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Center for Nonlinear Studies |
Bayesian statistics formalizes a procedure for using prior knowledge in the analysis of current or new data. Frequently this takes the form of using prior parameter estimates for a function in the process of analyzing new data. In this procedure, a subject matter expert provides the prior knowledge, generally on the basis of quantitative data. The uncertainties in those parameters are less commonly documented, and the form of their probability distribution function is almost never formally documented. The expert, however, must provide all three of these. A common scenario involves deriving parameter estimates from prior published work, estimating uncertainty in these parameter frequently using ad hoc methods, and assuming the parameters form a multivariate normal distribution. This seminar demonstrates that this procedure can degrade the quality of estimates rather than enhancing it in many situations and for even the most basic of models, including a straight line. The issues associated with this procedure influence many models common to the sciences and engineering, and are unavoidable even for a straight line model when the independent variable is positive valued. Since mole fractions, absolute temperatures and pressures, concentrations, and most other quantities common to science are positive valued, the issues discussed in this seminar are of particular importance to in these fields. However, there are alternative Bayesian-based procedures for parameter estimates that avoid the problems indicated above. These procedures focus on documenting what is best known, which is generally the parameter values and the experimental error in the measurements, and avoid making unjustified assumptions about parameter uncertainty and especially parameter probability functions. The procedures produce parameter characterizations with uncertainty and probability density functions, as are produced in the common scenario described above, but none of the heuristic, ad hoc or assumed information in the common scenario is used and therefore it cannot corrupt the results. The results from the alternative Bayesian approach are demonstrably improved for several data sets used as examples.
Host: Jim Gattiker (gatt@lanl.gov)