All Questions
Tagged with optimal-transportation pr.probability
114 questions
1
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0
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256
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Sobolev variant of Wasserstein space
Let $\mathcal{P}(\mathbb{R}^n)$ be the set of Borel probability measures on the Euclidean space $\mathbb{R}^n$ and consider thereof consisting of all probability measures $\mathbb{P}$ satisfying $\int\...
- 5,405
1
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1
answer
1k
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Closed-form upper-bounds for Wasserstein distance between finite measures
Let $x_1,\dots,x_n,y_1,\dots,y_n\in \mathbb{R}$ and such that $x_i\neq x_j$ and $y_i\neq y_j$ if $i\neq j$. Let $a,b$ be elements of the probability n-simplex. Define the measures $\mu\triangleq \...
- 5,405
3
votes
1
answer
989
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About the metrizability of the space of Probability measures $\mathcal{P}(S)$
It is often proved in Books that the space of Probability measures $\mathcal{P}(S)$ on a Polish metric space $(S,\rho)$ endowed with the weak/narrow topology induced by declaring it to be be the ...
- 309
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0
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45
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Transport-type duality for preduals of $C^{k,1}$-functions
Let $\Omega$ be a non-empty, simply connected, and open subset of $\mathbb{R}^d$ for some positive integer $d$. Let $k$ be a non-negative integer. Consider the Banach space $C^{k,1}_0(\Omega)$ ...
- 5,405
1
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0
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71
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Hyperplanes which equalize the Radon transforms of two distributions
Let $p_1$ and $p_2$ be "nice" probability densities on $\mathbb R^m$, for example the densities of a multivariate Gaussians $N(\mu_1,\Sigma)$ and $N(\mu_2,\Sigma)$ with common covariance ...
- 6,853
2
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0
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131
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Eigenvalues of Witten Laplacian induced by log-concave probability measure on manifold
Let $M$ be a closed $n$-dimensional Riemannian manifold and let $\mu=e^{-V}d\mathrm{vol}_M$ be a log-concave probability measure on $M$, such that the pair $(M,\mu)$ verifies the so-called Bakry-Emery ...
- 6,853
28
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1
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6k
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1-Wasserstein distance between two multivariate normal
The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by
$$d_p(\nu_{1},\nu_{2})=\left(\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,...
- 1,256
2
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0
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302
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Simplify Kantorovich–Rubinstein duality when distributions share a common marginal
Consider the product of two metric spaces $X\times Y$, and two probability distributions $\mu$ and $\nu$ on this product space. By the Kantorovich-Rubinstein duality, I can write the Wasserstein-1-...
- 53
1
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1
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100
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$L^p$-barycenters via continuous selectors
Let $\mathbb{P}$ be a Borel probability measure on $\mathbb{R}^n$ with $\int_{x \in \mathbb{R}^n} \|x\|^p\mathbb{P}(dx)<\infty$. For which $p\in [1,\infty)$ (other than $p=2$) is the following ...
- 5,405
4
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0
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117
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Improving log-Sobolev inequalities via quadratic regularisation
Suppose that $\mu(dx) = \exp(-\psi(x)) \, \mathrm{dx}$ is a probability measure on $\mathbf{R}^d$.
For suitable functions $g \geqslant 0$, define
$$\text{Ent}(g) = \int \mu(dx) g(x) \log \left( \frac{...
- 801
1
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1
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241
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Continuity of pushforward operation
Let $X$ and $Y$ be compact metric spaces and let $f,g:X\rightarrow Y$ be $\epsilon$-uniformly close; i.e.:
$$
\sup_{x \in X} d_Y(f(x),g(x))<\epsilon.
$$
Then, are their push-forwards close in ...
- 5,405
2
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1
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173
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Is there an analogue of transportation-cost inequality under a weighted Log-Sobolev Inequality?
It is known that under the Log-Sobolev Inequality for $\pi$, i.e., if for all $\rho$,
$$H_\pi(\rho):=\int \rho(x)\log\frac{\rho(x)}{\pi(x)}dx \leq \frac{1}{2\beta}\int \rho(x)\left\|\nabla \log\frac{\...
- 21
2
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1
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114
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Reweighting probability measures by convex potentials, and contraction in transport distance
Let $W: \mathbf{R}^d \to \mathbf{R}$ be a convex function such that $\int \exp(-W) = 1$, and define probability measures $\mu_y$ by
$$\mu_y (dx) = \exp( - W (x - y)) \,dx,$$
i.e. each $\mu_y$ is a ...
- 801
0
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1
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267
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Intersection of projection of sets
Suppose that we have two arbitrary sets $\mathcal{X}$, $\mathcal{Y}$ and are given a function $c : \mathcal{X} \times \mathcal{Y} \rightarrow \mathbb{R}$
Consider the following inequality for ...
- 143
0
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1
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86
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Is integration against an indicator Wasserstein-Continuous
Let $\mathcal{P}_p(X)$ denote the Wasserstein space over a compact metric space $X$, and $1\leq p<\infty$. Fix a non-empty closed subset $C\subseteq X$. Then is the map:
$$
\mathbb{P} \mapsto \...
- 5,405
2
votes
1
answer
1k
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Wasserstein interpolation between two probability measures on a metric space
Question 1
Given probability measures $\mu$ and $\nu$ on the same metric space $X=(X,d)$, and $\alpha \in [0, 1]$, is it always possible to find another probability measure $\lambda_\alpha$ on $X$ ...
- 6,853
0
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0
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95
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Empirical estimation of Brenier map from data
Let $f:\mathbb R^d \to \mathbb R$ be a "nice" (say, continuous) function define $A = A_f := \{x \in \mathbb R^d \mid f(x) \ge 0\}$ and $B =B_f:= \{x \in \mathbb R^d \mid f(x) \le 0\}$, and ...
- 6,853
1
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2
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889
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Simplify Wasserstein distance between Gaussians with binary cost function
Let $\mu_1$ and $\mu_2$ be 1D gaussian distributions with means $m_1$ and $m_2$ respectively and common variance $\sigma$. Let $\Omega$ be a closed subset of $\mathbb R^2$, and consider the cost ...
- 6,853
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0
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123
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Closure of finite support measures in the Wasserstein metric
This is a follow-up question to this question:
Closure Wasserstein for pointmasses
Let $(X,d)$ be a metric space, and let $W_1(X)$ be the space of probability measures $\mu$ on $X$ having finite first ...
- 11.1k
2
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1
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228
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Ideas on how to prove Pythagorean identity involving Wasserstein distances?
I conjectured earlier that if $P$ and $Q$ were two probability measures, then we could show
$$W^2(P,Q) = \min_{T} [d^2(P,T_{\#}P) + W^2(T_{\#}P,Q)]$$ where $W^2(P,Q)$ denotes the squared Wasserstein-2 ...
- 383
4
votes
2
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415
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Effect of perturbing the atoms of a measure on the Wasserstein distance
Let $(X,d)$ be a metric space, $x_1,\ldots,x_N\in X$ and $x_1',\ldots,x_N'\in X$ be atoms, and $G=\sum_{i=1}^Np_i\delta_{x_i}$, $G'=\sum_{i=1}^Np_i'\delta_{x_i}$, and $G''=\sum_{i=1}^Np_i'\delta_{x_i'}...
- 710
1
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0
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96
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Relationship between Wasserstein projections and metric projections in a linear space
Let $(X,d)$ be a metric space, $x\in X$, and $Y\subset X$ is a closed set. Assume that $X$ is also a real vector space, so that we can form linear combinations over $\mathbb{R}$, but I do not assume ...
- 13
2
votes
1
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289
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On semi-discrete Wasserstein distance
Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$, where $\nu$ has a bounded support. Consider the $2-$Wasserstein distance below:
$$...
- 345
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0
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87
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How does one define weak convergence of probability measures in $L^{\infty}(\Omega)$?
I am reading the following article and on page 9/17 (above Eqn (4.9)) the authors state that if $\gamma_{\epsilon_k}|\_G_{\delta}\times \Omega\to \gamma|\_G_{\delta}\times \Omega$ as $\epsilon_k\to 0$ ...
- 547
0
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1
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141
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Arbitrarily bad rates of convergence in Wasserstein metric
Suppose $W_p(\mu_n,\mu)\to 0$ and $d(E(\mu_n),E(\mu))<r_n$. Here, $W_p$ is the $p$th-order Wasserstein distance (with respect to the metric $d$) and $\mu_n,\mu$ are probability measures on some ...
- 13
5
votes
1
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396
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Universal decay rate of the Fisher information along the heat flow
I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation
$$
\partial_t u=\Delta u
$$
...
- 5,380
4
votes
1
answer
597
views
Monge-Kantorovich duality with a $\{0,1\}$ cost function
Consider the usual Monge-Kantorovich transportation problem where $X$ and $Y$ are Polish spaces, $\mu$ and $\nu$ are probability measures on $X$ and $Y$, and $c:X\times Y \to \mathbb{R}^+ \cup \{+\...
- 4,059
3
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0
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163
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A new "adversarial" Wasserstein distance?
Let us consider $\mu_1, \mu_2$ and $\mu_3$ three probability measures living on $[0,1]^{k_1}, [0,1]^{k_2}$ and $[0,1]^k$respectively, with $k_1 +k_2=k$. Let us denote by $\Gamma(\mu,\nu)$ the set of ...
- 565
8
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3
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936
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Question about Wasserstein metric
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^n$ with finite first moment. Denote by $d:=W_1(\mu,\nu)$, where $W_1(\cdot,\cdot)$ stands for the Wasserstein distance of order $1$.
My ...
- 345
1
vote
1
answer
1k
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Monotonicity, Convexity, and Smoothness of the KL-Divergence between Two Brownian Motions with Different Initializers
We consider the two distributions
$$
p_t = p_0 * N(0, tI),\quad q_t = q_0 * N(0, t I),
$$
where $*$ denotes the convolution between two densities, while $p_0$ and $q_0$ have the same mean and ...
- 1,127
0
votes
1
answer
223
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A problem with the dual form of semi-discrete optimal transport
Consider the uniform distribution $\lambda$ on $[0,1]$, and a point measure $\rho$ with density $\frac{1}{2} (\delta_{x_1} + \delta_{x_2})$, where we have $0\le x_1 \le x_2 < 1/2$.
If our cost is ...
7
votes
2
answers
3k
views
The largest Wasserstein distance to uniform distribution among all probability distributions with uniform marginals
I am looking for the largest Wasserstein distance to the uniform distribution among all probability distributions with uniform marginals.
More specifically, let $\Xi=\{1,2,\ldots,N\}^2$, and let $\nu$...
- 422
3
votes
2
answers
165
views
continuity/ measurablity of optimal transport
given polish space $(X,d)$, consider weak* topology of probability. optimal transport of probability $u,v$ is defined by $\pi(u,v)$ such that $\pi(u,v)$ minimizes:
$\{\int d(x,y) d \pi(dx,dy): \pi \...
- 553
4
votes
1
answer
2k
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Earth movers distance (EMD) between two multivariate normals. Is it negative definite distance?
I was looking at the closed form formula for 2-Wassersteins distance for multivariate normal distribution on Wikipedia. https://en.wikipedia.org/wiki/Wasserstein_metric#Normal_distributions
It has a ...
- 141
9
votes
3
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2k
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2-Wasserstein (optimal transport) and extension to the set of all signed measures
Consider the 2-Wasserstein distance between probability measures $\mu$ and $\nu$ (on $\mathbb{R}^d$), defined as
$$
d_{W_2}(\mu,\nu) = \inf_{\gamma} \Big[\int \|x-y\|^2 d\gamma(x,y)\Big]^{1/2}
$$
...
- 1,741
0
votes
1
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211
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Relationship between a certain binary optimal transport and total-variation of modified distributions
Let $\mathcal X$ be a Polish space, and let $(N_x)_{x \in \mathcal X}$ be a system of closed neighborhoods in $\mathcal X$. Define $\Omega := \{(x,x') \in \mathcal X^2 \mid N_x \cap N_{x'} = \emptyset\...
- 6,853
1
vote
1
answer
242
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Barycenter Map on Wasserstein Space
Let $(X,d)$ be a complete separable metric space, $P_1(X,d)$ be the set of Radon probability measures on $X$ satisfying
$$
P_1(X,d)\triangleq \left\{
\nu:\,(\exists x_0\in X)\, \int_{x\in X} d(x,x_0)d\...
- 5,405
2
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1
answer
257
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Reference Request: 2-Wasserstein Metric on Wiener Space
Suppose that X is the subspace of the set of probability measures on the classical Wiener space $C[0,T]$, for some $T>0$, comprised of Gaussian measures.
In the finite-dimensional setting, the ...
- 5,405
2
votes
1
answer
194
views
Strong convexity of internal energy with respect to Wasserstein metric
It is well known that the internal energy (see, e.g., Definition 3.32 in and Proposition 3.33 in 1) is geodesically convex with the 2-Wasserstein distance. I was wondering under what condition, the ...
- 422
1
vote
0
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56
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Moduli of continuity and Wasserstein differentiability of functions between measures
Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...
- 11
2
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0
answers
328
views
Explicit formula for this distance between positive semi-definite matrices?
Let $A$ and $B$ in $\mathbb{R}^{d\times d}$ be positive semi-definite (psd) matrices and let $d\tau$ be the uniform probability distribution on the unit sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^d$. I ...
1
vote
0
answers
113
views
Metrics on the space of distributions in terms of p.d.fs
If two probability distributions (on the same measure space) are s.t they have p.d.fs and the $L^1$ distance between the p.d.f.s is large, then is there a choice of a ``nice" metric $d_{\rm ...
- 2,246
3
votes
1
answer
311
views
Kantorovich duality with pseudometrics
The usual framework for the Kantorovich duality in optimal transport theory uses Polish spaces as ground spaces for the distributions that should be transported. Are there results available that ...
3
votes
2
answers
758
views
Multi-marginal optimal transport
The notion of Wasserstein distance between two probability measures is well-studied and well-motivated in many different branches of math and stat.
Let $\mu$ and $\nu$ be any two probability measures ...
- 1,109
4
votes
0
answers
757
views
Tangent space and gradient on subspace of Wasserstein space given by finitely supported measures
Let $\mathcal{P}_2(M)$ be the 2-Wasserstein space over some Riemannian manifold $(M,g)$ (connected, complete, and without boundary). Let $\mathcal{FP}_2(M,n)$ be the subspace of probability measures ...
- 1,675
10
votes
1
answer
274
views
Cutting a Gaussian in two pieces that are maximally separated in the Wasserstein metric
Denote the standard Gaussian probability measure on $\mathbb R^n$ by $\gamma$. We partition $\mathbb R^n$ into two sets $A$ and $A^c$ such that $\gamma(A) = \gamma(A^c) = 1/2$.
Denote by $\gamma_{A}$...
- 1,034
5
votes
0
answers
244
views
Distribution of point knowing target in optimal matching
I am a young PhD student in statistics.
Recently, papers (Ambrosio, Stra and Trevisan; Talagrand; Ledoux to cite but a few) tackled the problem of finding the expected cost in an optimal matching, ...
- 565
3
votes
1
answer
170
views
Reformulation as optimization on probability distributions
This is a "soft" question, in the sense that I'm looking for historical remarks and general commentary rather than a definite answer.
For compact $X \in R^n$ and $f : R^n \to R$ consider the problem
...
3
votes
1
answer
752
views
Wasserstein convergence of conditional measures
Suppose $W_r(\mu_n,\mu)\to0$, where $\mu_n$ and $\mu$ are discrete probability measures on some metric space $\Omega$, and that all measures have the same number of atoms $d$ (but not the same atoms):
...
- 710
3
votes
1
answer
202
views
Is there a coupling that induces a given coupling via a transition kernel?
Let $X,Y$ be two measurable spaces, $\mu,\nu$ two probability measures on $X$, and $\kappa$ a transition kernel from $X$ to $Y$.
Define $\tilde\mu(dy)=\int_X\kappa(dy|x)\mu(dx)$ and $\tilde\nu(dy)=\...
- 1,675