  In this paper we address the problem of estimating the ratio $\frac{q}{p}$
where $p$ is a density function and $q$ is another density, or, more generally
an arbitrary function. Knowing or approximating this ratio is needed in various
problems of inference and integration, in particular, when one needs to average
a function with respect to one probability distribution, given a sample from
another. It is often referred as {\it importance sampling} in statistical
inference and is also closely related to the problem of {\it covariate shift}
in transfer learning as well as to various MCMC methods. It may also be useful
for separating the underlying geometry of a space, say a manifold, from the
density function defined on it.
  Our approach is based on reformulating the problem of estimating
$\frac{q}{p}$ as an inverse problem in terms of an integral operator
corresponding to a kernel, and thus reducing it to an integral equation, known
as the Fredholm problem of the first kind. This formulation, combined with the
techniques of regularization and kernel methods, leads to a principled
kernel-based framework for constructing algorithms and for analyzing them
theoretically.
  The resulting family of algorithms (FIRE, for Fredholm Inverse Regularized
Estimator) is flexible, simple and easy to implement.
  We provide detailed theoretical analysis including concentration bounds and
convergence rates for the Gaussian kernel in the case of densities defined on
$\R^d$, compact domains in $\R^d$ and smooth $d$-dimensional sub-manifolds of
the Euclidean space.
  We also show experimental results including applications to classification
and semi-supervised learning within the covariate shift framework and
demonstrate some encouraging experimental comparisons. We also show how the
parameters of our algorithms can be chosen in a completely unsupervised manner.
