_{1} $ factor, then $ \chi (S_{1},\dots ,S_{n}|B)=\frac{n}{2}\log 2\pi e>-\infty  $.

Let $ X_{1},\dots ,X_{p} $ be random variables. Assume that $ {X_{1},\dots ,X_{n}}$ are free from the algebra generated by $ X_{n+1},\dots ,X_{p} $ and $ B $. Then $$\label{eqn:additivity}
\chi ({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|B)=\chi ({X_{1},\dots ,X_{n}}|B)+\chi (X_{n+1},\dots ,X_{p}|B).$$

If $ B $ is embeddable into the ultrapower of the hyperfinite II$ _{1} $ factor, we have by Theorem \[thrm: whenfreefromB\] that $$\chi ({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|B)=\chi ({X_{1},\dots ,X_{n}})+\chi (X_{n+1},\dots ,X_{p}|B),$$ and also that $$\chi ({X_{1},\dots ,X_{n}}|B)=\chi ({X_{1},\dots ,X_{n}}).$$ If $ B $ is not embeddable into the ultrapower of the hyperfinite II$ _{1} $ factor, then the quantities on both sides of (\[eqn:additivity\]) are equal to $ -\infty  $.

The analogy between $ \chi ({X_{1},\dots ,X_{n}}|B) $ and $ \chi ^{*}({X_{1},\dots ,X_{n}}:B) $ makes it tempting to conjecture that (\[eqn:additivity\]) holds under the weaker assumption that $ {X_{1},\dots ,X_{n}}$ and $ X_{n+1},\dots ,X_{p} $ are free with amalgamation over $ B $; however, we were unable to prove this.

Separate change of variables formulas.
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\[thrm: onechangeofvar\]Let $ f_{i}:\mathbb {R}\to \mathbb {R} $ be diffeomorphisms, and let $ \mu _{i} $ be the distribution of $ X_{i} $. Then $$\chi 