Efficient Computing of Stochastic Complexity
Stochastic complexity of a data set is defined as the shortest possible code length for the data obtainable by using some fixed set of models. This measure is of great theoretical and practical importance as a tool for tasks such as model selection or data clustering. Unfortunately, computing the modern version of stochastic complexity, defined as the Normalized Maximum Likelihood (NML) criterion, requires computing a sum with an exponential number of terms. Therefore, in order to be able to apply the stochastic complexity measure in practice, in most cases it has to be approximated. In this paper, we show that for some interesting and important cases with multinomial data sets, the exponentiality can be removed without loss of accuracy. We also introduce a new computationally efficient approximation scheme based on analytic combinatorics and assess its accuracy, together with earlier approximations, by comparing them to the exact form. The results suggest that due to its accuracy and efficiency, the new sharper approximation will be useful for a wide class of problems with discrete data.