  In this paper, we consider a surrogate modeling approach using a data-driven
nonparametric likelihood function constructed on a manifold on which the data
lie (or to which they are close). The proposed method represents the likelihood
function using a spectral expansion formulation known as the kernel embedding
of the conditional distribution. To respect the geometry of the data, we employ
this spectral expansion using a set of data-driven basis functions obtained
from the diffusion maps algorithm. The theoretical error estimate suggests that
the error bound of the approximate data-driven likelihood function is
independent of the variance of the basis functions, which allows us to
determine the amount of training data for accurate likelihood function
estimations. Supporting numerical results to demonstrate the robustness of the
data-driven likelihood functions for parameter estimation are given on
instructive examples involving stochastic and deterministic differential
equations. When the dimension of the data manifold is strictly less than the
dimension of the ambient space, we found that the proposed approach (which does
not require the knowledge of the data manifold) is superior compared to
likelihood functions constructed using standard parametric basis functions
defined on the ambient coordinates. In an example where the data manifold is
not smooth and unknown, the proposed method is more robust compared to an
existing polynomial chaos surrogate model which assumes a parametric
likelihood, the non-intrusive spectral projection.
