Relation between the Perceptron and Bayes Classifier for a Gaussian Environment

Neural Network | Bayes Classifier?

Abstract

The Bayes Classifier is a probabilistic model based on Bayes’ Theorem used for classification, as is the Perceptron. Under the Gaussian Environment, where the data for each class is drawn from a multivariate normal distribution, the Perceptron and Bayes Classifiers share the classifying feature. This article dives into Mathematics to show the relationship between them.?

Introduction

The Bayes Classifier (or Bayesian classifier) is a probabilistic model based on Bayes’ theorem. It classifies a new observation by calculating the posterior probabilities of each class and assigning the observation to the class with the highest posterior probability.

The Perceptron is a linear classifier that aims to separate two classes by finding a linear decision boundary (hyperplane) that best divides the data. It does this by adjusting the weights based on misclassified examples using a simple iterative algorithm.

Perceptron

The weight and bias are adjusted by an iterative weight-updating algorithm. The mathematical form is:

y=sign(w^T * x +?b)

where:

• w is the weight vector,
• x is the input feature vector,
• b is the bias term.

Bayes’ theorem:?

In the case of binary classification, the posterior probability for class c given observation x is:

P(c∣x)=P(x∣c) * P(c) / P(x)

where:

• P(x∣c) is the likelihood, the probability of observing x given the class c,
• P(c) is the prior probability of class c,
• P(x) is the marginal likelihood of observing x.

Normal Distribution(Gaussian Distribution)

Named after mathematician Carl Friedrich Gauss, Gaussian Distribution is a bell-shaped curve that shows the probability distribution and how values of a variable are spread from the center(mean). Having mean, mode, and median at the center mean (μ) = 0 and standard deviation(σ) at the tail. It’s a symmetric bell curve centered at μ = 0, with σ shaping how wide or narrow the tails are!?

Multivariate Gaussian distribution

For multivariate Gaussian distribution, the equation extends to multiple dimensions. While the one-dimensional version deals with a single random variable, the multivariate version handles multiple random variables that may be correlated. So, it is a generalization of the one-dimensional Gaussian distribution to higher dimensions. If we are dealing with only one mountain it is Univariate Normal else it is multivariate Normal Distribution.

Handwritten Notes

I find it impossible to write down all the derivations without directly inserting the handwritten notes.

Conclusion

Dealing with the probabilities, misclassification cost, risk factor, inequality in the variable equation, constant terms in the risk equation with decision rule, and rejection overlapping areas, we can see the clear relation between the perceptron and Bayes classifier. The Perceptron can mimic the Bayes Classifier’s decision boundary in a Gaussian environment under specific conditions, but it lacks the probabilistic foundation and optimality guarantees of the Bayes approach. This relation can be used for real-world applications like spam detection, medical diagnostics, or financial risk assessment, where different misclassifications carry different costs

References:

1. Kafley, Sabin. Lecture Notes on Neural Networks. Feb 24, 2025.
2. Haykin, S. (1999). Neural networks: A comprehensive foundation (2nd ed.). Prentice Hall.