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.