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1.1 Bayesian Inference

Throughout this thesis I use the Bayesian paradigm for statistical inference. Bayesian
inference is characterized by a joint probability model on all unknown quantities,
including observable data y and parameters θ. Classical inference in contrast uses
only probability models for y indexed by θ. Under the Bayesian paradigm, all relevant
information after seeing the data is contained in the posterior distribution p{θ ∣ y).
The main challenges are the construction of appropriate prior probability models p(θ),
and the often computationally intensive assessment of relevant summaries of the high
dimensional posterior distribution ρ(θ ∣ y).

Over the last two decades a barrage of new methods commonly known as Markov
Chain Monte Carlo (MCMC) have been proposed to deal with the latter problem.
Most Bayesian inference can be represented as posterior expectation of appropriate
functions of the parameters. The main idea of MCMC is to approximate posterior
expectations by ergodic averages over Markov chain simulations that are set up to
have p(θ ∣ y) as asymptotic distribution. These developments are well summarized in,
among many other references, Cappé and Robert (2002) and Lopes and Gamerman
(2006) .

1.2 Non-parametric Bayesian Inference

The second big challenge concerns the choice of the prior probability model. Con-
ventional parametric priors are families of prior probability models p{θ ∣ η) indexed
by a finite dimensional parameter η. Typical examples of these types of priors are
normal models, Beta distributions, etc. In many applications this assumption of finite
dimensions turns out to be too restrictive. A typical situation is the specification of
random effects distributions. Assuming a parametric random effects model implies a
very homogeneous population of experimental units (patients, peptides, etc). It does
not properly reflect the population heterogeneity that is typical for many biomedical

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