Page 234 - Special Topic Session (STS) - Volume 1

STS426 Tanuka C.
and is a prior distribution on the parameters , , ∑ , which

includes other parameters denoted by . Under this setup, we try to
find a posterior mode or MAP (maximum a posteriori) estimate rather
than a ML-estimate for the mixture parameters. For further
discussions on regularization, we refer to ?. This method has been
implemented in the software mclust (https://cran.r-project.org/
web/packages/mclust/index.html) built under the R environment by
the same authors of the paper. We have used mclust for our work
and did not face issue with convergence of the EM algorithm (instead,
we found excellent computation speed with fast convergence) and
thus have not used any prior .

3.2.2 Model selection under GMMBC setup
As already discussed, the problems in applied cluster analysis:
selection of clustering method and that of number of clusters can be
postulated into one single problem of Statistical Model Selection
under the MBC setup (?). The approach taken to the problem is based
on Bayesian Model Selection via the use of Bayes’ Factors (?) and
posterior model probabilities. The idea goes like this: Let several
models {ℳ , ℳ , . . . , ℳ } are considered with prior probabilities of
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2

getting selected as (ℳ ), k = 1, . . . , K often taken to be equal. Now,

by applying Bayes’ Theorem, the posterior probability of ℳ getting

selected given the data (D) is:

(|ℳ ) ∝ (|ℳ )(ℳ )

where,

(|ℳ ) = ∫ (| , ℳ )( |ℳ )

where, ( |ℳ ) is the prior distribution of : the parameter vector

for the model ℳ . (|ℳ ) is known as Integrated Likelihood of the

model ℳ . This likelihood will help us in deciding the best model. We

will choose that model which is maximum likely a posteriori.
Assuming the priors (ℳ )s are equal, then we select the model with

highest integrated likelihood, i.e., if we are comparing ℳ and ℳ ,

then we calculate:

ℬ = (|ℳ )

(|ℳ )

With the comparison favouring ℳ , if:

ℬ > 1 ⇔ (|ℳ ) > (|ℳ )

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