Characterising the function space for Bayesian kernel models

We study a non parametric (bayesian) function estimation problem.  We
observe data $\{Y_i\}$ at points $\{X_i\} \in \mathbb{R}$, perhaps
with some noise. We assume that $Y_i = f(X_i)$, for some unknown
function $f$.  However we want to restrict our choice of functions to
be the elements of Reproducing kernel Hilbert space (RKHS) spanned by
a fixed kernel K. We model the function through an integral operator,
and show that putting prior over the RKHS is equivalent to putting
priors over the space of measures on $\mathbb{R}$. Once we have this
motivation, we implement using Levy processes. Our method for
implementation essentially follows that of Wolpert,R et. al.(See the
background paper). No background in RKHS, Levy process will be
assumed. After a brief introduction and motivation, we will devote
more time on the above topics, and we will look at detail on the
reversible jump MCMC involved in the implementation.

This is joint work with Qiang,Feng,Sayan,Robert