Bayesian inference under qualitative assumptions and its applications

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The problem of estimating functions under qualitative assumptions has a long history dating back at least to the last five decades. Shape constrained regression models are useful for analyzing data with specific shape responses, such as (monotone) dose-response curves, the (concave) utility functions of a risk averse decision maker, the (increasing) growth curves of children's height through time, that are particularly common in medicine, economic and epidemiological studies. Regression and density estimation under shape and smoothness constraints are of considerable interest both for theoretical and practical reasons. From a theoretical point of view, shape constraints and smoothing usually reduce the variance of the estimators. From a practical point of view, there are many situations in which it is necessary to take into account some prior knowledge on the shape of the function to estimate. The B-spline nice property of locally supported basis functions enables us to localize the constraint or to consider a combination of constraints like being increasing on the interval [1,2] and concave on the interval [3,4]. Such combinations of constraints have been undertaken in my PhD thesis and my recent papers. Recently, I proposed an extension to the the smoothing spline context by considering a stochastic simulated annealing method for localized shapes optimization and we generalized the idea to the free-knot adaptive Bayesian inference. Now, I have started to derive asymptotic analysis and model selection based on Bayes factor criteria to go along with this method. I have just come up with what I have called the Laplace-Bayes factor that works well to choose the best constraint that should enter the model.***Now my research program over the next few years builds upon these recent developments and consists in developing new tools to fit regression and density models to datasets with specific shape responses. The short term objectives will be to fine tune the Laplace-Bayes Factor criterion to increase its efficiency when dealing with small datasets and massive databases. One key advantage of our method is that computations can easily be implemented by simulated annealing scheme. The analysis of acidification curves and global warming data will be the main applications of the new methodologies in the short term. In the medium term, I would like with my students to adapt my new techniques to the density and the mixture estimation problems and to develop model selection procedures that will be implementable in this context. Industrial partners will likely get involved at this stage and should provide data and internship opportunities for students. R packages to implement the methods will be made publicly available. In the long term, we would like to undertake the asymptotic analysis (consistency and convergence rates) of our Bayesian procedures in the presence of qualitative assumptions.**