How well do discrete Bayesian methods represent a true model for each class?

In this paper, mathematical formulas are developed for analytically determining how well discrete Bayesian classification methods, such as the combined Bayes test (CBT), estimate probability mass functions for each class. In general, an analytical solution for this problem is computationally not trivial and solutions are typically given with empirically generated results. In this case, the actual analytical probability of error expression is given for a two-class problem that is a function of the number of data samples, the true cell probabilities for each class, and the number of discretized cells. Results are shown by plotting the difference between the actual and optimal error probabilities, versus the optimal probability of error, where it is demonstrated that, as expected, discrete classifier performance depends on the number of data samples for each class. However, interesting additional results are given where it is shown that actual classifier performance depends on both the true value of the optimal error probability, and the overall quantization of the data.