"Riku, what time did you wake up today?"
Yuki asked. In the after-school club room, the three were chatting casually.
"6:30. Unusually early."
"Wow! I'm surprised," Yuki was amazed.
"I was surprised too. Woke up before the alarm."
Aoi interrupted with interest. "Just now, Yuki and Riku shared 'surprise.'"
"Shared?"
"Yuki was surprised by the information 'Riku woke up early.' Riku himself is surprised. So both are feeling information about the same event."
Yuki thought. "Does this have information-theoretic meaning?"
"Very much. It's the concept of mutual information."
Aoi wrote an equation in the notebook. "I(X;Y) = H(X) - H(X|Y)"
"How much information X and Y share."
"Specifically?" Riku asked.
"What can Yuki learn from Riku's behavior. Conversely, what can Riku learn from Yuki's reaction."
"Hmm, abstract."
"Then an example," Aoi continued. "Riku woke up early (X). Yuki was surprised (Y). Are these independent?"
"Not independent," Yuki answered. "Because Riku woke up early, I was surprised."
"Yes. X and Y are correlated. So mutual information is positive."
"But if Riku always woke up early?"
"Yuki wouldn't be surprised. X and Y correlation weakens, mutual information decreases."
Riku understood. "So we could share surprise because it was rare?"
"Precisely. Mutual information measures correlation strength."
Yuki looked at her notebook. "Larger I(X;Y), more information shared."
"Conversely, I(X;Y)=0 means X and Y are independent. Knowing one tells nothing about the other."
Riku gave an example. "My waking up early and today's weather are independent."
"Yes. Unrelated."
"But," Yuki said, "Riku waking up early and having a test?"
"Related!" Riku admitted. "Before tests, probability of waking up early increases."
"Then mutual information is positive. If Yuki knows 'Riku woke up early,' she can infer 'there's a test.'"
"Information is transmitted."
Aoi supplemented. "In communication theory, this mutual information becomes the upper limit of information transmitted through channels."
"Shannon's channel capacity," Yuki recalled.
"Correct. C = max I(X;Y). Maximize mutual information between transmission X and reception Y."
Riku asked. "Can you also measure mutual information in human relationships?"
"Interesting perspective," Aoi showed interest.
"Closer friends have larger mutual information. From one person's behavior, you can infer the other's feelings."
Yuki continued. "Conversely, first meetings have mutual information near zero."
"So through conversation, exchange information. The process of increasing mutual information is relationship building."
Riku thought seriously. "Deep. 'Understanding each other' means raising mutual information?"
"As one interpretation, yes."
Yuki laughed. "We three probably have high mutual information already."
"Yeah. From Yuki's way of speaking, I can mostly tell what she's thinking," Riku said.
"Riku too. Your behavior patterns are becoming readable."
"Predictable? Is my entropy dropping?"
"Entropy doesn't change. But conditional entropy H(X|Y) is dropping."
Aoi explained. "Riku's behavior X is still random, but uncertainty after Yuki observes Riku decreases."
"I see. Mutual information I(X;Y)=H(X)-H(X|Y), so the amount uncertainty decreased by observation is mutual information."
"Perfect understanding."
The three nodded satisfactorily.
"Today too, we shared surprises," Yuki said.
"Accumulating mutual information," Riku laughed.
Outside the window, birds sang. Shared surprises become bonds. Information connects people.