Using expert knowledge when the data model is unknown with an application in modeling the Mixed Layer of the Atlantic Ocean

Oceanographers are interested in studying decadal variability of the ocean's heat content through the depth of the Mixed Layer M. This is the top layer of the ocean where the waters come together through turbulent mixing and convection, which creates a nearly uniform temperature layer in the ocean. It evolves with the season's annual cycle of surface temperatures and is smoothly varying in spaces. Understanding the sampling distribution of the data is difficult because of the spatio-temporal dependencies, because M does not uniquely determine the distribution of temperatures, and because the estimator Mˆ is not known. However, before and after observing the temperature profile, an expert has a prior and posterior belief. The novel idea of this thesis is how to translate the expert's uncertainty into a likelihood function.; We elicit the expert's posterior belief on a partition in the parameter space such that her prior probabilities may be considered constant across all elements of the partition. For any single profile, the expert's posterior belief is expressed by the function h(M| data). To model spatio-temporal dependencies, we suggest substituting the likelihood by h(M). The function h is not a product of a known density function, but it may be viewed as a distribution of the pivotal variables and is independent of the data rather than M, as usually defined. The pivotal variables have a strong scientific interpretation, and they serve as a transformation between the observed space and the transformed space. The posterior inference is given directly from the distribution of the transformed variables. The resulting posterior distribution is closely associated with direct probabilities of Dempster and fiducial distributions of Fisher. In general, there need not exist a marginal distribution for M which yields the posterior inference identical to the one arising from the pivotal quantities, but there always exists a likelihood function which, by the fiducial argument, gives the same posterior inference.; We also provide the inference using traditional methods when the sampling model is unknown. Although modeling known features in the data may provide valuable results, the posterior distribution of M for any single profile may not agree with the expert opinion. We provide a discussion as to why this is true. In the final chapter, we propose a flexible spatio-temporal model to help us understand the nature of the long-term changes in the depth of the Mixed Layer.