"Saying two people's feelings are correlated is a strange expression."
Yuki said while laughing.
"But statistically accurate," Aoi answered.
Mira quietly supplemented. "Correlation is the degree two variables move together."
"Move together?"
"When one increases, the other also increases. Or decreases," Aoi explained.
Yuki thought. "So close friends have correlated feelings?"
"Can interpret it that way. When one is happy, the other becomes happy too."
"That's empathy."
"Empathy is positive correlation of emotions."
Mira spoke. "Correlation coefficient ranges from -1 to 1."
"What's -1?" Yuki asked.
"Perfect negative correlation. When one goes up, the other always goes down."
"Opposing relationship?"
"Could say that," Aoi nodded. "Conversely, 1 is perfect positive correlation. Complete synchronization."
"And 0?"
"No correlation. Mutually independent."
Yuki pondered. "Completely independent relationships feel lonely, don't they?"
"Information-theoretically, zero mutual information," Aoi explained.
"Knowing one tells nothing about the other."
Mira supplemented. "But independence isn't bad. Healthier than excessive dependence maybe."
"Balance," Yuki understood.
Aoi continued. "Correlation and causation are different. This is important."
"How different?"
"Correlation is changing together. Causation is one causing the other."
"For example?"
"Ice cream sales and drowning deaths correlate. But one doesn't cause the other."
Yuki was surprised. "Then what's the cause?"
"Summer, a common factor. Called confounding variable."
Mira added. "In human relationships too, apparent correlation can deceive."
"What do you mean?"
Aoi explained. "Two people's similar behavior might actually be a common friend's influence."
"Not direct correlation but indirect?"
"Right. Statistical dependence is complex."
Yuki asked seriously. "So how do you identify true correlation?"
"There's a field called causal inference. But it's difficult."
"Why?"
"Can't experiment on humans. Can't do randomized controlled trials."
Mira said quietly. "So rely on observation and inference."
Aoi nodded. "Make best estimates from incomplete information."
Yuki thought for a moment. "Like detectives piecing together clues."
"Good analogy. We're all detectives of human behavior in a way."
"Trying to understand patterns without perfect information," Riku added.
Yuki thought. "Information theory and probability are sciences of inference."
"Beautiful expression," Aoi smiled.
"No complete certainty. But can find statistically meaningful relationships."
Yuki wrote in her notebook. "Correlated feelings of two people. Not perfect, but connected."
"Yes. Neither complete independence nor complete dependence."
Mira smiled unusually. "Moderate correlation creates comfortable relationships."
"Not too strong, not too weak."
Aoi summarized. "Relationships have parts measurable by correlation coefficient and parts unmeasurable."
"Unmeasurable parts?"
"Quality of emotions, history, expectations for the future. Things that don't become numbers."
Yuki nodded. "But correlation as a concept provides a clue for thinking about relationships."
"Yes. Not a perfect tool, but provides useful perspective."
The three smiled quietly. Correlation is one aspect of relationships, but an important aspect.