The BI file is the "UNDO log". It is used to back out transactions.
The AI file (actually a series of archived ai extents...) is the "REDO log". It is used to "roll forward" transactions against a backup in order to recover a database.
BI is mandatory. There is no way around it. Even the -i option (no-integrity) writes bi notes, just in a less recoverable manner.
AI is optional -- you have to set it up and configure it. While it is technically optional no one should ever run a production database without it. It is a DBA's best friend and it is negligent and irresponsible to not be running after-imaging.