Additive cubic spline regression with Dirichlet process mixture errors
The goal of this article is to develop a flexible Bayesian analysis of regression models for continuous and categorical outcomes. In the models we study, covariate (or regression) effects are modeled additively by cubic splines, and the error distribution (that of the latent outcomes in the case of categorical data) is modeled as a Dirichlet process mixture. We employ a relatively unexplored but attractive basis in which the spline coefficients are the unknown function ordinates at the knots. We exploit this feature to develop a proper prior distribution on the coefficients that involves the first and second differences of the ordinates, quantities about which one may have prior knowledge. We also discuss the problem of comparing models with different numbers of knots or different error distributions through marginal likelihoods and Bayes factors which are computed within the framework ofÂ Chib (1995) as extended to DPM models by Basu and Chib (2003). The techniques are illustrated with simulated and real data.
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- Tiwari, R C & Jammalamadaka, S R & Chib, Siddhartha, 1988. "Bayes Prediction Density and Regression Estimation--A Semiparametric Approach," Empirical Economics, Springer, vol. 13(3/4), pages 209-22.
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- Chib, Siddhartha & Greenberg, Edward, 1994. "Bayes inference in regression models with ARMA (p, q) errors," Journal of Econometrics, Elsevier, vol. 64(1-2), pages 183-206.
- Geweke, John & Keane, Michael, 2007. "Smoothly mixing regressions," Journal of Econometrics, Elsevier, vol. 138(1), pages 252-290, May.
- M. Vannucci & F. Corradi, 1999. "Covariance structure of wavelet coefficients: theory and models in a Bayesian perspective," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(4), pages 971-986.
- Chib S. & Jeliazkov I., 2001. "Marginal Likelihood From the Metropolis-Hastings Output," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 270-281, March.
- Basu S. & Chib S., 2003. "Marginal Likelihood and Bayes Factors for Dirichlet Process Mixture Models," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 224-235, January.
- Villani, Mattias & Kohn, Robert & Giordani, Paolo, 2007. "Nonparametric Regression Density Estimation Using Smoothly Varying Normal Mixtures," Working Paper Series 211, Sveriges Riksbank (Central Bank of Sweden).
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