The training set consists labelled and unlabelled patterns
72 Density estimation – parametric
where the first term in the product, p.z D jjD/, is
p.z D jjD/ D E[³ jjD] D  1  N  (2.47)  

X tDMC1 

N � M 
p.xjD; z D j/ ³  1  N  

X tDMC1 

N � M 
the sum of Student t distributions.
C1 D  � 5 3  1  C2 D  � 0 1  1 

C3 D  � 5 3  0 



5  5  1 
x j D Ci  �  w j  �  

1 � w j 
Using a normal model, with diagonal covariance matrix, for the density of each class, an MCMC approach using Gibbs sampling is taken. The training set consists of 1200 labelled and 300 unlabelled patterns. The priors for the mean and variances are normal and inverse gamma, respectively. The parameter values are initialised as samples from the prior. Figure 2.11 shows components of the mean and covariance matrix that are produced from the chain. The package WinBugs (Lunn et al., 2000) has been used to complete the MCMC sampling.
2.4.4 Example application study
Bayesian estimates 73
6
1
−1 1 2 3 4 5 6
3.0 3.1
2.9 3.0
sigma[1,1] sigma[2,2]
1.4 1.5
10850 10900 10950 10850 10900 10950
iteration iteration
complete rotation of the vehicle on a turntable. An ISAR image is an image generated
by processing the signals received by the radar. One axis corresponds to range (distance