Deep neural networks are typically represented by a much larger number of parameters than shallow models, making them prohibitive for small footprint devices. Recent research shows that there is considerable redundancy in the parameter space of deep neural networks. In this paper, we propose a method to compress deep neural networks by using the Fisher Information metric, which we estimate through a stochastic optimization method that keeps track of second-order information in the network. We first remove unimportant parameters and then use non-uniform fixed point quantization to assign more bits to parameters with higher Fisher Information estimates. We evaluate our method on a classification task with a convolutional neural network trained on the MNIST data set. Experimental results show that our method outperforms existing methods for both network pruning and quantization.