Predictive mathematical models normally follow the causal direction, forecasting consequences of known causes. The interpretation of measured data, on the other hand, moves in the opposite direction, looking for unknown causes of observed consequences. In practice, the interpretation of data entails solving an ill-posed inverse problem, The traditional way of approaching inverse problems is to use different regularization techniques. A versatile alternative is to consider the inverse problem as a Bayesian problem of inference, allowing a natural way to augment the data by complementary prior information. In this talk we present an overview the Bayesian approach and classical regularization methods for some classes of inverse problems, and point out how some of these ideas can be efficiently implemented using sophisticated numerical algorithms.