"Yesterday I explained something to Riku, but it didn't get through at all."
Yuki was dejected.
"What kind of explanation?" Aoi asked.
"About encoding. But Riku confused it with encryption."
Aoi wrote an equation in the notebook. "There's a concept called mutual information. I(X;Y). How much information two variables share."
"Shared information...?"
"Yes. How much does knowing X reduce the uncertainty about Y?"
Mira quietly approached and drew a diagram. A Venn diagram with two overlapping circles.
"The overlapping part is mutual information," Aoi continued explaining. "X's entropy, Y's entropy, and the shared part."
Yuki began to understand. "So the mutual information between my explanation and Riku's understanding was low?"
"Probably. If completely independent, mutual information is zero. If completely dependent, it equals one's entropy."
"How can we increase mutual information?"
Aoi thought. "Create common context. Confirm prior knowledge. Ask for feedback."
Mira showed a note. "I(X;Y) = H(X) - H(X|Y)"
"Right. X's entropy minus X's conditional entropy given Y. How much X's uncertainty decreased after knowing Y."
Yuki wrote in the notebook. "H(X|Y) is X's uncertainty after knowing Y..."
"Exactly. If Y tells us nothing about X, then H(X|Y) = H(X). Mutual information is zero."
"In Riku's case, even after hearing my explanation, the uncertainty about encoding didn't decrease."
"That's it. But it's not just Riku's fault. The way of explaining also matters."
Mira wrote a new equation. "I(X;Y) = I(Y;X)"
"Mutual information is symmetric," Aoi supplemented. "The amount X teaches about Y equals what Y teaches about X."
Yuki's eyes sparkled. "So it's a problem for both the sender and receiver?"
"Exactly. Communication is bidirectional."
"Then what does perfectly getting through mean?"
"When X = Y. Mutual information equals H(X). Complete redundancy."
Riku entered the club room. "What are you talking about?"
"Mutual information," Yuki answered.
"Oh, yesterday's thing. Actually, I looked it up later. I figured out encoding and encryption are different."
Yuki was surprised. "Really?"
"Yeah. Encoding is for efficiency, encryption is for secrecy. Different purposes."
Aoi smiled. "Riku, your mutual information increased."
"Huh?"
"More shared knowledge with Yuki than yesterday. That's mutual information."
Riku laughed. "Then what's the mutual information among us three?"
"I(X;Y;Z). Mutual information of three variables. More complex, but same principle."
Mira drew a diagram on the whiteboard. Three circles overlapping in a complex way.
"The center part is information all three of us share," Aoi pointed.
Yuki pondered. "So are club activities activities to increase mutual information?"
"Good metaphor. Common experiences, common knowledge. That's what grows mutual information."
"Even if things don't get through, we shouldn't give up."
"Right. By trying repeatedly, mutual information gradually increases."
Riku said, "I couldn't understand yesterday, but today I feel like I understand a little."
"That's the essence of communication," Aoi said quietly.
Mira smiled and left a small note. "Communication = increasing mutual information"
Yuki read it and nodded. She thought it was getting through, but it wasn't. But that's just the beginning. Mutual information is something to cultivate from now on.