  We consider the situation where the observed sample contains some
observations whose class of origin is known (that is, they are classified with
respect to the g underlying classes of interest), and where the remaining
observations in the sample are unclassified (that is, their class labels are
unknown). For class-conditional distributions taken to be known up to a vector
of unknown parameters, the aim is to estimate the Bayes' rule of allocation for
the allocation of subsequent unclassified observations. Estimation on the basis
of both the classified and unclassified data can be undertaken in a
straightforward manner by fitting a g-component mixture model by maximum
likelihood (ML) via the EM algorithm in the situation where the observed data
can be assumed to be an observed random sample from the adopted mixture
distribution. This assumption applies if the missing-data mechanism is
ignorable in the terminology pioneered by Rubin (1976). An initial likelihood
approach was to use the so-called classification ML approach whereby the
missing labels are taken to be parameters to be estimated along with the
parameters of the class-conditional distributions. However, as it can lead to
inconsistent estimates, the focus of attention switched to the mixture ML
approach after the appearance of the EM algorithm (Dempster et al., 1977).
Particular attention is given here to the asymptotic relative efficiency (ARE)
of the Bayes' rule estimated from a partially classified sample. Lastly, we
consider briefly some recent results in situations where the missing label
pattern is non-ignorable for the purposes of ML estimation for the mixture
model.
