"You two get along well, senpai and Riku."
Yuki said while watching them.
"Really?" Riku tilted his head.
Aoi answered. "Probably because mutual information is high."
"Another difficult topic?" Riku smiled wryly.
"It's simple. We share a lot of each other's information. That's mutual information."
Yuki showed interest. "Can closeness be measured by information sharing?"
"To some extent. I(X;Y), how much of X's information is contained in Y."
Aoi drew a diagram on the whiteboard.
"Two circles overlapping. That overlap is mutual information."
"The bigger the overlap, the more commonalities?" Yuki asked.
"Yes. Shared experiences, knowledge, understanding. All increase mutual information."
Riku thought. "So with strangers, mutual information is zero?"
"Theoretically close to zero. But it increases through conversation."
Yuki wrote in the notebook. "Becoming friends means increasing mutual information?"
"Beautiful interpretation," Aoi nodded. "Understanding each other is information exchange."
Riku asked. "But if you know everything, wouldn't it be boring?"
"Sharp point," Aoi was impressed. "Complete mutual information has no surprise."
"Surprise?"
"The essence of information is unpredictability. If completely predictable, information is zero."
Yuki understood. "So having a bit of mystery is more interesting?"
"You could say that. Optimal mutual information is neither complete nor zero."
Aoi wrote equations.
"I(X;Y) = H(X) - H(X|Y)"
"The uncertainty reduced when knowing Y. That's mutual information."
Riku gave an example. "I can predict about half of Aoi-senpai's actions."
"That's moderate mutual information," Aoi laughed. "If everything was predicted, it wouldn't be fun."
Yuki asked seriously. "So best friends have maximum mutual information?"
"Not necessarily. Closeness and mutual information don't perfectly align."
"What do you mean?"
"Trust also depends on information quality. Not just quantity."
Aoi drew another diagram.
"Surface information: Weather, hobbies, daily life Deep information: Values, fears, dreams"
"Sharing deep information creates higher intimacy."
Yuki nodded. "That's why even long-time friends may not know each other deeply."
"Yes. Even with high mutual information, important parts may not be shared."
Riku looked outside. "How much mutual information do we have?"
"Measurement is difficult," Aoi admitted. "But the fact we can have this conversation is evidence of it being reasonably high."
"That's a nice thing to say," Riku said shyly.
Yuki laughed. "I want to keep increasing our mutual information."
"Good goal. But trying to increase it too rapidly doesn't work well."
"Why not?"
"Take time and gradually increase overlap. That's natural relationship building."
Riku pondered. "Information theory can be applied to human relationships too."
"Information theory is also the science of relationships," Aoi answered.
Yuki closed the notebook. "I think I understand a bit about the definition of closeness."
"It can't be defined, but there are metrics to measure it. That's mutual information."
The three quietly left the classroom. Sharing each other's information, deepening understanding.
The journey to increase mutual information continues.