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# Bayesian Curve Fitting with Free-Knot Splines

**Ilaria DiMatteo, Christopher R. Genovese, and Robert E. Kass**

### Abstract:

Reversible-jump Markov chain Monte Carlo may be used to fit
scatterplot data with cubic splines having unknown numbers of knots
and knot locations. Key features of the implementation investigated
here are (i) a fully Bayesian formulation that puts priors on the
spline coefficients and (ii) Metropolis-Hastings proposal densities
that attempt to place knots close to one another. Simulation results
indicate this methodology can produce fitted curves with substantially
smaller mean squared-error than competing methods. The
reversible-jump implementation requires ratios of marginal densities
for the data (integrated likelihood ratios). We approximate these
using the Bayes Information Criterion and thereby obtain a general
approach to Bayesian nonparametric regression for arbitrary
response-variable distributions. We illustrate with an application to
Poisson nonparametric regression modeling of neuron firing patterns.

*Heidi Sestrich*

*5/4/2000*
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