On the Stochastic Complexity for Order-1 Markov Chains and the Catalan Constant

The stochastic complexity (SC) selects from a given family of parametric models the one that yields the shortest code length for the available measurements. Theoretical developments have made possible the evolution of the SC from the “two-part code” formula to the most recent expression based on the so-called Normalized Maximum Likelihood (NML) distribution. The application of the NML criterion is recommended especially in the case of small sample size, but high computational burdens prevent its general use. During recent years increasing interest has been growing in obtaining an approximation of the NML-based SC by use of the Fisher information matrix. We show in this note that the most important step in working out such an approximate expression for order-1 Markov chains is the calculation of an integral that leads to the Catalan constant. We evaluate the accuracy of the approximation for small, moderate, and large samples, and we illustrate the use of the formula in model selection.