# Decomposition of Total Variation Distance

Let $$X$$, $$Y$$, and $$Z$$ be three random variables on a common measurable space. I am interested in the total variation distance between the two joint distributions $$(X,Z)$$ and $$(Y,Z)$$, i.e. $$$$\label{eq:tv-distance} d_{TV} \left( (X, Z) , (Y, Z) \right) \quad (\dagger)$$$$

I am looking for a "law of total probability" for the total variation distance which may be used to decompose the total variation distance above into the total variation distance between the conditional distributions $$X \mid Z = z$$ and $$Y \mid Z = z$$. In particular, define the function $$h: \mathbb{R} \to \mathbb{R}$$ as $$h(z) = d_{TV} \left( X \mid Z = z , Y \mid Z = z \right) \enspace.$$

Is there a "law of total probability" way to decompose the total variation $$(\dagger)$$ to the conditional total variation distance function $$h(z)$$ and the distribution of $$Z$$? Say, for example, $$d_{TV} \left( (X, Z) , (Y, Z) \right) = \mathbb{E} \left[ h(Z) \right]$$

We have \begin{align*} \mathrm{d}_{\rm TV}((X,Z),(Y,Z)) &= \frac{1}{2}\sum_{v,z} \left|\Pr[ X=v, Z=z ]-\Pr[ Y=v, Z=z ]\right| \\ &= \frac{1}{2}\sum_{v,z} \left|\Pr[ X=v \mid Z=z ]-\Pr[ Y=v \mid Z=z ]\right|\cdot \Pr[Z=z] \\ &=\sum_z\Pr[Z=z]\sum_{v} \frac{1}{2}\left|\Pr[ X=v \mid Z=z ]-\Pr[ Y=v \mid Z=z ]\right| \\ &=\sum_z\Pr[Z=z]\mathrm{d}_{\rm TV}(X\mid Z=z,Y\mid Z=z) \\ &= \mathbb{E}_Z[h(Z)] \end{align*} where $$h\colon \mathbb{R}\to\mathbb{R}$$ is defined as in your post, $$h(z) = \mathrm{d}_{\rm TV}(X\mid Z=z,Y\mid Z=z)$$.