Nonparametric adaptive Bayesian regression using priors with tractable normalizing constants and under qualitative assumptions

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. This paper proposes a new adaptive Bayesian approach towards constructing prior distributions, with known normalizing constants, which enables us to take into account combinations of shape constraints and to localize each shape constraint on a given interval. Our strategy enables us to compute the simulation from the posterior distribution using a reversible jump Metropolis-Hastings scheme. The major advantages of the proposal are its flexibility achieved by adjusting local shape restrictions to detect better the high and low variability regions of the data and to facilitate the control of the regression function shape when there is no data at all in some regions. We give asymptotic results that show that our Bayesian method provides consistent function estimator from the adaptive prior. The performance of our method is investigated through a simulation study with small samples. An analysis of two real data sets is presented to illustrate the new approach.