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We present a probabilistic block-constant biclustering model that simultaneously clusters rows and columns of a data matrix. All entries with the same row cluster and column cluster form a bicluster. Each cluster is part of a mixture having a nonparametric Bayesian prior. The number of biclusters is therefore treated as a nuisance parameter and is implicitly integrated over during simulation. Missing entries are completely integrated out of the model, allowing us to completely bipass the common requirement for biclustering algorithms that missing values be filled before analysis, but also makes it robust to high rates of missing values. By using a Gaussian model for the density of entries in bliclusters, an efficient sampling algorithm is produced because bicluster parameters are analytically integrated out. We present several inference procedures for sampling cluster indicators, including Gibbs and split-merge moves. We show that our method is competitive, if not superior, to existing imputation methods, especially for high missing rates, despite imputing constant values for entire blocks of data. We present imputation experiments and exploratory biclustering results.