  Let $M$ be a $B$-probability space. Assume that $B$ itself is a
$D$-probability space; then $M$ can be viewed as $D$-probability space as well.
Let $X$ be in $M$. We look at the question of relating the properties of $X$ as
$B$-valued random variable to its properties as $D$-valued random variable. We
characterize freeness of $X$ from $B$ with amalgamation over $D$: (a) in terms
of a certain factorization condition linking the $B$-valued and $D$-valued
cumulants of $X$, and (b) for $D$ finite-dimensional, in terms of linking the
$B$-valued and the $D$-valued Fisher information of $X$. We give an application
to random matrices. For the second characterization we derive a new
operator-valued description of the conjugate variable and introduce an
operator-valued version of the liberation gradient.
