I know there are a lot of responses, but I'll respond to the first post so everyone sees it.
The problem has nothing to do directly with iOS 9, 9.0.1, or 9.0.2. What caused it is the process of updating, and it has happened to a few phones (yes, really a few) after every update that has ever been released. It can also happen spontaneously, but it is more noticeable after an update because it is natural to see the correlation as causation.
What causes it is an app (or apps, but usually only one) that was interrupted in the middle of doing something requiring sending or receiving data. If the app isn't properly coded to deal with this it keeps attempting to complete the send/receive, but fails every time. But it uses up processing time, battery, and sometimes data as it tries. To fix it the app must be identified and either killed, or, in some cases, deleted and reinstalled. Go to Settings/Battery, scroll to the bottom, and look at Usage and Standby. If they are exactly the same it is a real runaway app. If they are not the same, but Usage is much higher than your memory of how much you have used the phone it is an app running in background.
Next, read the list of apps and the amount of energy each has used. This should tell you which app you have to deal with.
There is one special case. If the app is Mail and you have a Microsoft Exchange account on the phone it is the Exchange account that is causing the problem. This is a bug in Microsoft's ActiveSync protocol. To fix it either delete the Exchange account, reboot the phone, and add it back, or turn off all of the switches for it in Settings/Mail, Contacts, Calendars. Reboot, then turn them back on.
These are the first, easy steps to resolve sudden battery drain. There are more, but try these first.