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A Method for Bayesian Monotonic Multiple Regression

Authors


Olli Saarela, Department of Chronic Disease Prevention, National Institute for Health and Welfare, P.O. Box 30 (Mannerheimintie 166), 00271 Helsinki, Finland.
E-mail: olli.saarela@thl.fi

Abstract

Abstract.  When applicable, an assumed monotonicity property of the regression function w.r.t. covariates has a strong stabilizing effect on the estimates. Because of this, other parametric or structural assumptions may not be needed at all. Although monotonic regression in one dimension is well studied, the question remains whether one can find computationally feasible generalizations to multiple dimensions. Here, we propose a non-parametric monotonic regression model for one or more covariates and a Bayesian estimation procedure. The monotonic construction is based on marked point processes, where the random point locations and the associated marks (function levels) together form piecewise constant realizations of the regression surfaces. The actual inference is based on model-averaged results over the realizations. The monotonicity of the construction is enforced by partial ordering constraints, which allows it to asymptotically, with increasing density of support points, approximate the family of all monotonic bounded continuous functions.

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