  Let $(W,H,\mu)$ be the classical Wiener space, assume that $U_\la=I_W+u_\la$
is an adapted perturbation of identity where the perturbation $u_\la$ is an
$H$-valued map, defined up to $\mu$-equivalence classes, such that its Lebesgue
density $s\to \dot{u}_\la(s)$ is almost surely adapted to the canonical
filtration of the Wiener space and depending measurably on a real parameter
$\la$. Assuming some regularity for $u_\la$, its Sobolev derivative and
integrability of the divergence of the resolvent operator of its Sobolev
derivative, we prove the almost sure and $L^p$-regularity w.r. to $\la$ of the
estimation $E[\dot{u}_\la(s)|\calU_\la(s)]$ and more generally of the
conditional expectations of the type $E[F\mid\calU_\la(s)]$ for nice Wiener
functionals, where
  $(\calU_\la(s),s\in [0,1])$ is the the filtration which is generated by
$U_\la$. These results are applied to prove the invertibility of the adapted
perturbations of identity, hence to prove the strong existence and uniqueness
of functional SDE's; convexity of the entropy and the quadratic estimation
error and finally to the information theory.
