  We are concerned with the issue of how to calculate the normalized maximum
likelihood (NML) code-length. There is a problem that the normalization term of
the NML code-length may diverge when it is continuous and unbounded and a
straightforward computation of it is highly expensive when the data domain is
finite . In previous works it has been investigated how to calculate the NML
code-length for specific types of distributions. We first propose a general
method for computing the NML code-length for the exponential family. Then we
specifically focus on Gaussian mixture model (GMM), and propose a new efficient
method for computing the NML to them. We develop it by generalizing Rissanen's
re-normalizing technique. Then we apply this method to the clustering issue, in
which a clustering structure is modeled using a GMM, and the main task is to
estimate the optimal number of clusters on the basis of the NML code-length. We
demonstrate using artificial data sets the superiority of the NML-based
clustering over other criteria such as AIC, BIC in terms of the data size
required for high accuracy rate to be achieved.
