Figure 24.1 illustrates the modelling potential of mixture distributions: twocomponent normal mixture densities accurately fit histognnns of different shapes. The data corresponding to these histograms have been analysed in Stuchlik et al. (1994) and the modelling of these histograms is part of a Bayesian character recognition system. In this approach, handwritten words are autovnatically decomposed into charncters which are then represented by a vector of 164 characteristics describing concavities and other geometric features of the pixel maps corresponding to the characters. A training sample of identified characters is analysed by creating pseudoindependence through a principal components analysis, and then applying mixture models independently to each principle component. Some of the principle components provide fairly regular histograms which can be

q;[j;(x) "°'z ' L.,£=a qlfu(x)

24.1.3 Estimation methods

there is usually a positive probability that one component f(xl0;) generates none of the observations { x j}, i.e. that the sample brings no information on this particular component. This phenomenon does not necessarily prevent the derivation of a maximum likelihood estimator when the likelihood is bounded, but it contributes to its instability. This peculiar identifiability problem of mixture models motivates the use of Bayesian methods to produce acceptable estimators or to devise tests about the number of mixture components.