  We develop a fully Bayesian nonparametric regression model based on a L\'evy
process prior named MLABS (Multivariate L\'evy Adaptive B-Spline regression)
model, a multivariate version of the LARK (L\'evy Adaptive Regression Kernels)
models, for estimating unknown functions with either varying degrees of
smoothness or high interaction orders. L\'evy process priors have advantages of
encouraging sparsity in the expansions and providing automatic selection over
the number of basis functions. The unknown regression function is expressed as
a weighted sum of tensor product of B-spline basis functions as the elements of
an overcomplete system, which can deal with multi-dimensional data. The
B-spline basis can express systematically functions with varying degrees of
smoothness. By changing a set of degrees of the tensor product basis function,
MLABS can adapt the smoothness of target functions due to the nice properties
of B-spline bases. The local support of the B-spline basis enables the MLABS to
make more delicate predictions than other existing methods in the
two-dimensional surface data. Experiments on various simulated and real-world
datasets illustrate that the MLABS model has comparable performance on
regression and classification problems. We also show that the MLABS model has
more stable and accurate predictive abilities than state-of-the-art
nonparametric regression models in relatively low-dimensional data.
