Bayesian metrology include the following. The Bayesian approach provides a formal approach to utilizing scientific knowledge and past information to yield better design of experiments and testing strategies. By utilizing prior information, Bayesian methods can often be more efficient than other methods. The Bayesian approach can provide model conformance to physical laws. In many complex problems such as analyzing high throughput measurement and high-dimensional data and complex dynamical systems, the Bayesian method provides the only viable solution, and it has been the most powerful tool in many recent scientific and technological innovations. The Bayesian approach provides a scientific basis for codifying and combining knowledge and information from different sources.

What are the Cautions in Applying Bayesian Methodolgy to Metrology Problems?	As with any powerful tool, there are cautions that should be kept in mind when applying Bayesian methodolgy to metrology problems. There should be a judicious way of choosing and documenting prior information, and when an improper prior is used, checking the propriety of the posterior distribution may be difficult. Sensitivity analysis involving different priors is strongly recommended in order to verify the robustness of the conclusion. If little reliable prior information is available, default priors should be used as much as possible. More computational efforts may be involved, especially when the Markov Chain Monte Carlo (MCMC) method is used to simulate from the posterior distribution. It is crucial to assess convergence and to design efficient simulation experiments. There are many current activities in software developments for general Bayesian statistics in both the academic and commercial worlds. There needs to be a maturing and testing period before reliable and general Bayesian software can emerge.

Date created: 8/28/2001 Last updated: 8/28/2001 Please email comments on this WWW page to sedwww@nist.gov.

Bayesian Metrology: Overview Topics