A method of estimating a variety of curves by a sequence of piecewise polynomials is proposed, motivated by a Bayesian model and an appropriate summary of the resulting posterior distribution. A joint distribution is set up over both the number and the position of the knots de®ning the piecewise polynomials. Throughout we use reversible jump Markov chain Monte Carlo methods to compute the posteriors. The methodology has been successful in giving good estimates for`smooth' functions (i.e. continuous and differentiable) as well as functions which are not differentiable, and perhaps not even continuous, at a ®nite number of points. The methodology is extended to deal with generalized additive models.

http://stat.tamu.edu/people/faculty/bmallick.html.

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Summary We propose an efficient way to sample from a class of structured multivariate Gaussian distributions. The proposed algorithm only requires matrix multiplications and linear system solutions. Its computational complexity grows linearly with the dimension, unlike existing algorithms that rely on Cholesky factorizations with cubic complexity. The algorithm is broadly applicable in settings where Gaussian scale mixture priors are used on high-dimensional parameters. Its effectiveness is illustrated through a high-dimensional regression problem with a horseshoe prior on the regression coefficients. Other potential applications are outlined.

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