This paper introduces a hierarchical Dirichlet process (HDP) for modeling grouped data where each group has its own mixture model and clusters are shared across groups. The HDP is a nonparametric Bayesian approach that extends the Dirichlet process (DP) to allow for hierarchical clustering. The DP is a prior over probability measures, and the HDP is a prior over a set of DPs, where the base measure of each DP is itself a DP. This allows for sharing of mixture components across groups, as the atoms of the DPs are shared. The HDP is defined as follows: the base measure $ G_0 $ is distributed according to a DP with concentration parameter $ \gamma $ and base measure $ H $, and each group-specific DP $ G_j $ is distributed according to a DP with concentration parameter $ \alpha_0 $ and base measure $ G_0 $. This hierarchical structure allows for sharing of mixture components across groups, as the atoms of the DPs are shared. The paper discusses representations of the HDP in terms of a stick-breaking process and a generalization of the Chinese restaurant process called the "Chinese restaurant franchise." It also presents Markov chain Monte Carlo (MCMC) algorithms for posterior inference in HDP mixtures and describes applications to problems in information retrieval and text modeling. The HDP is a flexible nonparametric Bayesian model that allows for clustering of grouped data with shared clusters. It is particularly useful in applications where the number of clusters is unknown and needs to be inferred from the data. The HDP is a natural extension of the DP to allow for hierarchical clustering, and it provides a framework for modeling complex data structures with multiple levels of grouping.This paper introduces a hierarchical Dirichlet process (HDP) for modeling grouped data where each group has its own mixture model and clusters are shared across groups. The HDP is a nonparametric Bayesian approach that extends the Dirichlet process (DP) to allow for hierarchical clustering. The DP is a prior over probability measures, and the HDP is a prior over a set of DPs, where the base measure of each DP is itself a DP. This allows for sharing of mixture components across groups, as the atoms of the DPs are shared. The HDP is defined as follows: the base measure $ G_0 $ is distributed according to a DP with concentration parameter $ \gamma $ and base measure $ H $, and each group-specific DP $ G_j $ is distributed according to a DP with concentration parameter $ \alpha_0 $ and base measure $ G_0 $. This hierarchical structure allows for sharing of mixture components across groups, as the atoms of the DPs are shared. The paper discusses representations of the HDP in terms of a stick-breaking process and a generalization of the Chinese restaurant process called the "Chinese restaurant franchise." It also presents Markov chain Monte Carlo (MCMC) algorithms for posterior inference in HDP mixtures and describes applications to problems in information retrieval and text modeling. The HDP is a flexible nonparametric Bayesian model that allows for clustering of grouped data with shared clusters. It is particularly useful in applications where the number of clusters is unknown and needs to be inferred from the data. The HDP is a natural extension of the DP to allow for hierarchical clustering, and it provides a framework for modeling complex data structures with multiple levels of grouping.