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.

Wed Nov 27 14:16:58 JST 1996