"I can't decide..."
Yuki was troubled before three choices. It was a career path survey form.
"Do they all look good?" Aoi asked.
"Opposite. They're all uncertain..."
Riku peered over. "Science, humanities, undecided. I chose 'undecided'!"
"That's just not deciding," Yuki smiled wryly.
At that moment, Professor S. entered the club room. An earlier visit than usual.
"Seems you're troubled about handling uncertainty."
"Professor, were you listening?"
"In the hallway. Information theory has ways to mathematically handle uncertainty."
Aoi opened the notebook. "Probability distributions, right?"
"Yes. Express an undetermined state as probability. For example, in Yuki's heart, maybe 40 percent science, 30 percent humanities, 30 percent undecided."
"Can feelings be quantified too?"
"At least modeled. There's a field called Bayesian statistics."
Riku showed interest. "Bayesian?"
Professor S. explained quietly. "Prior probability and posterior probability. Update initial predictions with new information."
Aoi supplemented. "For example, suppose three are equally probable initially. But if you get information that science classes were fun, science probability increases."
"As information increases, the probability distribution changes?" Yuki began to understand.
"Exactly. Bayes' theorem. P(A|B) = P(B|A)P(A) / P(B). When there's evidence B, how does probability of A change?"
Riku tilted his head. "Sounds difficult..."
"Think of a concrete example," Professor S. said. "Imagine weather forecast. Last night's prediction, 70 percent sunny. But this morning, the sky is cloudy. Update sunny probability with this new information."
"I see. The initial 70 percent is prior probability, and the result updated with cloudy evidence is posterior probability," Yuki organized.
"Accurate. And to make the best decision amid uncertainty, continuously updating probability is important."
Aoi continued. "You can't decide because information is insufficient. But complete information is never obtainable. So make the best choice with the current probability distribution."
"Does that mean never being able to decide?" Riku worried.
"No. When probability becomes sufficiently skewed at some point, decide. Or when the decision deadline comes, choose based on the distribution at that time."
Professor S. added. "In decision theory, maximize expected utility. Multiply each option's value by probability and choose the highest expected value."
Yuki pondered. "But I don't really know the value of any option."
"That's also part of uncertainty. But thinking clarifies it gradually. That itself is information gathering."
Riku said. "So 'undecided' is also a probability distribution? Yuki is still gathering information."
"Good interpretation," Aoi nodded. "'Undecided' is when the probability distribution is still flat. As information increases, the distribution sharpens."
Yuki felt a bit relieved. "I don't have to feel bad about not deciding."
"Right. Uncertainty is natural. What's important is recognizing it and updating while gathering information," Professor S. said quietly.
Yuki looked at the career path survey again. For now, the probability distribution is flat. But that's okay. While gathering information, gradually update the probability.
"Living Bayesian-ly," Yuki smiled.
"You understand quickly. That flexibility is talent for learning information theory."
Professor S. quietly left the club room. As usual, leaving only precise suggestions.
"Undecided is also one state," Aoi said.
Riku laughed. "I'll also enjoy being Bayesian-ly undecided!"
Yuki wrote in the notebook. "Undetermined is possibility. Probability is flexibility."
Today again, new information obtained.