# Can mutual information gain value be greater than 1

I have a very basic doubt. Sorry if this irritates few. I know that Mutual Information value should be greater than 0, but should it be less than 1 ? Is it bounded by any upper value ?

Thanks, Amit.

The mutual information is bounded from above by the Shannon entropy of probability distributions for single parties, i.e. $I(X,Y) \leq \min \left[ H(X), H(Y) \right]$ .
• If the two parties $X,Y$ are binary variables i.e. each has only two possible outcomes {0,1}, then entropies $H(X), H(Y)$ max out at $1$ when $P(X)=0.5$ and $P(Y)=0.5$. Thus, maximum mutual information for two binary variables is $1$ Jul 4, 2019 at 14:43
It depends on whether the alphabet of interest is finite with a known finite cardinality $$K$$, a finite but unknown cardinality $$K$$, or an infinite countable alphabet. If you are talking about mutual information (there is a confusion in names, for example, mutual information, information gain, information gain ratio, etc.), then the answer is YES if $$K$$ is known, NO if$$K$$ is unknown or infinite - mutual information is unbounded on a countable alphabet!
The answer provided above is incorrect because $$I(X,Y)\leq \min(H(X),H(Y))$$ is an incorrect statement. This is easily seen since $$H(X)$$ and $$H(Y)$$ may be arbitrarily large. That said however it must be mentioned that the author of the answer above may be thinking $$K$$ being a known integer.