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Just a partial answer, but the proposed inequality doesn't hold. Take $p = [0.2, 0.8], q = [0.1, 0.9]$. Then $H(p) = 0.2 \log(5) + 0.8 \log(1/0.8) \approx 0.5$, $H(q) = 0.1 \log(10) + 0.9 \log(1/0. …

Take $\pi^N$ with $AW(\pi^N, \pi) \leq \frac{1}{N}$, where we denote by $\mu^N$ and $\nu^N$ the marginals of $\pi^N$. Note that by the backward induction for $AW$ (cf. here), it holds $$ AW(\pi, \pi^N …

I find the $d$-separation criterion (see, e.g., Theorem 2 here; note however the preceding definition, which basically means we are treating discrete random variables) a really useful sufficient crite …

I think I found an elementary proof of Question 2/3 for arbitrary probability distributions. In fact, it is not required that the components in the sums are squares, but general i.i.d. non-negative ra …

Maybe to add to the point of calculating $\max_{P_\varepsilon} \mathbb{E}_{P_\varepsilon}[f]$: I will write this a bit more in line with the literature I will refer to. Let $(X, d)$ be some polish spa …

We are on $\mathbb{R}^d$ with Borel $\sigma$-algebra. Let $\mu_1, ..., \mu_d$ be probability measures on $\mathbb{R}$ and $\Pi(\mu_1, \mu_2, ..., \mu_d)$ be the set of probability measures on $\mathbb …

Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ endo …