prior distribution of π α has the following representation It is easy to verify that π α under the above construction is a well defined probability mass function. This is known as the stick-breaking representation originated from the Dirichlet process. We borrow this representation mostly due to its technical convenience for modeling a discrete distribution. The likelihood function (7) and the prior distributions (8) and (9) completely specify a Bayesian model. We adopt this model for several reasons. First, it does not require to specify the number of latent classes. The stick-breaking representation penalizes the “tail” latent classes. In addition, the posterior distribution of this model can be obtained through a sliced sampler [35] that is a Gibbs sampler via a data augmentation scheme without truncating the model to a finite mixture. Given that this is not the emphasis of this paper, we present the posterior simulation scheme in the appendix. Henceforth, we assume that the posterior distribution of the model parameters has been ob- tained numerically. We consider posterior mean as the point estimator, that is, It is well known that the posterior mean has the same asymptotic distribution as that of the max- imum likelihood estimator; see, for instance, [25]. The item-wise partial information is estimated by clustering the item response probabilities. For each item j , let ˆ p$_{j}$ α = (ˆ p 1 j $_{α}$, ..., ˆ p k$_{j}$ j $_{α}$) be its item response distribution of latent class α . We treat ˆ p$_{j}$$_{1}$ , ˆ p$_{j}$$_{2}$ , ˆ p$_{j}$$_{3}$ , ... as (infinitely many) samples, each of which is a k$_{j}$ dimensional vector, and apply the K -means clustering algorithm. Although there are seemingly infinitely many samples, ˆ p$_{j}$ α ’s are practically identical for α large enough. This is because the data do not provide information for all p$_{j}$ α . For the latent classes to which the data provide little information, their posterior means are essentially the same as the prior mean. In this π α = V α ∏ l< α (1 − V$_{l}$ ) . (9) 17 (ˆ p , ˆ π ) = E { ( p , π ) | y$_{1}$ , ..., y$_{n}$ } .