Authentication-Information-Request and Authentication-Information-Answer messages are exchanged between MME and HSS as part of Authentication procedure. MME fetches authentication information from HSS and it performs NAS authentication and security procedure with UE.
Once UE is attached successfully with a particular MME, it updates its location (current serving MME and other details) to its HSS.
This is done by exchanging Update Location Request and Update Location Answer messages.
Purge-UE-Request message is triggered by MME towards HSS to inform that UE is no longer connected to MME now. Reason could be detach or User inactivity from long time.
Hope, this answer may help you out.