be embedded into the ultrapower of the hyperfinite II$ _{1} $ factor). Lastly, the infinitesimal change of variables formula for $ \chi (\cdots |B) $ involves conjugate variables used to define $ \chi ^{*}(\cdots :B) $ (see [@dvv:entropy5]).

It is interesting to note that $ \chi (\cdot |B) $ has an interpretation as a relative entropy, which suggests a similar interpretation for $ \chi ^{*}(\cdot :B) $. Indeed, we show that if $ {Y_{1},\dots ,Y_{m}}$ are generators of $ B $, then $ \chi ({X_{1},\dots ,X_{n}}|B)=\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}) $, where the latter entropy has properties of a relative entropy of $ {X_{1},\dots ,X_{n}}$ and $ {Y_{1},\dots ,Y_{m}}$. We caution the reader that we use a definition of $ \chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}}) $ which may be different from the one used by Voiculescu in [@dvv:entropy2], although the two quantities are related.

Relative Free Entropy $ \chi (\cdot |B)\protect $.
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Let $ (M,\tau ) $ be a tracial non-commutative probability space, and consider self-adjoint non-commutative random variables $ X_{1},\dots ,X_{n} $, $ Y_{1},\dots ,Y_{m}\in M $. We denote by $ M_{k} $ the algebra of $ k\times k $ matrices, and by $ M_{k}^{{{\operatorname{sa}}}} $ the set of self-adjoint 