Estimating Probability Distributions using "Dirac" Kernels (via Rademacher-Walsh Polynomial Basis Functions)
This is an incremental improvement for applications like pattern recognition and bioinformatics where uncertainty assessment is crucial.
The paper tackles the problem of non-parametrically estimating probability distributions in binary spaces by showing that using Rademacher-Walsh polynomial basis functions is equivalent to using Dirac kernel functions, which can reduce computational bottlenecks and notational complexity, especially for large binary input spaces.
In many applications (in particular information systems, such as pattern recognition, machine learning, cheminformatics, bioinformatics to name but a few) the assessment of uncertainty is essential - i.e., the estimation of the underlying probability distribution function. More often than not, the form of this function is unknown and it becomes necessary to non-parametrically construct/estimate it from a given sample. One of the methods of choice to non-parametrically estimate the unknown probability distribution function for a given random variable (defined on binary space) has been the expansion of the estimation function in Rademacher-Walsh Polynomial basis functions. In this paper we demonstrate that the expansion of the probability distribution function estimation in Rademacher-Walsh Polynomial basis functions is equivalent to the expansion of the function estimation in a set of "Dirac kernel" functions. The latter approach can ameliorate the computational bottleneck and notational awkwardness often associated with the Rademacher-Walsh Polynomial basis functions approach, in particular when the binary input space is large.