"Aoi-senpai, do you trust Riku?"
At Yuki's sudden question, Aoi thought briefly.
"Difficult question. If trust could be quantified, it'd be easier to answer."
"Quantified?"
"There's a concept called Bayesian inference. Updating beliefs from observed data."
Riku approached. "Talking about me?"
"About trust," Yuki answered.
Aoi opened the notebook. "Consider the probability that Riku keeps promises. First, set a prior probability."
"Like 50 percent initially?"
"For example. But each time we observe Riku's behavior, we update that probability. This becomes the posterior probability."
Riku asked, "So what's my trust rating?"
"Calculating from lateness data..." Aoi tapped the calculator. "About 27 percent."
"Low!"
"But," Aoi continued, "this is updateable. If Riku arrives on time 10 times in a row, trust will surge."
Yuki became interested. "That's Bayesian inference?"
"Yes. P(hypothesis|data) = P(data|hypothesis)・P(hypothesis) / P(data). Bayes' theorem."
"Sounds difficult..."
Aoi explained simply. "Posterior probability is proportional to the product of likelihood and prior probability. Observing data allows rational belief updates."
Riku became serious. "Then I'll prove it. Starting tomorrow, I'll come on time every day."
"That becomes new data," Aoi smiled. "But Bayesian-wise, past data matters too."
"Harsh..."
Yuki said, "But the possibility isn't zero, right?"
"Of course. Bayesian inference's beauty is that any hypothesis retains possibility as long as the prior isn't zero."
Riku pondered. "So friendship is also Bayesian inference?"
Aoi looked slightly surprised. "Interesting perspective."
"Because, initially we don't know each other. But time spent together updates our beliefs about each other."
Yuki nodded. "Good analogy."
Aoi supplemented, "Yes. Initial prior probability might be vague. But with each interaction, posterior probability refines."
"So," Riku asked, "is your trust in me really 27 percent?"
Aoi laughed quietly. "For the specific behavior of lateness, yes. But friendship is multidimensional. There are other factors besides lateness."
"Like what?"
Yuki jumped in. "Like how Riku always helps when someone's in trouble. That must have high posterior probability."
"True," Aoi admitted. "Probably above 90 percent for that trait. See? We maintain different probability distributions for different aspects of a person."
"Probability of helping when in trouble, probability of keeping secrets, probability of being fun together."
Yuki said, "Each has prior and posterior probabilities."
"And overall, a composite belief called 'trust' forms."
Riku smiled. "Somehow, friendship became a complex formula."
"But," Aoi said gently, "Bayesian inference is rational, yet humans are more complex. Intuition and emotion matter too."
Yuki asked, "So what's the difference between information theory and humans?"
"Information theory is an ideal model. But humans have parts that can't be modeled."
Riku said, "So that's why we can be friends?"
"Probably," Aoi nodded. "If completely predictable, it'd be boring."
Yuki smiled. "Uncertainty brings surprise."
"Yes. A relationship with zero entropy is informationally dead."
Riku took out his planner. "But I'll try to become more predictable. From 27 percent to 50 percent."
"Looking forward to it," Aoi said.
Yuki added, "But 100 percent isn't necessary."
"Imperfection is humanness," Aoi acknowledged.
The three discussed probability models and friendship in the sunset club room.
Bayesian inference is a technique for updating beliefs.
But believing itself transcends formulas.