2018-02-01 - Larry Baxter - Structure in Prior PDFs and Its Effect on Bayesian Analysis
Bayesian statistics formalizes a procedure for combining established (prior) statistical knowledge with current knowledge to produce a posterior statistical description that presumably is better than either the prior or new knowledge by itself. Two common applications of this theory involve (a) combining established (literature) estimates of model parameter with new data to produce better parameter estimates, and (b) estimating model prediction confidence bands. Frequently, the prior information includes reasonable parameter estimates, poorly quantified and often subjective parameter uncertainty estimates, and no information regarding how the values of one parameter affect the confidence intervals of other parameters. All three of these parameter characteristics affect Bayesian analysis. The first two receive a great deal of attention. The third characteristic, the dependence of model parameters on one another, creates structure in the prior pdfs. This structure strongly influences Bayesian results, often to an extent that rivals or surpasses the parameter uncertainty best estimates. Nevertheless, Bayesian analyses commonly ignore this structure.
All structure stems primarily from the form of the model and, in linear models, does not depend on the observations themselves. Most models produce correlated parameters when applied to real-world engineering and science data. The most common example of structure is parameter correlation coefficients. Linear models produce linear parameter correlations that depend on the magnitude of the independent variable under analysis but that in most practical applications produce large, often close to unity, correlation coefficients. Nonlinear models also generally have correlated parameters. However the correlations can be nonlinear, even discontinuous, and generally involve more complexity than linear model parameter correlations. Parameter correlations profoundly affect the results of Bayesian parameter estimation and prediction uncertainty. Properly incorporated structure produces Bayesian results that powerfully illustrate the strength and potential contribution of the theory. Bayesian analyses that ignore such structure produce poor or even nonsensical results, often significantly worse than a superficial guess.
This seminar demonstrates the importance of prior structure in both parameter estimation and uncertainty quantification using real data from typical engineering systems. Perhaps most importantly, the discussion illustrates methods of incorporating parameter structure for any given model that does not rely on observations. These methods quantify parameter structure, including the lack of structure, for linear and nonlinear models.