Note that the GDCM itself is a Markov network. If we reparameterize Φ ∗ = Φ $^{∗}$( α , B , Φ ) , the GDCM can be rewritten as where Φ ∗ has φ$_{jj}$ ′ if j = j ′ and 2 β ⊤ j α if j = j $^{′}$, admitting a Markov network. It is clear from this notion that the diagonal elements of the design matrix account for the effects of the DCM (i.e., the interactions between the latent class and the items), while the off-diagonal entries describe the remaining item dependencies that cannot be explained by the latent class. Define a prior distribution of α such that the joint distribution of ( X , α ) has the form It follows that the complete-data likelihood of the parameters ( B , Φ ) is expressed as wherein the normalizing constant is given by Since the latent class is unobservable, inference about the item parameters is made via a marginal distribution f ( x | B , Φ ) by integrating out α from the joint probability distribution f ( x , α | B , Φ ) with respect to π$_{α}$ . Evaluating f ( x , α | B , Φ ), however, is computationally challenging because the normalizing constant requires computation of 2 J × 2 K terms, making it almost infeasible to evaluate the joint probability function for even realistic sizes of J and K . A viable alternative is to approximate the joint likelihood function through a surrogate version, for example, most notably the pseudo-likelihood (Besag, 1975) or the composite likelihood (Cox & Reid, 2004; Lindsay, 1988). The current study employs the pseudo-likelihood approach that has been standard practice for estimating the Markov network. The ensuing sections discuss the relevant parameter estimation scheme and the computational methods. 8 f ( X = x | α , B , Φ $^{∗}$) ∝ exp$^{(}$ 1 2 x $^{⊤}$Φ$^{∗}$x$^{)}$ , f ( X = x , α | B , Φ ) ∝ π$_{α}$ exp$^{(}$ x $^{⊤}$B α + 1 2 x $^{⊤}$Φx$^{)}$ , z ( B , Φ ) = ∑ x ∈{ 0 , 1 } J ∑ α ∈{ 0 , 1 } K π$_{α}$ exp$^{(}$ x $^{⊤}$B α + 1 2 x $^{⊤}$Φx$^{)}$ . f ( x , α | B , Φ ) = π$_{α}$ z ( B , Φ ) exp$^{(}$ x $^{⊤}$B α + 1 2 x $^{⊤}$Φx$^{)}$ . (4)