  The two parameter Poisson-Dirichlet Process (PDP), a generalisation of the
Dirichlet Process, is increasingly being used for probabilistic modelling in
discrete areas such as language technology, bioinformatics, and image analysis.
There is a rich literature about the PDP and its derivative distributions such
as the Chinese Restaurant Process (CRP). This article reviews some of the basic
theory and then the major results needed for Bayesian modelling of discrete
problems including details of priors, posteriors and computation.
  The PDP allows one to build distributions over countable partitions. The PDP
has two other remarkable properties: first it is partially conjugate to itself,
which allows one to build hierarchies of PDPs, and second using a marginalised
relative the CRP, one gets fragmentation and clustering properties that lets
one layer partitions to build trees. This article presents the basic theory for
understanding the notion of partitions and distributions over them, the PDP and
the CRP, and the important properties of conjugacy, fragmentation and
clustering, as well as some key related properties such as consistency and
convergence. This article also presents a Bayesian interpretation of the
Poisson-Dirichlet process based on an improper and infinite dimensional
Dirichlet distribution. This means we can understand the process as just
another Dirichlet and thus all its sampling properties emerge naturally.
  The theory of PDPs is usually presented for continuous distributions (more
generally referred to as non-atomic distributions), however, when applied to
discrete distributions its remarkable conjugacy property emerges. This context
and basic results are also presented, as well as techniques for computing the
second order Stirling numbers that occur in the posteriors for discrete
distributions.
