A Universal Prior for Integers and Estimation by Minimum Description Length

An earlier introduced estimation principle, which calls for minimization of the number of bits required to write down the observed data, has been reformulated to extend the classical maximum likelihood principle. The principle permits estimation of the number of the parameters in statistical models in addition to their values and even of the way the parameters appear in the models; i.e., of the model structures. The principle rests on a new way to interpret and construct a universal prior distribution for the integers, which makes sense even when the parameter is an individual object. Truncated real-valued parameters are converted to integers by dividing them by their precision, and their prior is determined from the universal prior for the integers by optimizing the precision.