Title: Neuron ranking - an informed way to compress convolutional neural networks. Abstract: Convolutional neural networks (CNNs) in recent years have made a dramatic impact in science, technology and industry, yet the theoretical mechanism of CNN architecture design remains surprisingly vague. The CNN neurons, including its distinctive element, convolutional filters, are known to be learnable features, yet their individual role in producing the output is rather unclear. The thesis of this work is that not all neurons are equally important and some of them contain more useful information to perform a given task. Hence, we propose to quantify and rank neuron importance, and directly incorporate neuron importance in the objective function under two formulations: (1) a game theoretical approach based on Shapley value which computes the marginal contribution of each filter; and (2) a probabilistic approach based on what-we-call, the importance switch using variational inference. Using these two methods we confirm the general theory that some of the neurons are inherently more important than the others. Various experiments illustrate that learned ranks can be readily useable for structured network compression and interpretability of learned features. 