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If $\mu_1$ and $\nu_1$ are probability distributions on the finite state space $\Omega_1$, $\mu_2$ and $\nu_2$ are probability distributions on the finite state space $\Omega_2$, $\mu_1 \times \mu_2 (x,y) = \mu_1(x)\mu_2(y)$ and $\nu_1 \times \nu_2 (x,y) = \nu_1(x)\nu_2(y)$ are probability measures on $\Omega_1 \times \Omega_2$.

It follows from the triangle inequality that

$$\|\mu_1\times\mu_2 - \nu_1\times\nu_2 \|_{TV} \leq \|\mu_1-\nu_1\|_{TV}+\|\mu_2-\nu_2\|_{TV}$$

I heard it mentioned in a few places that this can also be proved using coupling, but could not figure that. Could someone walk me through or point me to the proof?

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    $\begingroup$ Proof: If $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$, then $$P(X\ne Y)\leqslant P(X_1\ne Y_1)+P(X_2\ne Y_2)$$ End of the proof. $\endgroup$
    – Did
    Commented Apr 10, 2018 at 16:31
  • $\begingroup$ But $P(X_1 \neq Y_1) \geq \|\mu_1 - \nu_1\|$. So how can I get that $P(X\neq Y) \leq \|\mu_1 - \nu_1\| + \|\mu_2 - \nu_2\|$ from this? $\endgroup$
    – The Hagen
    Commented Apr 10, 2018 at 17:05
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    $\begingroup$ By considering the infima of the probabilities $P(X_i\ne Y_i)$ on the couples $(X_i,Y_i)$ with the correct marginals. $\endgroup$
    – Did
    Commented Apr 10, 2018 at 19:01
  • $\begingroup$ Thanks! Why post this as a comment? It answered my question $\endgroup$
    – The Hagen
    Commented Apr 10, 2018 at 19:13
  • $\begingroup$ To let you the opportunity to expand these hints into a full-fledged answer, which you would then post it here. $\endgroup$
    – Did
    Commented Apr 10, 2018 at 19:15

1 Answer 1

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There are two parts to the Coupling Lemma:

If $\mu$ and $\nu$ are two probability measures over a finite set $\Omega$, then:

  1. For any coupling $\omega$ of $(\mu,\nu)$, if the random variable $(X,Y)$ is distributed according to $\omega$, then $$\|\mu-\nu\|_{TV} \leq P(X\neq Y)$$

  2. There exists an optimal coupling $\omega^*$ of $(\mu,\nu)$ for which $$\|\mu-\nu\|_{TV} = P(X\neq Y)$$

Picking $\omega_1^*$ as the optimal coupling on $(\mu_1, \nu_1)$, and $\omega_2^*$ as the optimal coupling on $(\mu_2, \nu_2)$, take the random variable $(X_1, Y_1)$ distributed according to $\omega_1^*$ and $(X_2, Y_2)$ distributed according to $\omega_2^*$, we have

$$P(X_1\neq Y_1) + P(X_2\neq Y_2) = \|\mu_1-\nu_1\|_{TV} + \|\mu_2-\nu_2\|_{TV}$$

by part 2.

Defining $X=(X_1,X_2)$, $Y=(Y_1,Y_2)$, we further have that

$$P(X\neq Y) = 1-P(X=Y) = 1-P(X_1=Y_1)P(X_2=Y_2) \leq 1-P(X_1=Y_1) + 1-P(X_2=Y_2) = P(X_1\neq Y_1) + P(X_2\neq Y_2)$$

where the inequality comes from the fact that for $0\leq a, b, \leq 1$:

$$a(1-b)\leq 1- b \Rightarrow a-ab\leq 1-b \Rightarrow -ab\leq 1-(a+b) \Rightarrow 1-ab \leq (1-a) + (1-b)$$

and noting that the law of $(X,Y)$ is a coupling of $(\mu_1\times\mu2, \nu_1\times\nu2)$, by part 1 we are done:

$$\|\mu_1\times\mu2 - \nu_1\times\nu2\|_{TV} \leq P(X\neq Y) \leq P(X_1\neq Y_1) + P(X_2\neq Y_2) = \|\mu_1-\nu_1\|_{TV} + \|\mu_2-\nu_2\|_{TV}$$

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  • $\begingroup$ Why is the (X;Y) a coupling of $(\mu, \nu)$? We do not know that $X_1$ and $X_2$ are independent. $\endgroup$
    – mark
    Commented Nov 4, 2019 at 19:40

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