α , that is, This is known as the local independence assumption. The π α is the probability mass function of the latent class membership α . Suppose conditional distribution is parameterized through where θ$_{j}$ is the item parameter specific to item j . Thus, the joint marginal distribution of Y can be expressed as Diagnostic classification models in general admit the same distribution as in (1). However, it often imposes additional parametric or low-dimensional structures so that the item response function f$_{j}$ and the latent variable α have practical interpretations. Here, we adopt the following parameterization. The latent variable α is parameterized by a multidimensional binary or multi- category vector, i.e. α ∈ ( α $^{1}$, ..., α $^{K}$) where α k takes d$_{k}$ possible values. Each α k is known as an attribute or trait indicating the presence/absence or level of a latent characteristic. In addition to the multidimensional structure of the latent class space, the item response function f$_{j}$ also admits some parametric and usually low-dimensional structures that distinguish themselves from the general latent class models. To elaborate on this feature, we need to introduce the Q - matrix that is a J × K matrix taking binary values and indicating the item-attribute association. Each row of Q corresponds to an item and each column corresponds to an attribute. We write where q$_{j}$ = ( q$_{j}$$_{1}$, ..., q$_{jK}$ ) is a K -dimensional row vector. As mentioned previously, q$_{jk}$ = 1 or 0, P ( Y j = y $^{j}$, j = 1 , ..., J | α ) = J ∏ j =1 P ( Y j = y $^{j}$| α ) . P ( Y j = y $^{j}$| α ) = f$_{j}$ ( y $^{j}$| θ, α ) , P ( Y 1 = y $^{1}$, ..., Y $^{J}$, = y $^{J}$) = ∑ α ∈ A { J ∏ j =1 P ( Y j = y $^{j}$| α ) π α } . (1) Q =    q$_{1}$ . q$_{J}$   $^{,}$ 5