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Suppose that I have two random processes. Process $X$ has probability 0.5 to be in state A and probability 0.5 to be in state B. Process $Y$ has probability 0.4 to be in state A and probability 0.6 to be in state B.

State B has two substates: $B_1$ and $B_2$. For both processes, the two substates are equally likely.

Now if I choose to only consider the states $A$ and $B$, then process $X$ has higher Shannon entropy. But if I consider states $A$, $B_1$ and $B_2$, then process $Y$ has higher Shannon entropy.

What is the correct way to chose the states for which you calculate the Shannon entropy?

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Shannon entropy computes the average number of bits needed to represent the outcomes.

Your first probability distributions are: D1: 0.5, 0.5 and D2: 0.4, 0.6, the first of which has higher shannon entropy (maximal in fact).

In the second situation your distributions are: D3: 05, 0.25 and 0.25 and D4: 0.4, 0.3 and 0.3.

The situation is now reversed: fewer bits are needed to represent a process with outcomes following distribution D3 (0.5, 0.25 and 0.25).

The intuition is that a more optimal encoding (using bits) exploits the highest frequencies (= highest probabilities of occurrence) equipping those outcomes with shorter codes (i.e. shorter binary numbers) so that the overall average encoding cost (the average number of bits used) is lower.

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