Bandwidth

At this point in any discussion of KDE it should be obvious that the one major problem is that of choosing the bandwidth $h$. In the literature there are many good sources for estimation of bandwidth, see for example articles by Jones et al. (1992); Park et al. (1994); Jones et al. (1994) and books by Scott (1992,1983); Silverman (1986) and Wand and Jones (1995).

However, for the current project, a series of projections from some density is required (along with a KDE from each of them) quickly enough for a human operator to perceive continuity. This implies that an automatic (no human intervention) system, with no delay caused by deliberation concerning the smoothness of the density, is needed. Speed of estimation here will improve the flow of images, thus helping the operator assess the density.

If the kernel density estimator is written in terms of some function where is a location parameter and , which may, more generally, be a vector, is a scale parameter, an estimate of $f_X(\cdot)$ based on the sample , at the point, can then be written as

where $n$ is the number of data items. Note that ${\mbox{\boldmath$\theta$}}$ may be considered to be a generalisation of bandwidth. It now becomes obvious that the bandwidth may be considered as merely another parameter in the specification of the problem. Belief in the posterior for ${\mbox{\boldmath$\theta$}}$ given some data can be written in the form