"Aoi-senpai, I have a question."
At Yuki's words, Aoi turned around.
"That question seems to have high mutual information."
"Mutual information?"
"An indicator measuring how much information two variables share."
Mira quietly approached and wrote an equation in her notebook.
"I(X;Y) = H(X) - H(X|Y)"
"Mutual information shows how much X's uncertainty decreases by knowing Y," Aoi explained.
Yuki thought. "So the stronger the relationship, the greater the mutual information?"
"Precise. If completely independent, mutual information is zero. If completely identical, it equals entropy."
Mira wrote another example. "Weather and umbrella"
"Good example," Aoi continued. "Weather and umbrella use. These have high mutual information. If it rains, probability of using an umbrella increases."
Yuki wrote in the notebook. "Mutual information measures correlation?"
"It's broader than correlation. It captures any statistical dependence, not just linear relationships."
"For example?"
Aoi drew a diagram.
"Even if X and Y have no linear relationship, if there's nonlinear dependence, mutual information is high."
Mira added. "Correlation ⊂ Mutual Information"
"Correlation is a special case of mutual information," Aoi explained.
Yuki got excited. "Can we use it in conversation?"
"Exactly. Good questions have high mutual information."
"What do you mean?"
"If the mutual information between question Q and answer A is high, asking Q teaches us much about A."
Yuki understood. "That's why you said 'That question seems to have high mutual information.'"
"Yes. Your question strongly relates to my answer. Meaning it becomes beneficial information exchange."
Mira smiled and wrote. "Communication = information exchange"
"Communication is information exchange," Aoi translated. "The higher the mutual information, the more efficient the exchange."
Yuki thought of another example. "So the question 'What day is today?'"
"If the other person already knows the day, mutual information is low. But if they don't know, it's high."
"So the question's value depends on the other's knowledge."
"Correct. This is subjective mutual information."
Aoi drew another diagram.
"In a Venn diagram, mutual information is the overlapping part of X and Y."
"The larger the overlap, the more shared information."
Mira added to the notebook. "I(X;Y) = I(Y;X)"
"Mutual information is symmetric," Aoi explained. "The amount X tells us about Y equals the amount Y tells us about X."
Yuki was impressed. "This is different from KL divergence."
"Sharp observation. KL divergence is asymmetric. But mutual information is symmetric. That's beautiful."
"Like friendship," Yuki said.
Aoi smiled. "Good metaphor. Good friendship involves mutual information exchange. A relationship with high mutual information."
Mira nodded. Unusually, she spoke.
"The three of us also have high mutual information."
Yuki and Aoi were surprised. It was rare for Mira to speak on her own.
"Mira, that's the highest compliment," Aoi said.
Yuki smiled. "It's wonderful to discuss friendship through information theory."
The three laughed. Their mutual information continues to increase today.