Shape restriction of the multi-dimensional Bernstein prior for density functions

Random Bernstein polynomials induces a probability measure on the space of multivariate density functions on a unit cube. For density estimation, it is important that the Bernstein prior can be restricted to an admissible class of densities with certain geometric properties of the target density. In this article, we study the shape properties such as monotonicity, convexity, and symmetry of the Bernstein prior.