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In Harry Potter and the Prisoner of Azkaban, Harry, Hermione and Dumbledore have a conversation.
Dumbledore: [...]We have no chance of overturning Sirius' sentence.
Harry and Hermione: But you believe us.
Dumbledore: Yes, I do.
Does this mean Dumbledore knew that Black was innocent? If he did, how did he know?
As stated by user I N T E R E S T I N G in his answer to a similar question, he couldn't possibly have believed that Sirius Black was innocent, because Albus Dumbledore had presented evidence that Sirius Black was the Potters' Secret Keeper:
‘There is not a shred of proof to support Black's story, except your word — and the word of two thirteen–year–old wizards will not convince anybody. A street full of eye-witnesses swore they saw Sirius murder Pettigrew. I myself gave evidence to the Ministry that Sirius had been the Potters' Secret Keeper.’
Harry Potter and the Prisoner of Azkaban - chapter 21 - page 287
He's only sure that Sirius Black is innocent after the three trustworthy protagonists say as such:
‘Sirius has not acted like an innocent man. The attack on the Fat Lady — entering Gryffindor Tower with a knife — without Pettigrew, alive or dead, we have no chance of overturning Sirius' sentence.’
‘But you believe us.’
‘Yes, I do,’ said Dumbledore quietly.
Harry Potter and the Prisoner of Azkaban - chapter 21 - page 287
It's quite possible, however, that Albus Dumbledore had his suspicions before then (relevant question: Why did Dumbledore believe Sirius' version of events in PoA?).
