"You two really click, don't you?"
Riku said while looking at Yuki and Aoi.
"You think so?" Yuki tilted their head.
"No, seriously. You often think the same thing at the same time."
Aoi laughed with interest. "That can be explained with information theory."
"Information theory again," Riku laughed.
"There's a concept called mutual information," Aoi opened their notebook. "It measures how much shared information two variables have."
Yuki showed interest. "Shared information?"
"Yes. If variables are independent, mutual information is zero. If completely dependent, it becomes large."
Aoi drew a diagram.
"For example, Riku's lateness and weather. If completely unrelated, knowing the weather doesn't help predict Riku's lateness."
"Actually, I'm more likely to be late on rainy days," Riku admitted.
"So there's correlation. The strength of that correlation is mutual information."
Yuki wrote in their notebook. "Is there a formula?"
"I(X;Y) = H(X) - H(X|Y)
X's entropy minus X's conditional entropy after knowing Y"
"Conditional entropy?"
"The uncertainty of X that remains after observing Y. If Y tells us a lot about X, conditional entropy becomes small."
Riku pondered. "So what's the mutual information of my and Yuki's behavior?"
Aoi laughed. "Worth measuring. If you two have similar behavior patterns, mutual information is high."
"How do you measure it?" Yuki asked.
"Gather actual data and take statistics. For example, times you read the same book simultaneously, percentage of topics where you had similar opinions, etc."
Riku took out his smartphone. "Location data could work too."
"There are privacy concerns though," Aoi cautioned. "But as a concept it's interesting. Close friends should have high mutual information."
Yuki suddenly thought of something. "Do completely independent people exist?"
"Complete independence is difficult. Living in the same society, same culture, some correlation naturally emerges."
Riku said. "What about Mira? Her behavior pattern doesn't match anyone's."
"Mira is special," Aoi acknowledged. "Maybe she's learning from different information sources than us."
Yuki asked with interest. "Can mutual information be an indicator of friendship?"
"Interesting perspective. High mutual information shows shared understanding. But that's not all friendship is."
"What do you mean?"
"Two people thinking completely identically have high mutual information, but don't exchange new information. Diversity matters too."
Riku nodded. "True. The three of us are totally different types, but maybe that's why it's interesting."
Aoi supplemented. "Moderate correlation and moderate independence. That's a well-balanced relationship."
Yuki reviewed their notebook. "I(X;Y) = I(Y;X), it's symmetric."
"Yes. Mutual information has symmetry. Information from X to Y is the same as from Y to X."
"Different from KL distance," Riku remembered.
"Good memory. Mutual information measures true informational overlap."
Outside the window, two birds took flight at the same moment.
"Those birds seem to have high mutual information," Riku pointed.
"Birds flying in flocks do indeed have high correlation," Aoi said.
Yuki smiled. "We're also sharing invisible information."
"Yes. That might be the meaning of being together."
The three quietly felt a common "surprise."