Inference and Learning from Data Inference

Maximum likelihood (ML) is a powerful statistical tool that determines model parameters θ in order to fit probability density functions (pdfs) onto data measurements. The estimated pdfs can then be used for at least two purposes. First, they can help construct optimal estimators or classifiers (such as the conditional mean estimator, the maximum a-posteriori (MAP) estimator, or the Bayes classifier) since, as we already know from previous chapters, these optimal constructions require knowledge of the conditional or joint probability distributions of the variables involved in the inference problem. Second, once a pdf is learned, we can sample from it to generate additional observations. For example, consider a database consisting of images of cats and assume we are able to characterize (or learn) the pdf distribution of the pixel values in these images. Then, we could use the learned pdf to generate “fake” cat-like images (i.e., ones that look like real cats). We will learn later in this text that this construction is possible and some machine-learning architectures are based on this principle: They use data to learn what we call a “generative model,” and then use the model to generate “similar” data. We provide a brief explanation to this effect in the next section, where we explain the significance of posterior distributions.