Forests of Latent Tree Models to Decipher Genotype-Phenotype Associations - Biomedical Engineering Systems and Technologies - page 94

Fig. 1. Latent tree model and forest of latent tree models. The light shade indicates the observed
variables whereas the dark shade points out the latent variables.

Fig. 2. Illustration of key terms specific to our approach. A nodes: ancestor nodes of the causal
SNP; O nodes: other latent nodes categorized in OT nodes (in causal tree) and OO nodes (outside
the causal tree). See Figure 1 for node nomenclature.

For further understanding, we now briefly recall some definitions, including that
of another branch of PGMs, the Markov random fields, which will be mentioned in
Section 3.

Definition 3 (Markov Random Field) . Given an undirected graph G ( X,E ) ,thesetof
random variables X form a Markov random field (MRF) with respect to G if it satisfies
the local Markov property, stating that a variable is conditionally independent of all
other variables given its neighbours.
In this case, the joint distribution can be factorized over the cliques of the graph:

( X )= C∈ cl( G ) φ C ( X C ) ,where cl ( G ) is the set of cliques of G and the functions
φ C are the so-called potentials.
A commonly used class of MRFs, the decomposable MRFs, represents those MRFs
whose graph is triangulated (i.e. no cycle of length strictly greater than 3 is allowed).

P

Definition 4 (Entropy, Mutual Information) . The entropy of variable X writes as:

n

H ( X )= − P ( x i )log P ( x i )

i =1

where ( x i ) is the probability mass function of outcome x i .

Given two variables X i and X j , the mutual information measures the dependence of
the two variables, expressing the difference of entropies between the independent model

P