We present a novel method of measurement covariance estimation that models measurement uncertainty as a function of the measurement itself. Existing work in predictive sensor modeling outperforms conventional fixed models, but requires domain knowledge of the sensors that heavily influences the accuracy and the computational cost of the models. In this work, we introduce Deep Inference for Covariance Estimation (DICE), which utilizes a deep neural network to predict the covariance of a sensor measurement from raw sensor data. We show that given pairs of raw sensor measurement and ground-truth measurement error, we can learn a representation of the measurement model via supervised regression on the prediction performance of the model, eliminating the need for hand-coded features and parametric forms. Our approach is sensor-agnostic, and we demonstrate improved covariance prediction on both simulated and real data.