41 research outputs found
Conjugate Projective Limits
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.Comment: 49 pages; improved version: revised proof of theorem 3 (results
unchanged), discussion added, exposition revise
Uniform estimation of a class of random graph functionals
We consider estimation of certain functionals of random graphs. The random
graph is generated by a possibly sparse stochastic block model (SBM). The
number of classes is fixed or grows with the number of vertices. Minimax lower
and upper bounds of estimation along specific submodels are derived. The
results are nonasymptotic and imply that uniform estimation of a single
connectivity parameter is much slower than the expected asymptotic pointwise
rate. Specifically, the uniform quadratic rate does not scale as the number of
edges, but only as the number of vertices. The lower bounds are local around
any possible SBM. An analogous result is derived for functionals of a class of
smooth graphons
Nonparametric Bayesian Image Segmentation
Image segmentation algorithms partition the set of pixels of an image into a specific number of different, spatially homogeneous groups. We propose a nonparametric Bayesian model for histogram clustering which automatically determines the number of segments when spatial smoothness constraints on the class assignments are enforced by a Markov Random Field. A Dirichlet process prior controls the level of resolution which corresponds to the number of clusters in data with a unique cluster structure. The resulting posterior is efficiently sampled by a variant of a conjugate-case sampling algorithm for Dirichlet process mixture models. Experimental results are provided for real-world gray value images, synthetic aperture radar images and magnetic resonance imaging dat