## Defining Priors for Distributions Using Dirichlet
Diffusion Trees

**Radford M. Neal,
Dept. of Statistics and Dept. of Computer Science, University of Toronto**

I introduce a family of prior distributions over univariate or
multivariate distributions, based on the use of a ``Dirichlet
diffusion tree'' to generate exchangeable data sets. These priors can
be viewed as generalizations of Dirichlet processes and of Dirichlet
process mixtures. They are potentially of general use for modeling
unknown distributions, either of observed data or of latent values.
Unlike simple mixture models, Dirichlet diffusion tree priors can
capture the hierarchical structure that is present in many
distributions. Depending on the ``divergence function'' employed, a
Dirichlet diffusion tree prior can produce discrete or continuous
distributions. Empirical evidence is presented that some divergence
functions produce distributions that are absolutely continuous, while
others produce distributions that are continuous but not absolutely
continuous. Although Dirichlet diffusion trees are defined in terms
of a continuous-time stochastic process, inference for finite data
sets can be expressed in terms of finite-dimensional quantities, which
should allow computations to be performed by reasonably efficient
Markov chain Monte Carlo methods.

Technical Report No. 0104, Dept. of Statistics, University of Toronto
(March 2001), 25 pages:
postscript,
pdf.

The results in this paper were produced using software available on-line.

**Associated references:**
Parts (but not all) of this technical report became part of the following
paper:
Neal, R. M. (2003) ``Density modeling and clustering using Dirichlet diffusion
trees'', in J. M. Bernardo, *et al.* (editors)
*Bayesian Statistics 7*, pp. 619-629:
abstract,
postscript,
pdf,
associated software.