"I thought probability was for prediction."
Yuki was talking with Aoi on the rooftop.
"That's also correct. But information theory has a different perspective."
At that moment, Professor S unusually appeared. A person who rarely showed up.
"Am I interrupting a discussion?" the professor asked gently.
"About the relationship between probability and information," Aoi answered.
"Good topic," the professor leaned against the railing. "Probability distributions represent the shape of the world."
"Shape of the world?" Yuki asked curiously.
"For example, a die. A uniform distribution represents perfect symmetry. A biased distribution indicates some cause."
Aoi supplemented. "Information resides in the shape of the distribution."
The professor nodded. "Entropy is a measure of that shape's complexity."
"Complexity..." Yuki murmured.
"A uniform distribution is most complex. Because it's unpredictable. A biased distribution is simple. Because there's a pattern."
"But complex yet simple?"
The professor smiled. "Paradoxical. That's where information theory's interest lies."
Aoi drew a diagram in the notebook. "Two dice. One is uniform, the other has high probability for six."
"Which is more complex?"
"The uniform one," Yuki answered.
"Yes. But which is easier to understand?"
"The biased one. I can imagine a reason. Like it's loaded."
"Exactly," the professor said. "Patterns create understandability. The lower the entropy, the more compressible and explainable."
Aoi continued. "Probability distributions are also models. Expressions of how we view the world."
"Models?"
"For example, tomorrow's weather. 70 percent sunny, 30 percent rain. This is a prediction and a model."
The professor gave another example. "Language too. In English, 'e' appears most frequently. This is the shape of the English language."
"Shape..." Yuki pondered.
"Different probability distributions mean different languages. Japanese has many 'no' and 'wa'."
Aoi added a diagram. "In information theory, we view everything as probability distributions. Messages, signals, data."
"And measuring the difference between two distributions is KL divergence," the professor said.
"I learned that before," Yuki recalled.
"By comparing distributions, understanding deepens. The gap between 'actual' and 'predicted.' The difference between 'Language A' and 'Language B.'"
Aoi's expression became serious. "Probability might be the only way to handle uncertainty."
"The only way?" Yuki asked.
"The world is complex and can't be perfectly predicted. But using probability, we can quantify uncertainty."
The professor looked up at the sky. "The law of large numbers. With repeated trials, probability becomes reality."
"Flip a coin 1000 times, and you'll get roughly 500 heads."
"Yes. Probability guarantees long-term stability."
Aoi said to Yuki, "Information theory is a discipline that views the world through probability."
"Data is born from probability distributions. Communication is a stochastic process. Learning is distribution estimation."
The professor quietly added, "And entropy is at the center of everything."
Yuki looked out the window. The city spread below.
"Can that city be represented by probability distributions?"
"It can," Aoi answered. "Flow of people, traffic patterns, store locations. All can be analyzed probabilistically."
"The world is a collection of probabilities."
The professor smiled. "But individual events occur deterministically. Probability is a tool to capture the overall picture."
"Determinism and probabilism..." Yuki murmured.
"They coexist. Deterministic at the micro level, probabilistic at the macro level."
Aoi summarized. "Probability is a language for handling complexity. Information is measured by probability."
The professor stood up. "Good discussion. Continue learning."
"Thank you very much," Yuki bowed.
After the professor left, Aoi said, "The professor rarely appears, but when they do, they hit the core."
"Viewing the world through probability. A new perspective."
"Learning information theory changes how you see the world."
Yuki looked up at the sky. Clouds drifted by. That movement too is a stochastic process. Wind speed, humidity, air pressure. All interact probabilistically.
"The world is filled with information."
Aoi nodded. "And probability holds the key."
The rooftop wind ruffled their hair. The journey of information theory had only just begun.