  Logistic Gaussian process (LGP) priors provide a flexible alternative for
modelling unknown densities. The smoothness properties of the density estimates
can be controlled through the prior covariance structure of the LGP, but the
challenge is the analytically intractable inference. In this paper, we present
approximate Bayesian inference for LGP density estimation in a grid using
Laplace's method to integrate over the non-Gaussian posterior distribution of
latent function values and to determine the covariance function parameters with
type-II maximum a posteriori (MAP) estimation. We demonstrate that Laplace's
method with MAP is sufficiently fast for practical interactive visualisation of
1D and 2D densities. Our experiments with simulated and real 1D data sets show
that the estimation accuracy is close to a Markov chain Monte Carlo
approximation and state-of-the-art hierarchical infinite Gaussian mixture
models. We also construct a reduced-rank approximation to speed up the
computations for dense 2D grids, and demonstrate density regression with the
proposed Laplace approach.
