2010/12/02 by Peter Orbanz, Orbanz, Peter
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #stat.ML #stat.TH
paper · pdf · doi:10.48550/arxiv.1012.0363
49 pages; improved version: revised proof of theorem 3 (results unchanged), discussion added, exposition revised
openalex publication_date 2010/12/02 · arxiv created 2011/01/07 · arxiv updated 2011/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We characterize conjugate nonparametric Bayesian models as projective limits of conjugate, finite-dimensional Bayesian models. In particular, we identify a large class of nonparametric models representable as infinite-dimensional analogues of exponential family distributions and their canonical conjugate priors. This class contains most models studied in the literature, including Dirichlet processes and Gaussian process regression models. To derive these results, we introduce a representation of infinite-dimensional Bayesian models by projective limits of regular conditional probabilities. We show under which conditions the nonparametric model itself, its sufficient statistics, and -- if they exist -- conjugate updates of the posterior are projective limits of their respective finite-dimensional counterparts. We illustrate our results both by application to existing nonparametric models and by construction of a model on infinite permutations.