Probabilistic models of sequences play a central role in most machine translation, automated speech recognition, lossless compression, spell-checking, and gene identification applications to name but a few. Unfortunately, real-world sequence data often exhibit long range dependencies which can only be captured by computationally challenging, complex models. Sequence data arising from natural processes also often exhibit power-law properties, yet common sequence models do not capture such properties. The sequence memoizer is a new hierarchical Bayesian model for discrete sequence data that captures long range dependencies and power-law characteristics while remaining computationally attractive. Its utility as a language model and general purpose lossless compressor is demonstrated. 1.

We propose a novel class of Bayesian nonparametric models for sequential data called fragmentation-coagulation processes (FCPs). FCPs model a set of sequences using a partition-valued Markov process which evolves by splitting and merging clusters. An FCP is exchangeable, projective, stationary and reversible, and its equilibrium distributions are given by the Chinese restaurant process. As opposed to hidden Markov models, FCPs allow for flexible modelling of the number of clusters, and they avoid label switching non-identifiability problems. We develop an efficient Gibbs sampler for FCPs which uses uniformization and the forward-backward algorithm. Our development of FCPs is motivated by applications in population genetics, and we demonstrate the utility of FCPs on problems of genotype imputation with phased and unphased SNP data. 1