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Let be a sequence of observations satisfying where are real numbers, θ is a fixed but unknown regression function, and the errors are independent and identically distributed observations from a N(0,v) density. We develop a prior probability model for the regression function θ which allows for the possibility of jump discontinuities in the mth derivative of θ. Neither the number of jumps nor their locations are assumed known. A Bayesian analysis is implemented using Markov Chain Monte Carlo methods. The new method is compared to smoothing splines in a simulation study and in the analysis of a set of data representing the average weight to height ratio of a group of boys recorded at one month intervals.

Consider a sequence of independent random variables X 1, X 2,…,X n observed at n equally spaced time points where X i has a probability distribution which is known apart from the values of a parameter θ i ∈ R which may change from observation to observation. We consider the problem of estimating θ = (θ1, θ2,…,θ n ) given the observed values of X 1, X 2,…,X n . The paper proposes a prior distribution for the parameters θ for which sets of parameter values exhibiting no change, or no change apart from a few sudden large changes, or lots of small changes, all have positive prior probability. Markov chain sampling may be used to calculate Bayes estimates of the parameters. We report the results of a Monte Carlo study based on Poisson distributed data which compares the Bayes estimator with estimators obtained using cubic splines and with estimators derived from the Schwarz criterion. We conclude that the Bayes method is preferable in a minimax sense since it never produces the disastrously large errors of the...

Observations yi are made at points ti according to the model i=θ(ti)+ei where the ei are independent normals with constant variance. It is conjec tured that the function θ lies in a set G of functions spanned by the basis functions φ1,φ2,…,φp. A prior distribution is developed whereby the proba bility assigned to a function θ is a decreasing function of a particular measure of the distance of θ from the set G. Bayes' theorem is used to construct an estimateθ. A marginal likelihood is derived which is used to estimate the parameter of the prior and also for testing the null hypothesis Ho θ ∊ G. The new methodology is tested in a Monte Carlo study and applied to a set of data representing the average weight to height ratio of a group of boys recorded at one month intervals.

Given a vector of N observations sampled from a multivariate normal density with mean vector of the form Xθ and covariance matrix vI where is a vector of p ≤ N parameters and X is an NX p design matrix, we consider the estimator of θ defined as the value which minimises where D is a pXp symmetric non-negative definite matrix and λ ≥ 0 is a smoothing parameter. For fixed values of v and λ, we define a prior probability distribution for θ for which the posterior mean of θ given the data y is . We show that Jeffreys’ prior for the parameters v and λ satisfies and m is the rank of the matrix D. We report the results of a simulation study comparing the performance of the Bayes estimate using Jeffreys’ prior with that of the estimate obtained using Generalized Cross Validation in the contexts of ridge regression and smoothing spline regression.

Let satisfy where for each i, F is a fixed but unknown regression function and the errors {e.} are uncorrelated with mean zero and variance v. In this paper we consider an estimate r of F where r is chosen to minimise the residual sum of squares plus a roughness penalty depending on the first and second order partial derivatives of F. Upper bounds are obtained on the integrated mean squared error defined by In particular for data on a uniform grid it is shown that .