The structure may / might + perfect infinitive is used to talk about the possibility that something happened in the past. It could also be used to say that something was true in the past.
‘Ann hasn’t arrived yet.’ ‘She may have missed the train.’ (= It is possible that she missed the train.)
‘What was that noise?’ ‘It might have been an airplane.’
The structure might + perfect infinitive is also used to talk about past events or situations that were possible but did not happen.
You were stupid to fight with him. He might have killed you. (It was possible but fortunately it didn’t happen.)
May is not normally used to express this idea although it is sometimes possible in British English.
You were stupid to fight with him. He may have killed you. (Possible, but not very common)
The structure may / might + perfect infinitive can also be used to refer to the present or future. In this case, may and might show possibility.
Compare:
By the end of this month, I may have finished this work. (Strong possibility)
By the end of this month, I might have finished this work. (Weak possibility)
By the end of this month, I will have finished this work. (Certainty)
