It's not completely any one thing, rather it’s a combination of many things
In Azkaban, Dumbledore walks through and checks every unit. Bella remarks afterward that it was Dumbledore doing the check. That means he was close enough for her to identify him. She was not in great shape at the time and had almost no magic left. He was the most powerful wizard in several centuries doing an active search. It seems very unlikely he didn't notice everyone in the room. So with no tricks or deductions, I think there’s a chance Dumbledore finds out right here, and we see some hints that he did.
Returning from his search Dumbledore remarks that a prisoner had the same magic as a first year. As a Hogwarts prof, he knows very well how much magic a first year has.
Returning from the search Dumbledore changes tune from "Bella must not leave alive" to "let us not be the first to use deadly curses". (Ostensibly because they didn't kill Barry)
But if Dumbledore didn’t find Harry with his search, there are still various clues out there. And Dumbledore is the most powerful wizard in the world, in a world where power comes from solving mysteries, remembering details, and piecing together clues.
He knows that the individual in question didn't kill Barry
Dumbledore checks and discovers that the intruder has a patronus that hides him from dementors in a way never before seen, a week after having watched Harry invent such a charm.
When the intruder dismisses the patronus, the intruder shows up to the dementors but Bella remains hidden. Hence there is a second way to hide. Dumbledore has given Harry the true cloak of invisibility.
The dementors later refuse to cooperate as if they have been threatened / commanded. Later in the story, Harry scares / commands a dementor in Dumbledore's presence.
The intruder leaves Azkaban on a rocket powered broom!!!!
The broom in question is nonstandard with two seats. Later in the story Harry asks that he and all his friends be given multi seat brooms.
It turns out the rocket was transfigured out of ice to protect people breathing the smoke. (Note we don't see Dumbledore get this detail on screen, but it’s fair to assume intelligence sharing given other scenes)
The breakout happened when Harry was out of Hogwarts.
Checking on Harry before the breakout would cause a paradox.
The night after the breakout, Harry has an emotional breakdown and flips out on Dumbledore over how terrible conditions are in Azkaban and how much of a jerk Dumbledore is for going there and doing nothing about it.
Dumbledore may act crazy, but he is not actually an idiot. From Quirrell “"Mr. Potter, one of the requisites for becoming a powerful wizard is an excellent memory. The key to a puzzle is often something you read twenty years ago in an old scroll, or a peculiar ring you saw on the finger of a man you met only once.” Again, Dumbledore is the most powerful wizard in the world; he doesn’t realistically fail to put the above together.
Its 55% and not higher because 1) I can’t see exactly why Dumbledore acts like he doesn’t know if he knows. 2) some odds that Dumbledore’s mind is clouded by the hand of god.
Dumbledore clearly suspects Harry might be behind Azkaban. In fact, he says as much. However he doesn't "know". He also suspects it might be Voldemort. (He probably hasn't considered the possibilities that it was Harry and Voldemort working together.)
This is a new thread to discuss Eliezer Yudkowsky’s Harry Potter and the Methods of Rationality and anything related to it. This thread is intended for discussing chapter 94. The previous thread has passed 200 comments.
There is now a site dedicated to the story at hpmor.com, which is now the place to go to find the authors notes and all sorts of other goodies. AdeleneDawner has kept an archive of Author’s Notes. (This goes up to the notes for chapter 76, and is now not updating. The authors notes from chapter 77 onwards are on hpmor.com.)
The first 5 discussion threads are on the main page under the harry_potter tag. Threads 6 and on (including this one) are in the discussion section using its separate tag system. Also: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22.
Spoiler Warning: this thread is full of spoilers. With few exceptions, spoilers for MOR and canon are fair game to post, without warning or rot13. More specifically: