Abstract:

In this tutorial, we will dive deep into the concepts and implementation details of Probabilistic Neural Networks (PNNs) with Maximum Likelihood Training, as described in the paper "Maximum Likelihood Training of Probabilistic Neural Networks" by Roy L. Streit and Tod E. Luginbuhl. We will cover the mathematical foundations, key excerpts from the paper, and a comprehensive implementation in Python using TensorFlow.

Overview

Probabilistic Neural Networks (PNNs) offer a robust approach for classification tasks by combining the principles of neural networks with probabilistic models. The key idea behind PNNs is to estimate the probability density function (PDF) of each class using Gaussian mixtures and then classify inputs based on the maximum likelihood criterion. This is a streamlined way of realizing the maybe back then hard optimization problem of fitting a GMM.

Mathematical Foundations

1. Gaussian Mixture Models (GMMs)

A Gaussian Mixture Model is a probabilistic model that assumes all data points are generated from a mixture of several Gaussian distributions with unknown parameters. The probability density function for a GMM is given by:

$$ ⁍ $$

where:

$\pi_k$ are the mixing coefficients.
$\mu_k$ are the means of the Gaussian components.
$\Sigma_k$ are the covariance matrices of the Gaussian components.

2. Maximum Likelihood Estimation (MLE)

Maximum Likelihood Estimation aims to find the parameters that maximize the likelihood of the observed data. For a PNN, this involves maximizing the log-likelihood function:

$$ ⁍ $$