A Bayesian Kernel Density Estimator

A Kernel Density Estimator with kernel $K(\cdot)$ is defined by

$$\hat f (x) = \frac{1}{n}\sum_{i=1}^n\frac{1}{h}K\left(\frac{x-x_i}{h}\right)$$ (1)

where $h$ is the window width, smoothing parameter or bandwidth. An approach is proposed in which $h$ is a parameter of the problem, so avoiding both the specification of the bandwidth and the assumption that all projections from a density have the same smoothness.