There exist, however, competing forms of determining the law of a signal. Customarily, signals are studied in terms of transition probabilities, which determine the probability of the next bit given the history. Nothing prevents, however, to characterize the distribution of this bit by studying past *and future*. The former approach is related to the theory of stochastic processes, while the latter corresponds to the theory of Gibbs measures in statistical mechanics.

The project involves a comparative study of these approaches. Depending on the interest of the student, the study can be done at the foundational level, with emphasis on rigorous results, or at a more applied level, with emphasis in empirical (numerical) comparisons. A mixed approach, and even the work of two students as a team, is also possible.