"Aoi-senpai, how well do you understand me?"
Yuki suddenly asked. They were alone in the Information Theory Club room.
"What's this all of a sudden?" Aoi stopped what she was doing.
"No reason, really. It's just that recently, your explanations have been so easy to understand."
Aoi pondered. "Understanding level, huh. That can also be measured with information theory."
"Really?"
"There's a concept called mutual information," Aoi opened her notebook. "It measures how related two random variables X and Y are."
"Random variables?"
"For example, think of it as my words X and your understanding Y."
Yuki began to understand. "What you say and what I understand."
"Right. If completely independent, mutual information is zero. Whatever I say has no effect on your understanding."
"That would be sad."
"Conversely, if perfectly correlated, mutual information is maximum. What I say gets through as is."
Aoi wrote an equation.
"I(X;Y) = H(X) - H(X|Y)
Subtract the conditional entropy of X after knowing Y from the entropy of X."
"Conditional entropy?"
"After knowing Y, how much uncertainty about X still remains."
Yuki thought. "So, after hearing your words, how much doubt still remains in me?"
"Exactly! Mutual information measures the uncertainty reduced by knowing the other."
"Deep..."
Aoi continued. "Human relationships are the same. The more we understand each other, the more mutual information increases."
"But is perfect understanding possible?"
"Theoretically difficult. People each have their own unique context."
Yuki suddenly thought. "Then how can we increase mutual information?"
"Communication. Through repeated dialogue, common context increases."
"Common context?"
"Shared experiences, knowledge, values. The more they increase, the more you can convey with fewer words."
Yuki understood. "So the more I talk with you, the easier your explanations become."
"That might be it," Aoi smiled. "Because you learned the basics of information theory, my explanations became more efficient too."
"Efficient?"
"Can convey the same content with fewer words. Because shared knowledge increased."
Yuki wrote in her notebook. "Mutual information = shared understanding"
"Good interpretation," Aoi acknowledged.
A moment of silence.
"Senpai," Yuki said quietly. "Does this mutual information also apply to friendship and such?"
Aoi looked serious. "I think it applies. Relationships where you deeply understand each other have high mutual information."
"But it can't be measured."
"It can't. But it can be felt."
Yuki nodded. "Sometimes you know without saying anything."
"That's because there's lots of shared information. You can infer much from the other's state."
A bird chirped outside the window.
"I thought information theory was a cold discipline," Yuki murmured.
"It's not?"
"That it can explain human relationships like this. It's warm."
Aoi was a bit embarrassed. "I'm happy you say that."
"I want to keep increasing mutual information with you, senpai."
"Me too."
They both laughed. The distance told by information content. It can be expressed in formulas, but felt in the heart. Today, one more piece of shared information increased.