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Stochastic Model and Parameter Estimation

The stochastic model for Bayesian self-organizing maps (BSOM) [1][2] is given by a Gaussian mixture model with a Gaussian smoothing prior probability for the centroid parameters. For simplicity, we consider a mixture model whose components are spherical Gaussian densities with a common variance and have an identical prior selection probability. For a data set consisting of data points , the likelihood function for the centroid parameters , is given by

where is a vector made by concatenating all centroid parameters. The prior probability for the centroid parameters is given by

where are commutated vectors for the centroid vectors. The matrix is a discretized Laplacian on a specified topology. We referred to the rank of as l and the positive eigenvalues of as . From the likelihood and the prior, a posterior probability for the centroid parameters can be calculated using Bayes' theorem.

For this Bayesian model, there are several algorithms for the maximum a posteriori (MAP) estimation of the centroid parameters. For example, the elastic net algorithm [3] is a MAP estimation algorithm using the gradient ascent method. A MAP estimation algorithm by the EM algorithm was also presented [2][4].

First, we calculate posterior selection probabilities for the components:

These are also called fuzzy memberships. Next, we obtain the means of data weighted by the fuzzy memberships. These are given by

where . Then, the centroid parameters are updated by

where is a diagonal matrix whose entries are , and . This updating is iterated until convergence.

Utsugi Akio;(6730)
Wed Nov 27 14:16:58 JST 1996