We prove the universality of covariance matrices of the form $H_{N \times N}
= {1 \over N} \tp{X}X$ where $[X]_{M \times N}$ is a rectangular matrix with
independent real valued entries $[x_{ij}]$ satisfying $\E \,x_{ij} = 0$ and $\E
\,x^2_{ij} = {1 \over M}$, $N, M\to \infty$. Furthermore it is assumed that
these entries have sub-exponential tails. We will study the asymptotics in the
regime $N/M = d_N \in (0,\infty), \lim_{N\to \infty}d_N \neq 1$.

We study random walk based algorithms for posterior simulation in a large of
class of Bayesian nonparametric problems with Gaussian random field or Gaussian
process priors. Our emphasis is both on developing practical guidelines for the
design and implementation of efficient algorithms for these naturally high
dimensional problems, and on the development of rigorous underpinning theory.
We illustrate via an example that, in designing algorithms for nonparametric
problems, it is important to take advantage of the infinite dimensional
structure inherent in both the prior and likelihood.

Approximate Bayesian computation (ABC), also known as likelihood-free
methods, have become a favourite tool for the analysis of complex stochastic
models, primarily in population genetics but also in financial analyses. We
advocated in Grelaud et al.

We investigate the properties of the Hybrid Monte-Carlo algorithm (HMC) in
high dimensions. HMC develops a Markov chain reversible w.r.t. a given target
distribution $\Pi$ by using separable Hamiltonian dynamics with potential
$-\log\Pi$. The additional momentum variables are chosen at random from the
Boltzmann distribution and the continuous-time Hamiltonian dynamics are then
discretised using the leapfrog scheme. The induced bias is removed via a
Metropolis-Hastings accept/reject rule.