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A Bayesian approach is presented for spatially adaptive nonparametric regression where the regression function is modelled as a mixture of splines. Each component spline in the mixture has associated with it a smoothing parameter which is defined over a local region of the covariate space. These local regions overlap such that individual data points may lie simultaneously in multiple regions. Consequently each component spline has attached to it a weight at each point of the covariate space and, by allowing the weight of each component spline to vary across the covariate space, a spatially adaptive estimate of the regression function is obtained. The number of mixing components is chosen using a modification of the Bayesian information criteria. We study the procedure analytically and show by simulation that it compares favourably to three competing techniques. These techniques are the Bayesian regression splines estimator of Smith & Kohn (1996), the hybrid adaptive spline estimator of Luo & Wahba (1997) and the automatic Bayesian curve fitting estimator of Denison et al. (1998). The methodology is illustrated by modelling global air temperature anomalies. All the computations are carried out efficiently using Markov chain Monte Carlo.