"Once you know, you can't go back."
Yuki murmured. Mira nodded quietly.
"That's the essence of information," Aoi said. "Once you know, uncertainty decreases."
"Conditional entropy?"
"Yes. H(Y|X) is Y's uncertainty after knowing X."
Aoi wrote an equation in the notebook.
"H(Y|X) = H(X,Y) - H(X)"
"Knowing X reduces entropy."
Yuki pondered. "But sometimes there are things we don't want to know."
"That's human emotion," Aoi admitted. "Information theory assumes information always has value."
"Is that really true?"
Mira wrote something in her notebook and showed it to Yuki.
"Information can hurt"
"Mira's right," Yuki said. "Knowing can cause suffering too."
Aoi answered quietly. "Information theory measures the 'quantity' of information. But not 'quality' or 'impact.'"
"Quantity and quality differ."
"Exactly. Sometimes small amounts of important information are more valuable than large amounts of meaningless information."
Yuki approached the whiteboard.
"Then what does mutual information I(X;Y) represent?"
"The amount of information X and Y share. How much they can tell each other."
Aoi drew a diagram. Two overlapping circles.
"The overlap is mutual information."
"I(X;Y) = H(Y) - H(Y|X)"
"Subtract Y's entropy after knowing X from Y's original entropy."
Yuki understood. "The reduction is information gained from X."
"Exactly. Mutual information is always non-negative."
Mira wrote another equation.
"I(X;Y) = I(Y;X)"
"There's symmetry," Aoi explained. "The amount X tells about Y equals the amount Y tells about X."
"But," Yuki said, "in human relationships, it's not symmetric, right?"
"Sharp observation. The information person A knows about B differs from what B knows about A."
"A limitation of information theory?"
"No, a matter of definition. Mutual information measures statistical dependence. Personal knowledge is a different concept."
Mira stood up and looked out the window.
"Sometimes knowing less is peaceful"
"Ignorance for peace?" Yuki asked.
Mira nodded.
Aoi continued. "But in the long term, having information is often advantageous."
"Short term and long term differ," Yuki summarized.
"Yes. In decision theory, we can quantify information value."
"Information value?"
"How much expected utility increases before and after gaining information."
Aoi gave an example.
"Knowing the weather forecast lets you decide whether to carry an umbrella. The risk of getting wet decreases."
"Practical value," Yuki understood.
"But," Mira said softly, "Some information has negative value"
"Negative value?"
"Knowing can constrain actions," Aoi explained. "For example, knowing disease risk might narrow life choices."
Yuki said quietly, "Conditional entropy decreases. But happiness doesn't necessarily increase."
"Information theory and life happiness are different scales."
The three fell silent for a while.
"Still," Yuki said, "I want to know. Even if melancholy."
"Why?" Aoi asked.
"Rather than living in uncertainty, I want to know the truth and face it."
Mira smiled. "Courage to face information"
"Courage," Yuki nodded.
Aoi said, "Information is a tool. Depending on how you use it, it can help or harm."
"Conditional entropy decreases," Yuki summarized. "But how we use it is up to us."
Dusk approached. The melancholy of knowing, and the courage to know.
Today too, they gained new information. And grew a little bit.