Forests of Latent Tree Models to Decipher Genotype-Phenotype Associations - Biomedical Engineering Systems and Technologies - page 98

In the first category, the algorithms explore the search space through a local search
strategy and optimize a score, such as the BIC score [15]. Zhang proposed a greedy
algorithm, to navigate in the structure search space [16]. This algorithm is coupled
with a hill climbing procedure, to adjust the cardinality of the latent variables. A more
efficient variant of this algorithm has recently been proposed [17]. In this first cate-
gory, structural expectation maximization (SEM) has also been adapted to the case of
Bayesian networks with latent variables. SEM successively optimizes θ , conditionally
on the structure

conditionally on θ . Parameter learning being a
time-consuming step, in this framework, Zhang and Kocka have adapted a procedure,
called local EM, to optimize the variables whose connection or cardinality has been
modified in the transition from former to current model [18]. In one of the two LTM-
based approaches dedicated to LD modeling, Zhang and Ji use a set of LCMs and apply
a SEM strategy [1]. The number of LCMs has to be specified. To avoid getting trapped
in local optima while running the EM algorithm to learn a set of latent models, these
authors have adapted a simulated annealing approach.
The above score-based approaches require the computation of the maximum likeli-
hood in presence of latent variables, a prohibitive task regarding computational burden.
Thus, various methods based on the clustering of variables have been implemented.
They all construct the model following an ascending strategy; they all rely on the mu-
tual information (MI) criterion, to identify clusters of dependent variables. In their turn,
these methods may be sub-categorized into binary- and non binary-based approaches.
Hwang and co-workers' learning algorithm is dedicated to binary trees and binary
latent variables [19]. It has to be noted that the trees are possibly augmented with con-
nections between siblings, that is nodes sharing the same parent into the immediate
upper layer. Also confining themselves to binary trees, Harmeling and Williams have
proposed two learning algorithms [20]. One of them approximates MI between a latent
variable H and any other variable X , based on a linkage criterion (single, complete or
average) applied for X and the variables in the cluster subsumed by H .Avariantofthis
first algorithm locally infers the data corresponding to any latent variable; therefore it is
possible to achieve an exact computation of the MI criterion between a latent variable
and any other variable.
Two approaches have been proposed to circumvent binary tree-based structures.
Wang and co-workers first build a binary tree; then they apply regularization and simpli-
fication transformations which may result in subsuming more than two nodes through
a latent variable [21]. In their approach devoted to LD modeling, Mourad and colla-
borators implement the clustering of variables through a partitioning algorithm [3]; the
latter yields cliques of pairwise dependent variables. Besides, this method imposes the
control of information fading as the level increases in the hierarchy, which generally
results in the production of a forest of LTMs (instead of a single LTM).
Two of the above cited methods have been shown to be tractable. For these two non
binary-based approaches, some reports are available: Hwang and co-workers' approach
was able to handle 6000 variables and around 60 observations. The scalability of the
FLTM construction by Mourad et al. has been shown for benchmarks describing 10 5
variables and 2000 individuals.

S , then optimizes S