# On the definition of internal and external information protocol

I feel like many places don't explicitly mention the definition of the internal and external information of a protocol (including the original paper in academia introducing it...), and would like to verify my understanding. I thought of posting to theory exchange, but it isn't a research question.

If I understand correctly:

We have a distribution $$u$$ on $$A\times B$$, the inputs.

We also have private for for our two players Alice and Bob, $$R_1,R_2$$

We also have public randomness $$R_p$$.

Finally, there is the random variable that is the transcript of the protocol, $$\pi$$- it contains what bits were sent.

Now consider the randomness over $$u,R_1,R_2,R_p$$, let $$X,Y$$ be the two inputs (so that $$(X,Y)\in A\times B$$). Then the external information is $$H(X,Y;(\pi,R_p))$$ (the random variables is over the large probability space I mentioned above, notice that the private randomness doesn't take part).

Similiarly, the internal information is $$H(X;(\pi,R_p)|Y)+H(Y;(\pi,R_p)|X)$$, again where each summand comes from the probability space mentioned above.