Mutual Information as probability Could the mutual information over the joint entropy:
$$
 0 \leq \frac{I(X,Y)}{H(X,Y)} \leq 1$$
be defined as:"The probability of conveying a piece of information from X to Y"?
I am sorry for being so naive, but I have never studied information theory, and I am trying just to understand some concepts of that.   
 A: Here is the definition of a probability space. Let us use the notations there. IQR is a function of a tuple $(\Omega,\mathscr F,P,X,Y)$ (The first three components form the probability space the two random variables are defined on). A probability measure has to be a set function that satisfy all the conditions of the definition listed in Tim's answer. One will have to specify $\Theta:=(\Omega,\mathscr F,P,X,Y)$ as some subset of a set $\tilde\Omega$. Moreover, the set of $\Theta$'s has to form a field of subsets of $\tilde\Omega$, and that $\text{IQR}(\Omega,\mathscr F,P,X,Y)$ has to satisfy all three properties listed in the definition of probability measure listed in Tim's answer. Until one constructs such an object, it is wrong to say IQR is a probability measure. I for one do not see the utility of such a complicated probability measure (not the IQR function itself but as a probability measure). IQR in the paper cited in Tim's answer is not called or used as probability but as a metric (The former is one kind of the latter, but the latter is not one kind of the former.).
On the other hand, there is a trivial construction that allows any number on $[0,1]$ to be a probability. Specifically in our case, consider any given $\Theta$. Pick a two-element set as the sample space $\tilde\Omega:=\{a,b\}$, let the field be $\tilde{\mathscr F}:=2^{\tilde\Omega}$ and set the probability measure $\tilde P(a):=\text{IQR}(\Theta)$. We have a class of probability spaces indexed by $\Theta$.
A: Going back in history a bit, the role of $\frac{I(X,Y)}{H(X,Y)} $ as a measure of probability can be seen, in part, in the 1961 article by Rajski: A Metric Space of Discrete Probability Distributions.  This article outlines the development of the Rajski Distance ${(D_R)}$ is:
$${D_R}=1 - \frac{I(X,Y)}{H(X,Y)} $$
A: The measure you are describing is called Information Quality Ratio [IQR] (Wijaya, Sarno and Zulaika, 2017). IQR is mutual information $I(X,Y)$ divided by "total uncertainty" (joint entropy) $H(X,Y)$ (image source: Wijaya, Sarno and Zulaika, 2017).

As described by Wijaya, Sarno and Zulaika (2017),

the range of IQR is $[0,1]$. The biggest value (IQR=1) can
  be reached if DWT can perfectly reconstruct a signal without losing of
  information. Otherwise, the lowest value (IQR=0) means MWT is not
  compatible with an original signal. In the other words, a
  reconstructed signal with particular MWT cannot keep essential
  information and totally different with original signal
  characteristics.

You can interpret it as probability that signal will be perfectly reconstructed without losing of information. Notice that such interpretation is closer to subjectivist interpretation of probability, then to traditional, frequentist interpretation.
It is a probability for a binary event (reconstructing information vs not), where IQR=1 means that we believe the reconstructed information to be trustworthy, and IQR=0 means that opposite. It shares all the properties for probabilities of binary events. Moreover, entropies share a number of other properties with probabilities (e.g. definition of conditional entropies, independence etc). So it looks like a probability and quacks like it. 

Wijaya, D.R., Sarno, R., & Zulaika, E. (2017). Information Quality Ratio as a novel metric for mother wavelet selection. Chemometrics and Intelligent Laboratory Systems, 160, 59-71.
