Page 232 - Special Topic Session (STS) - Volume 1

STS426 Tanuka C.
its mean vector and covariance matrix ∑ having the following explicit

form:

(| ) = (| , ∑ ) ≡ exp {−1/2(− ) ∑ −1 (− )} (5)

−1/2
det (2∑ )

This set-up of finite-mixture models are called Gaussian-Mixture-
Model based Clustering (GMMBC). Estimation of the parameters :=
{ , , ∑ } under such setup is done via the Expectation-Maximization

Algorithm (EM) (? & ?).

3.2.1 The EM-Algorithm for the parameter estimates under GMMBC
setup
Under the above discussed setup, the EM algorithm has the
following steps:
(a) Obtain some initial values (randomly) of the parameters and the
mixing proportions for the Gaussian mixtures:

∗
∗
∗
{ , , : = 1,2, … , }

(b) E-Step: Given the initial values, the E-step calculates the
ℎ
conditional probability that the observation comes from the
group as,
ℎ

∗

∗ = ∑ ( | ,∑ ) (6)

∗
( | ,∑ )

=1
this follows from direct application of the Bayes’ Rule.
(c) M-Step: Now, use the estimate of the mixing proportions
obtained from E-Step, calculate new parameter values. Let, =

ℎ
∑ ∗ i.e., the sum of the mixing proportions for the

=1
component :− this is the effective number of data points assigned
to the component k. M-Step gives the following estimates:

=1
= ∑ ∑ ∑ ∗ ∗ = (7)

=1
=1
∗

= ∑ =1 (8)

and

∗

Σ = ∑ =1 ( − )( − ) (9)

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