number of latent classes), the model adopts a prior distribution for the attribute population via the stick-breaking representation that is originated from the derivation of the Dirichlet processes. In the analysis, we adopt a full Bayesian setting. The posterior distribution can be obtained by means of a Gibbs sampling scheme without truncation to a finite mixture model. An estimator of the partial information structure is proposed based on a clustering algorithm combined with a Bayesian estimator. We illustrate the identifiability results by fitting this model to simulated data generated from several diagnostic classification models and a real data set. For the latter, we reconstructed a parametric loading structures and its Q -matrix. 

The rest is organized as follows. In Section 2, we present basic concepts including latent class models, diagnostic classification models, identifiability, and estimation consistency. In Section 3, the main results are presented including the identifiability of item parameters, attribute distribution, and the partial information structure. The latent class model with Dirichlet allocation along with its inference is presented in Section 4. Simulation studies and real data analysis are presented in Sections 5 and 6, respectively. 

## 2 Preliminaries 

## 2.1 Latent class models and diagnostic classification models 

We start with descriptions of the latent class models and the diagnostic classification models and establish their connections. Consider a J -dimensional multivariate categorical response random vector Y = ( Y $^{1}$, ..., Y $^{J}$). We use subscript to index independent replications, that is, Y$_{i}$ and Y j i . The random vector ( Y ) or variable ( Y $^{j}$) without a subscript denotes a generic random vector or variable. Let Y j be a discrete random variable taking k$_{j}$ possible values. Without loss of generality, let Y j ∈ { 1 , ..., k$_{j}$ } . In the formulation of latent class models, the dependence among Y $^{1}$,..., Y J is induced by a discrete latent variable (latent class) α taking values in a discrete set A . We similarly use α i to denote its independent replications. Conditional on α , the distribution of Y falls into some simple form. In this paper, we assume that Y $^{1}$, ..., Y J are conditionally independent given 

