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Abstract:
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This paper investigates the ability for Bayesian model selection to identify the correct shape constraints in nonparametric regression. Nonparametric regression models with Guassian process priors offer maximal flexibility in estimating regression functions. However, researchers often believe, either through theory or previous analyses, that the regression function has shape constraints, such as monotonicity or curvature. These constraints can be introduced through the Gaussian process priors. The paper explores the feasibility of the log-integrated likelihood (LIL) to confirm correctly the appropriate constraints. When a constraint is incorrect, e.g. the model wrongly assumes an increasing function, the LIL readily dismisses the model with the incorrect constraint. The analysis is more nuanced when two or more models are consistent with the true function. If the models have the same number of parameters, the LIL tends to favor the true and more restrictive model, especially in low information conditions. As the information increases, the simpler model is selected more often. If the more restrictive model has more parameters, then the LIL often selects the simpler model.
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