Modern measurement, probability, and statistics: Some generalities and multivariate illustrations

In broad terms, effective probability modeling of modern measurement requires the development of (usually parametric) distributions for increasingly complex multivariate outcomes driven by the physical realities of particular measurement technologies. "Differences" between measures of distribution center and truth function as "bias." Model features that allow hierarchical compounding of variation function to describe "variance components" like "repeatability," "reproducibility," "batch-to-batch variation," etc. Mixture features in models allow for description (and subsequent downweighting) of outliers. For a variety of reasons (including high-dimensionality of parameter spaces relative to typical sample sizes, the ability to directly include "Type B" considerations in assessing uncertainty, and the relatively direct path to uncertainty quantification for the real objectives of measurement), Bayesian methods of inference in these models are increasingly natural and arguably almost essential. We illustrate the above points first in an overly simple but instructive example. We then provide a set of formalisms for expressing these notions. Then we illustrate them with real modern measurement applications including (1) determination of cubic crystal orientation via electron backscatter diffraction, (2) determination of particle size distribution through sieving, and (3) analysis of theoretically monotone functional responses from thermogravimetric analysis in a materials study.