Séminaires, congrès et conférences
Titre: Consistency and convergence rate of posterior distribution under combinations of shape constraints
Heure: 13 h 30
Conférencier: Professeur Khader Khadraoui, Département de mathématiques et de statistique, Université Laval
The Bayesian approach in statistics has become quite popular in recent years as the main appeal of the Bayesian methodology is its conceptual simplicity. B-splines are useful building blocks when constructing priors on non-parametric models indexed by functions. Recently it has been established by Abraham and Khadraoui (2014) that the local support property of B-splines enables us to take into account combinations of shape constraints and to localize each shape constraint on a given interval using the Bayesian regression inference. It is now known that hierarchical adaptive priors based on splines with a random number of knots and random coefficients in the B-spline basis lead, under some conditions, to optimal posterior contraction rates, over Hölder, Sobolev and Besov spaces. In this talk we extend these results for when the location of the knots is endowed with a prior together with the presence of combinations of shape constraints. This has already been a common practice in Markov chain Monte Carlo applications, however an asymptotic analysis in terms of adaptive consistency and contraction rate was missing. Under some mild assumptions, we establish a result that provides sufficient conditions for adaptive consistency in a range of shape constraints, over Hölder space. We also present some numerical results illustrating how such a prior adapts to shape constraints. Applications to a real data sets in food industry and Global Warming are provided.