  Consider the $\mathcal{B}$-valued probability space $(\mathcal{A}, E,
\mathcal{B})$, where $\mathcal{A}$ is a tracial von Neumann algebra. We extend
the theory of operator valued free probability to the algebra of affiliated
operators $\tilde{\mathcal{A}}$. For a random variable $X \in
\tilde{\mathcal{A}}^{sa}$ we study the Cauchy transform $G_{X}$ and show that
the operator algebra $(\mathcal{B} \cup \{X\})"$ can be recovered from this
function. In the case where $\mathcal{B}$ is finite dimensional, we show that,
when $X, Y \in \tilde{\mathcal{A}}^{sa}$ are assumed to be $\mathcal{B}$-free,
the $\mathcal{R}$-transforms are defined on universal subsets of the resolvent
and satisfy $$ \mathcal{R}_{X} + \mathcal{R}_{Y} = \mathcal{R}_{X + Y}. $$
Examples indicating a failure of the theory for infinite dimensional
$\mathcal{B}$ are provided. Lastly, we show that the class of functions that
arise as the Cauchy transform of affiliated operators is, in a natural way, the
closure of the set of Cauchy transforms of bounded operators.
