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This article describes a new class of prior distributions for nonparametricfunction estimation. The unknown function is modeled as a limit of weightedsums of kernels or generator functions indexed by continuous parameters thatcontrol local and global features such as their translation, dilation,modulation and shape. L\'{e}vy random fields and their stochastic integrals areemployed to induce prior distributions for the unknown functions or,equivalently, for the number of kernels and for the parameters governing theirfeatures. Scaling, shape, and other features of the generating functions arelocation-specific to allow quite different function properties in differentparts of the space, as with wavelet bases and other methods employingovercomplete dictionaries. We provide conditions under which the stochasticexpansions converge in specified Besov or Sobolev norms. Under a Gaussian errormodel, this may be viewed as a sparse regression problem, with regularizationinduced via the L\'{e}vy random field prior distribution. Posterior inferencefor the unknown functions is based on a reversible jump Markov chain MonteCarlo algorithm. We compare the L\'{e}vy Adaptive Regression Kernel (LARK)method to wavelet-based methods using some of the standard test functions, andillustrate its flexibility and adaptability in nonstationary applications.

Stochastic expansions using continuous dictionaries: L\'{e}vy adaptive regression kernels