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General Points on Parameter and Hyperparameter Estimation

Most of the topics addressed in this thesis can be reduced to parameter and hyperparameter estimation problems. The purpose of this chapter is to give the theory and computational tools for solving such problems. As its name may indicate, a parameter estimation problem is the problem of finding the parameters of a model so that the considered data set is correctly modelled. We first give the theoretical building blocks underlying parameter estimation problems. We then review some of the most common statistical methods (such as Maximum Likelihood Estimation and Maximization A Posteriori) used to solve such problems. In particular, we show how these methods may be formally derived from statistical considerations. Related to the problem of estimating parameters is the problem of estimating hyperparameters. Hyperparameters are additional parameters such as the weighting of different terms in a compound cost function that also influence the result but that, for reasons that will become clear, cannot be estimated the same way as classical parameters. In this chapter, we give the fundamental definitions and concepts related to the hyperparameters. We finally review the standard and general techniques in hyperparameter estimation.