"What exactly does mutual information measure?"
Yuki asked Aoi. Sunset light streamed through the club room window.
"How related two variables are," Aoi opened her notebook. "How much knowing one reduces the uncertainty of the other."
Mira quietly spoke. "The amount of information shared."
"For example," Aoi continued explaining. "Weather and carrying an umbrella. Knowing it's raining increases the probability of carrying an umbrella."
"So weather information reduces umbrella uncertainty."
"Precisely. That's mutual information. Represented as I(X;Y)."
Yuki thought. "Can it measure human relationships too?"
"Interesting perspective," Aoi smiled. "The more you can predict someone's thoughts, the higher the mutual information."
Mira supplemented. "Long-time friends accumulate mutual information."
"Because time spent together shares information."
Yuki said somewhat sadly. "So with someone you just met, mutual information is zero?"
"Theoretically yes. But it grows through conversation."
"How do you increase it?"
Aoi organized her thoughts. "By asking questions. Observing reactions. Sharing experiences."
"Questions are a means to know the other's state," Mira said.
"Reactions update probability distributions."
Yuki wrote in her notebook. "So asking 'What's your favorite food?' is also an act of increasing mutual information?"
"Right. The answer reveals their preference distribution."
"But," Yuki pondered, "if the answer is as expected, is the information small?"
"Sharp observation. Predictable answers have little surprise. Little information."
Mira said quietly. "That's why unexpected answers deepen relationships."
"Surprise brings new information."
Aoi continued. "Mutual information is symmetric. I(X;Y) = I(Y;X). The information I gain about you equals what you gain about me."
"It's fair."
"Information theory is fair. But human perception might be asymmetric."
Yuki suddenly asked. "How much mutual information do I have with you, Aoi-senpai?"
Aoi laughed. "Hard to measure, but I've been able to predict your questions more often."
"Is that evidence of increasing mutual information?"
"I'd say so. We're starting to have common context."
Mira looked at them and said. "High mutual information relationships are stable."
"Why?"
"Because predictability increases. Surprise decreases."
Aoi supplemented. "But zero surprise might be boring. Some new information is needed."
"Balance," Yuki understood.
"Yes. Balance between known and unknown. That's a healthy relationship."
Mira spoke longer than usual. "Wanting to measure mutual information means wanting to understand someone. That itself is proof of valuing the relationship."
Yuki nodded. "There are things information theory can't measure, but it's interesting as a framework for thinking."
"Mathematics can't directly capture emotions. But it helps understand structure."
"Thank you, Aoi-senpai."
"Ask anytime. Let's increase our mutual information."
The three laughed. Sunset gently illuminated the club room. Perhaps relationships are a process of continuously sharing information.