  A major challenge in cluster analysis is that the number of data clusters is
mostly unknown and it must be estimated prior to clustering the observed data.
In real-world applications, the observed data is often subject to heavy tailed
noise and outliers which obscure the true underlying structure of the data.
Consequently, estimating the number of clusters becomes challenging. To this
end, we derive a robust cluster enumeration criterion by formulating the
problem of estimating the number of clusters as maximization of the posterior
probability of multivariate $t_\nu$ distributed candidate models. We utilize
Bayes' theorem and asymptotic approximations to come up with a robust criterion
that possesses a closed-form expression. Further, we refine the derivation and
provide a robust cluster enumeration criterion for data sets with finite sample
size. The robust criteria require an estimate of cluster parameters for each
candidate model as an input. Hence, we propose a two-step cluster enumeration
algorithm that uses the expectation maximization algorithm to partition the
data and estimate cluster parameters prior to the calculation of one of the
robust criteria. The performance of the proposed algorithm is tested and
compared to existing cluster enumeration methods using numerical and real data
experiments.
