Bayesian Kernel Density Estimation

Ordinary KDE selects a single bandwidth (typically by cross-validation or Silverman's rule) and uses it throughout. Bayesian KDE treats the bandwidth as a random variable, places a prior on it, and averages the density surface over the posterior distribution of the bandwidth, producing uncertainty estimates rather than a single surface.

A log-normal or half-normal prior on h is common because bandwidth must be positive and the log-scale reflects multiplicative uncertainty in smoothing. Setting the prior mean near a reference bandwidth (e.g., Silverman's rule-of-thumb value) and using a moderate standard deviation is a reasonable starting point, after which prior sensitivity should be assessed.

Yes. The kernel and bandwidth parameter can be extended to two dimensions (a bandwidth matrix H), and a prior is placed over H. The computational cost increases but the logic is identical. Most implementations simplify by using a scalar bandwidth and isotropic kernel.

Use kriging when your variable is a continuous measurement recorded at sample locations and you need to interpolate its value at unsampled locations. Use Bayesian KDE when you have a point pattern (event locations) and want to estimate the spatial intensity or density of events.

The leave-one-out cross-validation likelihood is evaluated on a grid of h values, combined with the prior, and normalised to approximate the posterior. For higher-dimensional bandwidth matrices, MCMC (e.g., Metropolis-Hastings) is used to draw samples from the posterior.