In statistics, especially in Bayesian statistics, the kernel of a probability density function (pdf) or probability mass function (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the variables in the domain are omitted. Note that such factors may well be functions of the parameters of the pdf or pmf. These factors form part of the normalization factor of the probability distribution, and are unnecessary in many situations. For example, in pseudo-random number sampling, most sampling algorithms ignore the normalization factor. In addition, in Bayesian analysis of conjugate prior distributions, the normalization factors are generally ignored during the calculations, and only the kernel considered. At the end, the form of the kernel is examined, and if it matches a known distribution, the normalization factor can be reinstated. Otherwise, it may be unnecessary (for example, if the distribution only needs to be sampled from).
For many distributions, the kernel can be written in closed form, but not the normalization constant.
The first requirement ensures that the method of kernel density estimation results in a probability density function. The second requirement ensures that the average of the corresponding distribution is equal to that of the sample used.
If K is a kernel, then so is the function K* defined by K*(u) = λK(λu), where λ > 0. This can be used to select a scale that is appropriate for the data.
Kernel functions in common use
All of the kernels below in a common coordinate system.
Several types of kernel functions are commonly used: uniform, triangle, Epanechnikov, quartic (biweight), tricube, triweight, Gaussian, quadratic and cosine.
In the table below, if is given with a bounded support, then for values of u lying outside the support.
^Named for Epanechnikov, V. A. (1969). "Non-Parametric Estimation of a Multivariate Probability Density". Theory Probab. Appl. 14 (1): 153–158. doi:10.1137/1114019.
^Altman, N. S. (1992). "An introduction to kernel and nearest neighbor nonparametric regression". The American Statistician. 46 (3): 175–185. doi:10.1080/00031305.1992.10475879. hdl:1813/31637.
^Cleveland, W. S.; Devlin, S. J. (1988). "Locally weighted regression: An approach to regression analysis by local fitting". Journal of the American Statistical Association. 83 (403): 596–610. doi:10.1080/01621459.1988.10478639.