In the mathematical theory of

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

, the dependent Dirichlet process (DDP) provides a non-parametric

prior Prior (or prioress) is an ecclesiastical title for a superior in some religious orders. The word is derived from the Latin for "earlier" or "first". Its earlier generic usage referred to any monastic superior. In abbeys, a prior would be l ...

over evolving

mixture model In statistics, a mixture model is a probabilistic model for representing the presence of subpopulations within an overall population, without requiring that an observed data set should identify the sub-population to which an individual observation ...

s. A construction of the DDP built on a Poisson point process. The concept is named after Peter Gustav Lejeune Dirichlet. In many applications we want to model a collection of distributions such as the one used to represent temporal and spatial

stochastic processes In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appe ...

. The

Dirichlet process In probability theory, Dirichlet processes (after the distribution associated with Peter Gustav Lejeune Dirichlet) are a family of stochastic processes whose realizations are probability distributions. In other words, a Dirichlet process is a pro ...

assumes that observations are exchangeable and therefore the data points have no inherent ordering that influences their labeling. This assumption is invalid for modelling temporal and spatial processes in which the order of data points plays a critical role in creating meaningful clusters.

Dependent Dirichlet process

The dependent Dirichlet process (DDP) originally formulated by MacEachern led to the development of the DDP mixture model (DDPMM) which generalizes DPMM by including birth, death and transition processes for the clusters in the model. In addition, a low-variance approximations to DDPMM have been derived leading to a dynamic clustering algorithm.T. Campbell, M. Liu, B. Kulis, J. P. How, and L. Carin
Dynamic clustering via asymptotics of the Dependent Dirichlet Process.
Neural Information Processing Systems (NIPS), 2013. Under time-varying setting, it is natural to introduce different DP priors for different time steps. The generative model can be written as follows: :

D_t \sim \operatorname(\alpha, H_t)

\theta_ \mid D_t \sim D_t ~~~ \texti=1,\ldots,n_t,~t=0,\ldots,T

X_ \mid \theta_ \sim F(\theta_) ~~~ \text i=1,\ldots, n_t,~t=0, \ldots,T

A Poisson-based construction of DDP exploits the connection between Poisson and Dirichlet processes. In particular, by applying operations that preserve complete randomness to the underlying Poisson processes: superposition, subsampling and point transition, a new Poisson and therefore a new Dirichlet process is produced.

References

{{reflist *S. N. MacEachern, "Dependent Nonparametric Processes", in ''Proceedings of the Bayesian Statistical Science Section'', 1999 Nonparametric statistics Bayesian statistics Stochastic processes