2 research outputs found
Efficient Bayesian shape-restricted function estimation with constrained Gaussian process priors
This article revisits the problem of Bayesian shape-restricted inference in
the light of a recently developed approximate Gaussian process that admits an
equivalent formulation of the shape constraints in terms of the basis
coefficients. We propose a strategy to efficiently sample from the resulting
constrained posterior by absorbing a smooth relaxation of the constraint in the
likelihood and using circulant embedding techniques to sample from the
unconstrained modified prior. We additionally pay careful attention to mitigate
the computational complexity arising from updating hyperparameters within the
covariance kernel of the Gaussian process. The developed algorithm is shown to
be accurate and highly efficient in simulated and real data examples
Sequential construction and dimension reduction of Gaussian processes under inequality constraints
Accounting for inequality constraints, such as boundedness, monotonicity or
convexity, is challenging when modeling costly-to-evaluate black box functions.
In this regard, finite-dimensional Gaussian process (GP) models bring a
valuable solution, as they guarantee that the inequality constraints are
satisfied everywhere. Nevertheless, these models are currently restricted to
small dimensional situations (up to dimension 5). Addressing this issue, we
introduce the MaxMod algorithm that sequentially inserts one-dimensional knots
or adds active variables, thereby performing at the same time dimension
reduction and efficient knot allocation. We prove the convergence of this
algorithm. In intermediary steps of the proof, we propose the notion of
multi-affine extension and study its properties. We also prove the convergence
of finite-dimensional GPs, when the knots are not dense in the input space,
extending the recent literature. With simulated and real data, we demonstrate
that the MaxMod algorithm remains efficient in higher dimension (at least in
dimension 20), and has a smaller computational complexity than other
constrained GP models from the state-of-the-art, to reach a given approximation
error