content_image/326244.jpg content_image/326245.jpg content_image/326246.jpg content_image/326247.jpg
on the above argument, we have that for each j , 

<img src='content_image/326244.jpg'>

as the sample size n → ∞ . Hence, the estimation of equivalence classes is the same as the true one up to a permutation. 

## A Sliced sampler for latent class model 

We now present the sliced sampler for the simulation from the posterior distribution of model (7), (8), and (9). The likelihood function is 

<img src='content_image/326245.jpg'>

For each observation, we augment an independent index u$_{i}$ following the uniform distribution in [0 , 1]. Thus, the complete data likelihood is 

<img src='content_image/326246.jpg'>

With this augmentation scheme, a Gibbs sampler iterates according the following conditional dis- tributions. 

1. Update u$_{i}$ , for i = 1 , ..., n , by sampling from the conditional posterior, U (0 , π α $_{i}$). 

2. For h = 1 , ..., M where M = max { α $_{1}$, ..., α n } , update p$_{j}$ α from the full conditional posterior distribution 

<img src='content_image/326247.jpg'>

3. For h = 1 , ..., M , update V α from the conditional distribution that is Beta(1 , β ) truncated to 

