Bayesian Classifier for a Gaussian Distribution, Decision Surface Equation, with Application

Abstract

Bayesian decision theory is a fundamental statistical approach to the problem of classification as for pattern recognition. It makes the assumption that the decision problem is posed in probabilistic term, and all of the relevant probability values are known. To minimize the error probability in classification problem, one must choose the state of nature that maximizes the posterior probability. Bayes formula allows us to calculate such probabilities given the prior probabilities, and the conditional densities for different classes or categories. Statistical classification is one of the most useful analysis tool which can be used for analyzing several kinds of data in various sciences. Its interesting in separating more than one class or category from each other, when their behaviors are near each other. The Bayesian surface decision equation is one of the classification rule that produces a linear separable equation for these interacted classes, specially where they contain vectors of random variables that are identically independently distributed (iid),(or all vectors have the same distribution with same parameters values). This was done as a special case by Muller, P. & Insua, D.R, (1995), This study is a trial to generalize (Bayesian Surface Decision Equation) to produce a such (Linear Separable Equation) as a linear classifier for those classes contain random vectors distributed identically Gaussian) but with different parameters values ( mi , Σ i ), moreover, in this study the researcher tried to search for equivalence between Bayesian Surface Decision Equation, and a linear perceptron for classification for this general case, with a numeric application, (encoded data vectors) that is illustrated latter.