For nonparametric Bayesian inference we use a prior which supports piecewise linear quantile functions, based on the need to work with a finite set of partitions, . Nils Lid Hjort, Chris Holmes, Peter Müller, and Stephen G. Walker the history of the still relatively young field of Bayesian nonparametrics, and offer some. Part III: Bayesian Nonparametrics. Nils Lid Hjort. Department of Mathematics, University of Oslo. Geilo Winter School, January 1/
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The conditional probability of a point process given a sample point has a number of specific properties that general conditional probabilities do not satisfy. This is one of the topics on which “the” book to read has been written; Kingman’s book on the Poisson process is certainly one of the best expository texts in probability.
Nonparametric Bayes Tutorial
Random Fields and Geometry. Walker Search this author in: Given the current dearth of books on BNP, this book will be an invaluable source of information and reference for anyone interested in BNP, be it a student, an established statistician, or a researcher in need of flexible statistical analyses.
An excellent introduction to Gaussian process models and many references can be found in the monograph by Bjort and Williams. Historical references The original DP paper is of course Ferguson’s article. Billingsley’s book is a popular choice. Hjorg applications, these models are typically used as priors on the mixing measure of a mixture model e.
If nonpara,etrics, look no further: If you are interested in the bigger picture, and in how exchangeability generalizes to other random structures than exchangeable sequences, I highly recommend an article based on David Aldous’ lecture at the International Congress of Mathematicians: Via the correspondence between random discrete measures and random partitions, the theory of Palm measures can be applied to partitions: 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: Annals of Statistics, 2 6: If a byesian discrete measure is represented as a point process, nonparrametrics posterior is represented by a Palm measure.
Springer, 3rd edition, This coherent text gives ready access both to underlying principles and to state-of-the-art practice.
Tutorials on Bayesian Nonparametrics
Bayesian Nonparametrics Series Number Tutorial talks available online as streaming videos. Annals of Statistics, 35 2: Posterior convergence A clear and readable introduction to the questions studied in this area, and to how they are addressed, is a survey chapter by Ghosal which is referenced above.
For applications to existing models, see Posterior analysis for normalized random measures with independent increments. Antoniak introduces the idea of using a parametric likelihood with a DP or MDP, which he refers to as “random noise” cf his Theorem 3 and as a sampling distribution cf Example 4.
Hjort , Walker : Quantile pyramids for Bayesian nonparametrics
Other books in this series. Fuzione caratteristica di un fenomeno aleatorio. For a clear exposition of the discreteness argument used by Blackwell, see Chapter baydsian.
Steven has kindly given me permission to make it available here: Be aware though hmort the most interesting work in this area has arguably been done in the past decade, and hence is not covered by the book. Two tales about Bayesian nonparametric modeling. With quantile pyramids we instead fix probabilities and use random partitions.
These are covered in every textbook on probability theory. Review quote “The book looks like it will be useful to a wide range of researchers. Annals of Statistics, 1 2: Annals of Statistics, hjrot 3: An introduction to the theory of point processes. This problem has motivated my own work on conjugate models since conjugacy is the only reasonably general way we know to get from the prior and data to the posterior ; see e.
Permanent link to this document https: For the construction of general random measures, see Projective limit random probabilities on Polish spaces. Gaussian Processes for Machine Learning.
If you are interested in this book, make sure you read the second edition. In his acknowledgments, Ferguson attributes the idea to David Blackwell. Random functions Distributions on random functions can be used as prior distributions in regression and related problems.
In Encyclopedia of Machine Learning Springer ,