Title: General Bayesian inference schemes in infinite mixture models,
Abstract: Bayesian statistical models allow us to formalise our knowledge about the
world and reason about our uncertainty, but there is a need for better
procedures to accurately encode its complexity. One way to do so is through
compositional models, which are formed by combining blocks consisting of
simpler models. One can increase the complexity of the compositional model by
either stacking more blocks or by using a not-so-simple model as a building
block. This thesis is an example of the latter. One first aim is to expand the
choice of Bayesian nonparametric (BNP) blocks for constructing tractable
compositional models. So far, most of the models that have a Bayesian
nonparametric component use a Dirichlet Process or a Pitman-Yor process because
of the availability of tractable and compact representations. This thesis shows
how to overcome certain intractabilities in order to obtain analogous compact
representations for the class of Poisson-Kingman priors which includes the
Dirichlet and Pitman-Yor processes.
A major impediment to the widespread use of Bayesian nonparametric building
blocks is that inference is often costly, intractable or difficult to carry
out. This is an active research area since dealing with the model's infinite
dimensional component forbids the direct use of standard simulation-based
methods. The main contribution of this thesis is a variety of inference schemes
that tackle this problem: Markov chain Monte Carlo and Sequential Monte Carlo
methods, which are exact inference schemes since they target the true
posterior. The contributions of this thesis, in a larger context, provide
general purpose exact inference schemes in the flavour or probabilistic
programming: the user is able to choose from a variety of models, focusing only
on the modelling part. Indeed, if the wide enough class of Poisson-Kingman
priors is used as one of our blocks, this objective is achieved.