Interpretability of Neural Networks with Probability Density Functions

Abstract

AbstractIt is an interesting topic to interpret artificial neural networks (ANNs) by considering some change various approaches. This paper explores the relationship between the input and output units of the simplest ANN, a single layer perceptron for the binary classification problem, from the probability point of view. If the feature variables of datasets follow independent normal distribution and outputs are activated by sigmoid function or smooth Relu function, we advocate that the probability density function (pdf) of the output variable is an exponential family distribution. Furthermore, by introducing an intermediate variable, the pdf of the output variable can be written as a linear combination of three normal distributions with same spread but different centers. Based on these results, the probability of the predicted class label can be written as a standard normal cumulative distribution function (cdf). The originality of this paper comes with interesting theoretical results to provide ANNs with a new description of the relationship between input variables to output variables, which can enable ANNs to be understood from a new perspective. Extensive experiments based on one artificial synthesized dataset and ten real‐world benchmark datasets validate the reasonability of those results.