Why did Dumbledore believe Sirius' version of events in TPoA

This question is not a duplicate of this

I want to know how Dumbledore knew that Sirius wasn't lying in The Prisoner of Azkaban. Dumbledore couldn't have used Leglimancy because Sirius might have used Occlumency. Dumbledore wouldn't risk Harry and Hermione's life(there was a werewolf on the loose) based solely on a hunch. So Sirius must have given some concrete evidence. Remember that Dumbledore couldn't have spoken to Remus either.So why would he believe in the Animagus version of what happened when Sirius supposedly blew up the whole street? As far as I am aware,this isn't mentioned fully in the books.

Here are a few explanations:

1)Dumbleddore realized that the whole story fitted together. It explained why Sirius had entered the castle but did not kill Harry. It explained why Sirius was in Ron's bed and not on Harry's. But there was this possibility that Sirius was mad.

2)He also realized why Buckbeak had disappeared earlier that night. But this can't be a sole reason for thinking some one is innocent. Maybe he used this in connection with (1).

So why did Dumbledore believe Sirius' version of events?