On the Identifiability of a Two-Factor Model with Correlated Residuals
Sergei Stafeev · 2002
where X = (X1 1 , ..., X1 n1 , X 2 1 , ..., X 2 n2) is a vector of observable variables with a vector of means mX and covariance matrix ΣX ; θX = (θ1 1, ..., θ1 n1 , θ 2 1, ..., θ 2 n2) is a vector of parameters; H = (H1,H2) is a vector of latent variables normally distributed with E(H1) = E(H2) = 0 and V ar(H1) = V ar(H2) = 1; Y = (Y 1 1 , ..., Y 1 n1 , Y 2 1 , ..., Y 2 n2) is a vector of residuals normally distributed with positive definite covariance matrix; Y and H are independent. We will assume that probabilistic relationships among components of the vector Y are represented by a Bayesian network [1] with a structure SY , that is a directed acyclic graph with nodes identified with components of the vector Y , and a vector of parameters θY . The model (1) is a generalization of the single-factor model with correlated residuals suggested in [2]. The model (1) is said to be identifiable if the vector of parameters θ = (θX , θY ) can be uniquely determined bymX and ΣX . Transform the structure SY by the following way. Connect by edges all disconnected by arcs nodes and then delete all arcs. Obtained graph SY is called complementary graph of the structure SY . Let C1 be a cycle that consists of nodes Y1, ..., Yk and C2 be a graph that consists of a cycle with nodes Y1, ..., Yi1 , a simple chain with nodes Yi1 , ..., Yi2 and a cycle with nodes Yi2 , ..., Yn. Define two functions: