"Aoi-senpai, I have a question."
Yuki was gazing at a graph showing the relationship between two variables by the club room window.
"What is it?"
"In information theory, how do we express when two variables are completely unrelated?"
Aoi opened the notebook. "There's a concept called mutual information. Written as I(X;Y). It measures how much information X and Y share."
"Share?"
"Yes. For example, even if I remain silent, if you can sense my mood, there's mutual information there."
Yuki pondered. "But when I can't sense it at all?"
"The mutual information is zero. That means X and Y are statistically independent."
At that moment, Mira quietly entered and wrote an equation on the whiteboard.
"I(X;Y) = H(X) - H(X|Y)"
"Mira's equation is perfect," Aoi nodded. "The entropy of X minus the conditional entropy after knowing Y. In other words, how much X's uncertainty decreases by knowing Y."
Yuki's eyes sparkled. "So the moment when inability to communicate communicates..."
"Is when mutual information goes from zero to positive."
Mira wrote another equation. "I(X;Y) = I(Y;X)"
"It's symmetric!" Yuki was surprised.
"Yes. The information X has about Y equals the information Y has about X. This is different from KL divergence."
Yuki began drawing a diagram in the notebook. "So two people were completely unrelated, but something triggers correlation..."
"Exactly. Mutual information increases. A communication channel opens."
Mira smiled faintly. She always seemed to enjoy these abstract discussions.
Aoi continued. "What's interesting is that mutual information is always non-negative. I(X;Y) ≥ 0. Observing a variable never increases uncertainty."
"Seems obvious, yet profound..."
"There's also something called the data processing inequality. When information is processed in the order X→Y→Z, I(X;Z) ≤ I(X;Y). Information can be lost during processing but never increases."
Yuki sketched a chain in her notebook. "So like a game of telephone? Each person in the chain can only reduce information, never add true information about the original message?"
"Precisely. That's why direct communication is generally better than through intermediaries. Each step of processing is a potential bottleneck."
Yuki's face became serious. "Then what is misunderstanding? In information theory terms?"
Aoi thought briefly. "Misunderstanding is when the receiver has incorrect conditional probabilities. They can't correctly calculate mutual information."
Mira wrote in her notebook. "Channel noise creates wrong mutual information"
"Noise creates false mutual information," Aoi translated.
Yuki looked out the window. "Between people, there's mutual information too."
"Yes. Conversation, expressions, actions. Everything is a signal. And the higher the mutual information, the better we understand each other."
"But complete understanding is..."
"Maybe impossible. But we can work to increase mutual information."
Aoi added thoughtfully, "In machine learning, maximizing mutual information is a common objective. The model tries to extract as much relevant information from the input as possible."
"So even AI studies how to communicate better," Yuki mused.
Mira stood up and handed Yuki a small note.
"Perfect communication = maximum mutual information. But humans are not perfect channels."
Yuki carefully received the note.
"The moment when inability to communicate communicates. It's the moment we realize mutual information isn't zero."
Aoi smiled. "Good understanding. Information theory is also the mathematics of human relationships."
Mira nodded. Among the three, there was certainly positive mutual information.
In the after-school club room, a quiet atmosphere of understanding flowed.