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Let $(M, d)$ be a metric space with its Borel sigma algebra $\mathcal{B} (M)$. Let $\mathcal{P} (M)$ denote the collection of all probability measures on the measurable space $(M, \mathcal{B} (M))$.

For a subset $A \subseteq M$, define the $ε$-neighborhood of $A$ by $$ A^{\varepsilon} := \{ p \in M ~|~ \exists q \in A, \ d(p, q) < \varepsilon \} = \bigcup_{p \in A} B_{\varepsilon} (p). $$ where $B_{\varepsilon} (p)$ is the open ball of radius $\varepsilon$ centered at $p$.

The Lévy–Prokhorov metric $\pi : \mathcal{P} (M)^{2} \to [0, + \infty)$ is defined by setting the distance between two probability measures $\mu$ and $\nu$ to be $$ \pi (\mu, \nu) := \inf \left\{ \varepsilon > 0 ~|~ \mu(A) \leq \nu (A^{\varepsilon}) + \varepsilon \ \text{and} \ \nu (A) \leq \mu (A^{\varepsilon}) + \varepsilon \ \text{for all} \ A \in \mathcal{B}(M) \right\}. $$

  1. I wonder what the purpose, motivation and intuition of the L-P metric are?
  2. Is the following alternative a reasonable metric or some generalized metric between measures $$ \sup_{A \in \mathcal{B}(M)} |\mu(A) - \nu(A)|? $$ If yes, is this one more simple and easy to understand and therefore maybe more useful than L-P metric?
  3. A related metric between distribution functions is the Levy metric:

    Let $F, G : \mathbb{R} \to [0, + \infty)$ be two cumulative distribution functions. Define the Lévy distance between them to be $$ L(F, G) := \inf \{ \varepsilon > 0 | F(x - \varepsilon) - \varepsilon \leq G(x) \leq F(x + \varepsilon) + \varepsilon \mathrm{\,for\,all\,} x \in \mathbb{R} \}. $$

    I wonder how to picture this intuition part:

    Intuitively, if between the graphs of $F$ and $G$ one inscribes squares with sides parallel to the coordinate axes (at points of discontinuity of a graph vertical segments are added), then the side-length of the largest such square is equal to $L(F, G)$.

Thanks and regards!

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Atleast one motivation for the Prokhorov metric is the metrization of weak convergence of measures. This topology is very used and fruitful for applications. It is not however the only useful metric on $P(M)$ that metrizises weak convergence. Wasserstein metrics, that arise from optimal transportation of measures, also metrizise weak convergence if $M$ is compact, or Polish and $d$ bounded. –  Thomas E. Apr 19 '12 at 8:35
@ThomasE.: Thanks! By Weak convergence, do you mean this link? –  Tim Apr 19 '12 at 12:40
Yeah, exactly. The first equivalent expression from the list is usually being used as a definition for the case of probability measures. For arbitrary measures the test functions need to have a compact support as well. I.e. $(\mu_{k})$ converges weakly to $\mu$ if $\int_{S}fd\mu_{k}\to \int_{S}fd\mu$ for all continuous, compactly supported $f:S\to \mathbb{R}$. –  Thomas E. Apr 19 '12 at 19:28
I have developed my answer somewhat here: math.stackexchange.com/a/358095/48890 –  zab Apr 18 '13 at 22:04
One can write $L(F,G)$ slightly differently as $$L(F,G) = \inf\{\epsilon > 0| F(x) \leq G(x+\epsilon) + \epsilon \text{ and } G(x) \leq F(x+\epsilon) + \epsilon, \forall x\in \mathbb{R}\}$$ so that more clearly L-P metric is a generalization of Lévy's metric –  Petite Etincelle Apr 6 at 4:45

2 Answers 2

Most of what occurs to me has already been said, but you may find the following picture useful.

If $d_C$ is the Chebyshev metric on $R^2$, i.e. with points $\mathbf{p} = (x_1,y_1)$ and $\mathbf{q} = (x_2,y_2)$ in $R^2$,

$d_C(\mathbf{p,q}) := |x_1-x_2| \vee |y_1-y_2|$,

and $h_C$ is the Hausdorff metric on closed subsets of $R^2$ induced by $d_C$, i.e. with $A$ and $B$ being closed subsets of $R^2$,

$h_C(A,B):= \sup_{\mathbf{p} \in A} d_C(\mathbf{p},B) \vee \sup_{\mathbf{q} \in B} d_C(\mathbf{q},A)$,

where as usual $d_C(\mathbf{p},B) = \inf_{\mathbf{r} \in B} d_C(\mathbf{p,r})$ etc,

then the Levy metric between two distribution functions $F$ and $G$ is simply the Hausdorff distance $d_C$ between the closures of the completed graphs of $F$ and $G$.

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+1 Thanks! What does the completed graph of a function F mean? –  Tim Dec 10 '12 at 1:25
Can the Levy-Prokhorov metric be interpreted in terms of or similar to Hausdorff metric in some ways? –  Tim Dec 10 '12 at 2:21
Hi zab, I have opened a new post about my comments above math.stackexchange.com/questions/313465/… –  Tim Feb 25 '13 at 18:02
  1. The Levy-Prokhorov metric does metrizises the weak convergence of measures. That is a quite cool thing since it allows you to conclude from the fact that $\mu_n$ weakly converges to $\mu$ that it "approaches $\mu$ with respect to some distance".

  2. I think this one is more related to the variation norm of the distance of the measures and hence, describes something like norm-convergence. Together with the first answer you see that both approaches are for completely different purposes.

One intuition I have for the Levy-Prokhorov metric is that two point-masses $\delta_x$ and $\delta_y$ have the distance of their points if the points are not too far away, i.e. for $d(x,y)\leq 1$ it holds that $$\pi(\delta_x,\delta_y) = d(x,y).$$ One the other hand, your term in 2. is always 2, regardless of $x$ and $y$. If you have a sequence $(x_n)$ converging to $x$ in $M$, then $(\delta_{x_n})$ converges to $\delta_x$ with respect to Levy-Prokhorov but not in norm.

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+1. Thanks, Dirk! I wonder about the intuition of L-P metric in general? –  Tim Apr 19 '12 at 12:41
@Dirk When you say $\pi(\delta_{x}, \delta_{y}) = d(x,y)$, are you assuming that $d(x,y)\leq 1$? Otherwise, shouldn't $\pi(\delta_{x}, \delta_{y})$ be the minimum of $d(x,y)$ and $1$? –  Quinn Culver Apr 23 '12 at 12:56
Oh yes, you are right! –  Dirk Apr 23 '12 at 19:25

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