  Missingness in categorical data is a common problem in various real
applications. Traditional approaches either utilize only the complete
observations or impute the missing data by some ad hoc methods rather than the
true conditional distribution of the missing data, thus losing or distorting
the rich information in the partial observations. In this paper, we propose the
Dirichlet Process Mixture of Collapsed Product-Multinomials (DPMCPM) to model
the full data jointly and compute the model efficiently. By fitting an infinite
mixture of product-multinomial distributions, DPMCPM is applicable for any
categorical data regardless of the true distribution, which may contain complex
association among variables. Under the framework of latent class analysis, we
show that DPMCPM can model general missing mechanisms by creating an extra
category to denote missingness, which implicitly integrates out the missing
part with regard to their true conditional distribution. Through simulation
studies and a real application, we demonstrate that DPMCPM outperforms existing
approaches on statistical inference and imputation for incomplete categorical
data of various missing mechanisms. DPMCPM is implemented as the R package
\texttt{MMDai}, which is available from the Comprehensive R Archive Network at
https://cran.r-project.org/web/packages/MMDai/index.html.
