  Let $H$ be a Hilbert space and $H_1,...,H_n$ be closed subspaces of $H$.
Denote by $P_k$ the orthogonal projection onto $H_k$, $k=1,2,...,n$. Following
Patrick L. Combettes and Noli N. Reyes, we will say that the system of
subspaces $H_1,...,H_n$ possesses the inverse best approximation property
(IBAP) if for arbitrary elements $x_1\in H_1,...,x_n\in H_n$ there exists an
element $x\in H$ such that $P_k x=x_k$ for all $k=1,2,...,n$. We provide
various new necessary and sufficient conditions for a system of $n$ subspaces
to possess the IBAP. Using the main characterization theorem, we study
properties of the systems of subspaces which possess the IBAP, obtain a
sufficient condition for a system of subspaces to possess the IBAP, and provide
examples of systems of subspaces which possess the IBAP.
  These results are applied to a problem of probability theory. Let
$(\Omega,\mathcal{F},\mu)$ be a probability space and
$\mathcal{F}_1,...,\mathcal{F}_n$ be sub-$\sigma$-algebras of $\mathcal{F}$. We
will say that the collection $\mathcal{F}_1,...,\mathcal{F}_n$ possesses the
inverse marginal property (IMP) if for arbitrary random variables
$\xi_1,...,\xi_n$ such that (1) $\xi_k$ is $\mathcal{F}_k$-measurable,
$k=1,2,...,n$; (2) $E|\xi_k|^2<\infty$, $k=1,2,...,n$; (3)
$E\xi_1=E\xi_2=...=E\xi_n$, there exists a random variable $\xi$ such that
$E|\xi|^2<\infty$ and $E(\xi|\mathcal{F}_k)=\xi_k$ for all $k=1,2,...,n$. We
will show that a collection of sub-$\sigma$-algebras possesses the IMP if and
only if the system of corresponding marginal subspaces possesses the IBAP. We
consider two examples; in the first example $\Omega=\mathbb{N}$, in the second
example $\Omega=[a,b)$. For these examples we establish relations between the
IMP, the IBAP, closedness of the sum of marginal subspaces and "fast
decreasing" of tails of the measure $\mu$.
