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.

**Answer**

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.

**Attribution***Source : Link , Question Author : luca maggi , Answer Author : Tim*