Bayesian Nonparametric Modeling Using Mixtures of Triangular Distributions

Abstract

Nonparametric modeling is an indispensable tool in many applications and its formulation in an hierarchical Bayesian context, using the entire posterior distribution rather than particular expectations, increases its flexibility. In this article, the focus is on nonparametric estimation through a mixture of triangular distributions. The optimality of this methodology is addressed and bounds on the accuracy of this approximation are derived. Although our approach is more widely applicable, we focus for simplicity on estimation of a monotone nondecreasing regression on [0, 1] with additive error, effectively approximating the function of interest by a function having a piecewise linear derivative. Computationally accessible methods of estimation are described through an amalgamation of existing Markov chain Monte Carlo algorithms. Simulations and examples illustrate the approach.

Bayesian Goodness of Fit Testing with Mixtures of Triangular Distributions

“…With priors such as these, verifying that the general conditions given in section 2 hold is far from trivial. In order to overcome this difficulty we consider in section 3 a class of priors based on certain mixtures of triangular distributions which were proposed in Perron & Mengersen (2001). Using these priors, we discuss in section 4 the consistency of the Bayes factor for testing a parametric family, giving simple conditions on the parametric family for the Bayes factor to be consistent.…”

Bayesian Modelling and Inference on Mixtures of Distributions

“…Depending on the range of the (α k , β k )'s, different behaviours can be observed in the vicinities of 0 and 1, with much more variability than with the Bernstein prior which restricts the (α k , β k )'s to be integers. An alternative to mixtures of Beta distributions for modelling unknown distributions is considered in Perron and Mengersen (2001) in the context of non-parametric regression. Here, mixtures of triangular distributions are used instead and compare favourably with Beta equivalents for certain types of regression, particularly those with sizeable jumps or changepoints.…”

Metropolis–Hastings algorithms with adaptive proposals

“…If the Markov chain is already close to stationarity, then these will be an almost iid sample of π and thus a nonparametric density estimate based on these m − 1 points will be a good approximation to the target density. Our idea for a nonparametric density estimate is based on Perron and Mengersen (2001) who show that any cumulative distribution function can be approximated by a mixture of triangular distributions. They give bounds on the accuracy of this approximation and show an improvement over the approximation by Bernstein polynomials, i.e.…”

“…Perron and Mengersen (2001) showed that amongst the best partitions for approximating any nondecreasing function, say F (x), by a mixture of triangulars with respect to the Kolmogorov supnorm distance, there is one that has weights given approximately by (4). Therefore, we use these optimal (in the above sense) fixed weights here.…”