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Theorem 2.1 in the quoted paper of Diaconis and Zabell actually states that boundedness of the ratios $Q(\omega)/P(\omega)$ is a necessary and sufficient condition for obtaining $Q$ from $P$ by condit …

It's true for a very simple reason: the entropy of a dynamical system with respect to a (not necessarily ergodic) invariant measure is the average of the entropies of its ergodic components.

I will begin with a reformulation of your question which makes it not only more symmetric, by also (at least for me) more natural and interesting. I will pass from your variables $(W,H,Q)$ to new vari …

The point is that the nature of these two distributions is completely different. The degree distribution is local: in order to find it one one just has to know how the 1-neighbourhoods of vertices loo …

It can be done in pretty much the same way as for a single vector by using the fact that if you fix $x$, then the conditional distribution of $y-x$ is uniform on the sphere of radius $\sqrt{1-\gamma^2 …

Such a family doesn't exist. If you fix the maximal atom (say, $p$) of a distribution $\mu$ supported by $n$ points, then its entropy is maximal when all the remaining atoms have the same weight $(1-p …

No. Any Markov operator is contracting in the total variation norm, whereas your function $F$ is subject to a much weaker condition of weak continuity. It is easy to construct a counterexample. For in …

The example one should have in mind is the Fubini theorem for the unit square (endowed with the Lebesgue measure) projected onto the horizontal base. The conditional measures are then just the Lebesgu …

To begin with, endowing the set of integers with the upper density is quite far from making it a probability space. Nonetheless, the question you ask still makes sense. Namely, you consider the $G$-va …

Let $f$ be a piecewise constant function which takes value $f_i$ on an interval of length $p_i$, and let $q_i=f_i p_i$. Then $$ H(f) = - \int f \log f\,dx = - \sum q_i \log\frac{q_i}{p_i} = - D_{KL} ( …

The easiest is to look just at the situation when the quotient measures on $G/H$ are purely atomic, so that $$ \mu = \sum_h \alpha_h \mu_h $$ and $$ \nu = \sum_h \beta_h \nu_h \;, $$ where $\alpha$ an …

First, the definition of the KL divergence has nothing to do with compactness as it is defined entirely in terms of the density of one measure with respect to the other one. Second, the KL divergence …

You have too much notation here - which is quite confusing. For instance, what is your "combined probability space" $\Omega$ (it seems to be $X\times Y$ as this is where your Radon-Nikodym derivatives …

Yes - take $Y=-X$ if $X<0$ and $Y$ be "negative gaussian" independent of the value of $X$ if $X>0$. Then $X+2Y=-X>0$ for $X<0$ (i.e., $X+2Y$ is positive at least with probability 1/2); on the other ha …

No. One should realize that the transportation and the total variation distances metrize two quite different topologies. Even if the measures are equivalent (i.e., absolutely continuous with respect t …