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.