Distance between the product of marginal distributions and the joint distribution

Question

Given a joint distribution $P(A,B,C)$, we can compute various marginal distributions. Now suppose: \begin{align} P1(A,B,C) &= P(A) P(B) P(C) \\ P2(A,B,C) &= P(A,B) P(C) \\ P3(A,B,C) &= P(A,B,C) \end{align} Is it true that $d(P1,P3) \geq d(P2,P3)$ where $d$ is the total variation distance?

In other words, is it provable that $P(A,B) P(C)$ is a better approximation of $P(A,B,C)$ than $P(A) P(B) P(C)$ in terms of the total variation distance? Intuitively I think it's true but could not find out a proof.

Answer 1

I just find the following counter-example. Suppose $A,B,C$ are discrete variables. $A,B$ can each take two values while $C$ can take three values. The joint distribution $P(A,B,C)$ is:

\begin{array}{cccc} A & B & C & P(A,B,C) \\ 1 & 1 & 1 & 0.1/3 \\ 1 & 1 & 2 & 0.25/3 \\ 1 & 1 & 3 & 0.25/3 \\ 1 & 2 & 1 & 0.4/3 \\ 1 & 2 & 2 & 0.25/3 \\ 1 & 2 & 3 & 0.25/3 \\ 2 & 1 & 1 & 0.4/3 \\ 2 & 1 & 2 & 0.25/3 \\ 2 & 1 & 3 & 0.25/3 \\ 2 & 2 & 1 & 0.1/3 \\ 2 & 2 & 2 & 0.25/3 \\ 2 & 2 & 3 & 0.25/3 \\ \end{array}

So the marginal distribution $P(A,B)$ is: \begin{array}{ccc} A & B & P(A,B) \\ 1 & 1 & 0.2 \\ 1 & 2 & 0.3 \\ 2 & 1 & 0.3 \\ 2 & 2 & 0.2 \\ \end{array}

The marginal distributions $P(A), P(B)$ and $P(C)$ are uniform.

So we can compute that: \begin{align} d(P1,P3) &= 0.1 \\ d(P2,P3) &= 0.4/3 \end{align}