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Estimation
Discriminant analysis using a MLMM with a normal mixture 147
The MLMM (1) can be regarded as a standard LMM (2) with X and Z matrices
supplemented by zeros. However, maximum-likelihood based estimation routines
that ignore the specific sparse structure of X and Z matrices are inefficient and encounter
numerical problems even with normally distributed random effects (K =1
in expression (3)). For example, Fieuws et al. 9 used a pairwise fitting approach of
Fieuws and Verbeke17 to avoid numerical problems. Another route would be to use
methods for sparse matrices, see, e.g., R package Matrix (Bates and Maechler18 ).
Here we adopt the Bayesian approach with Markov chain Monte Carlo (MCMC)
estimation. The MCMC approach proved to be a useful machinery for problems
involving hierarchically specified models (like linear mixed models) and models involving
mixture distributions. Our prior distributions for the model parameters are
weakly informative such that the posterior summary statistics correspond closely to
maximum-likelihood estimates.
Prior distributions
To specify the model from a Bayesian point of view, prior distributions have to be
assigned to model parameters. The vector θ of model parameters is supplemented
by latent quantities (values of random effects b =(b ′ 1,...,b ′ N) ′ ), variance hyperpa-
rameters γ b =(γb,1,...,γb,q) ′ , γ ε =(γε,1,...,γε,R) ′ (see below), and further, by
other parameters pertaining to the hierarchical structure of the model which simplify
the subsequent computations (component allocations u =(u1,...,uN) ′ , see below)
in the spirit of the Bayesian data augmentation approach of Tanner and Wong19 .
This leads to the vector of parameters
ψ =(θ ′ , b ′ , γ ′ b, γ ′ ε, u ′ ) ′ , (6)
for which the joint prior distribution will be specified hierarchically.
It is well known (see, e.g., Diebolt and Robert20 ) that, due to the problem of an
unbounded likelihood mentioned above, mixture models do not allow for improper
priors. Nevertheless, several proper, however weakly informative prior distributions
for mixture problems have been suggested in the literature leading to the proper pos-
terior distribution (see Diebolt and Robert20 , Roeder and Wasserman21 , Richardson
and Green22 ). In this paper, we exploit the approach of Richardson and Green22 adapted to our needs and to hierarchical linear models (see, e.g., Gelman et al. 23 ,
Chapter 15). Let p be a generic symbol for a distribution. The joint prior distribution