"Where is this relationship heading?"
Yuki murmured absently. Eating lunch on the rooftop.
"Whose relationship?" Riku asked curiously.
"That's a secret."
Aoi said quietly, "Predicting uncertain relationships is a conditional probability problem."
"Conditional probability?"
"Yes. P(Y|X). Probability of Y given X."
Yuki thought. "How is it related?"
"For example, observe someone's behavior X to predict their feelings Y."
"But you can't predict perfectly, right?"
"Of course. That's why conditional entropy H(Y|X) remains."
Aoi wrote a formula in the notebook. "H(Y|X) = H(X,Y) - H(X)"
"What does this mean?" Riku asked.
"After knowing X, the uncertainty of Y that remains."
"In other words, the part you still don't know even after observation."
Yuki understood. "So that's why it's uncertain."
"Yes. But observation reduces uncertainty."
Aoi continued explaining. "Initially, entropy of Y is H(Y). You know nothing about their feelings."
"But observing conversation and behavior X reduces it to H(Y|X)."
"The reduced part is?"
"Mutual information I(X;Y). Information shared between X and Y."
Riku asked for an example. "Specifically?"
"For example, the person often smiles (X). Observing that increases probability of affection (Y)."
"But it's not certain," Yuki said.
"Exactly. They might just be sociable. So conditional entropy remains."
"Then how can you be certain?"
Aoi smiled. "Gather more information. Observe multiple X's."
"Like?"
"Smiles (X1), contact frequency (X2), eye contact (X3)."
"With more observations, conditional entropy decreases."
Riku got excited. "So theoretically you can predict with certainty?"
"No," Aoi shook his head. "Human hearts have unobservable parts."
"Unobservable?"
"Internal state. Emotions they're not aware of themselves."
Yuki said thoughtfully, "So no matter how much you observe, uncertainty remains."
"Yes. That might be the essence of human relationships."
Riku asked, "Then what's the point of prediction?"
"Not perfect prediction, but estimating probability distribution."
"Probability distribution?"
Aoi explained. "Like 60 percent probability they have affection, 40 percent see you as a friend."
Yuki laughed. "That's vague."
"But that's realistic. Can't always make things black and white."
Riku asked seriously, "How do you deal with that ambiguity?"
"Accept conditional entropy," Aoi answered.
"Accept?"
"Acknowledge uncertainty exists and act within it."
Yuki pondered. "But isn't that anxious?"
"It is. But that doesn't have to prevent relationship progress."
Aoi continued. "In information theory, reducing entropy is the goal, but in life it's different."
"Different?"
"Moderate uncertainty enriches relationships."
Riku understood. "If you could predict everything, it'd be boring?"
"Exactly. No room for surprises or growth."
Yuki looked out the window. "It's interesting because it's uncertain."
"Yes. And you gradually reduce conditional entropy over time."
"That's the process of deepening relationships."
Riku laughed. "Information theory is useful for love."
"Useful, but that's not everything," Aoi warned.
"I know. The part that can't be reduced to formulas is important."
Yuki said quietly, "Maybe it's fresh every day because it's unpredictable."
"Good realization."
"Enjoying conditional entropy."
Aoi nodded. "If you can do that, uncertainty isn't an enemy."
The three looked up at the sky. The future is unpredictable. But that's why it's full of possibilities.
"Uncertain relationships aren't bad," Yuki smiled.
"Living with entropy," Riku said.
Aoi answered quietly, "That's life."
Conditional probability is a tool. But the heart is free. The three continue sailing the sea of uncertainty.