"What's the maximum entropy principle?"
Riku suddenly asked. Unusually serious for someone normally casual.
"The idea of choosing the distribution that maximizes entropy while following constraints," Aoi answered.
"Why maximize it?" Yuki asked.
"To make the most unbiased prediction when you have little information."
Aoi drew a diagram on the whiteboard.
"For example, suppose you know a die's average value is 3.5."
"That's normal for a die," Riku said.
"But knowing just the average of 3.5, you don't know the probability of each face."
"So what do you do?" Yuki showed interest.
"Under this constraint, maximize entropy. Then you get uniform distribution—1/6 for each face."
"That's obvious," Riku said.
"No, it's important," Aoi countered. "There's a theoretical basis for why we assume uniform distribution."
Aoi gave another example.
"What if you know the average value is 3?"
"Not uniform distribution?" Yuki pondered.
"Correct. The maximum entropy distribution takes the form of an exponential distribution."
Riku looked confused. "It changes with the average?"
"Different constraints mean different optimal distributions. That's what's interesting about the maximum entropy principle."
Aoi showed calculations.
"When mean and variance are fixed, the maximum entropy distribution is normal distribution."
"Oh, I heard that before," Yuki recalled.
"Yes. So normal distribution is the 'safest' choice when you have little information."
Riku looked out the window. "Should we live life at maximum entropy too?"
"What do you mean?" Aoi asked.
"There are constraints, but within them, most freely, without bias."
Yuki laughed. "That's philosophical."
"But not a bad way of thinking," Aoi admitted. "Accept constraints while maximizing possibilities."
"Like equilibrium," Yuki said.
"Exactly. Thermal equilibrium is also the state of maximum entropy."
Riku asked another question. "So even romance at maximum entropy?"
Yuki blushed slightly.
"Romance has constraints," Aoi answered calmly. "Time, distance, social norms."
"Within those, maximize entropy?"
"In the sense of keeping possibilities broad, maybe."
Yuki spoke up. "But in romance, focusing on one person is more..."
"That's adding a constraint," Aoi explained. "You're optimizing under the constraint of 'I like this person.'"
"I see," Riku understood. "More constraints change the distribution."
"At each stage of life, constraints change. Accordingly, the optimal way of living changes too."
Aoi continued quietly.
"Maximum entropy is also the humility of 'admitting what you don't know.'"
"Not knowing?" Yuki asked.
"Use only the information you have, and don't bring bias into what you don't know."
"Fair judgment," Riku understood.
"Yes. In Bayesian inference too, maximum entropy distributions are often used as prior distributions."
Yuki summarized in her notebook. "The most unbiased state while satisfying constraints."
"That's maximum entropy. The mathematical expression of equilibrium, fairness, and humility."
The three fell silent for a while.
"Senpai," Yuki said. "Maybe we're also living at maximum entropy."
"Within the constraints of school, clubs, and studies," Riku continued.
"Yet possibilities are expanding," Aoi smiled. "That's why each day is fresh."
Sunset illuminated the club room. Within their constraints, they found maximum freedom.