Improve the interpretability of convolutional neural networks with probability density function

Currently, Convolutional Neural Networks (CNNs) have demonstrated extensive success in numerous practical applications. Nevertheless, their limited interpretability remains a significant barrier to further advancement in certain crucial fields. Improving the interpretability of CNNs stands as an exceptionally compelling topic in the present time. This paper explores the interpretability of a basic CNN incorporating a convolution-pooled block and a fully connected layer from a statistical perspective. Assuming that the input variables adhere to a normal distribution and maintain independence from each other, the output variables subsequent to the convolution and pooling layers also conform to a normal distribution. Simultaneously, the probability density function (pdf) characterizing the final output variable belongs to an exponential family distribution. By introducing intermediate variables, the pdf of this output variable can be expressed as a linear combination of three distinct normal distributions. Furthermore, the likelihood of the predicted class label can be rewritten as a cumulative density function (cdf) of the standard normal distribution. The originality of this paper lies in its provision of a more innovative and intuitive perspective for dissecting the operational mechanism of CNNs, analyzing them layer by layer to improve their interpretability. Experimental results obtained from both an artificial dataset and the image datasets CIFAR-10 and ImageNet further validate the rationality of these conclusions.