Over the past few years, it has become much clearer which models exist, how they can be represented, and in which cases we can expect inference to be tractable. For applications to existing models, see Posterior analysis for normalized random measures with independent increments. An excellent introduction to Gaussian process models and many references can be found in the monograph by Rasmussen and Williams. In parametric models, this set of exceptions does not usually cause problems, but in nonparametric models, it can make this notion of consistency almost meaningless. Misspecification in infinite-dimensional Bayesian statistics. Google Scholar Project Euclid.
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I discuss applications to nonparametric Bayesian models of data not representable as exchangeable sequences in this preprint: Nonparametric priors on complete separable metric spaces. P Orbanz. Roughly speaking, an urn model assumes that balls of different colors are contained in an urn, and are drawn uniformly at random; the proportions of balls per color determine the probability of each color to be drawn.
A specific urn is defined by a rule for how the number of balls is changed when a color is drawn. For Bayesian nonparametrics, urns provide a probabilistic tool to study the sizes of clusters in a clustering model, or more generally the weight distributions of random discrete measures. They also provide a link to population genetics, where urns model the distribution of species; you will sometimes encounter references to species sampling models.
If you are interested in urns and power laws, I recommend that you have a look at the following two survey articles in this order : Notes on the occupancy problem with infinitely many boxes: General asymptotics and power laws. Probability Surveys, , A survey of random processes with reinforcement. R Pemantle. There are a few specific reasons why Bayesian nonparametric models require more powerful mathematical tools than parametric ones; this is particularly true for theoretical problems.
One of the reasons is that Bayesian nonparametric models do not usually have density representation, and hence require a certain amount of measure theory. Since the parameter space of a nonparametric model is infinite-dimensional, the prior and posterior distributions are probabilities on infinite-dimensional spaces, and hence stochastic processes. If you are interested in the theory of Bayesian nonparametrics and do not have a background in probability, you may have to familiarize yourself with some topics such as stochastic processes and regular conditional probabilities.
These are covered in every textbook on probability theory. Probability and Measure.
Tutorials on Bayesian Nonparametrics
Karamar I stumbled upon the O. We include many common tasks, including data management, descriptive summaries, inferential procedures, regression analysis, multivariate methods, and the creation of nonparanetrics. The theory with careful proofs of the asymptotic results is fully developed. The distinction between exploratory and confirmatory factor analysis is also touched on, but is explored in more depth in Chapter jjort.