Let's assume we have a physical experiment which runs continuously and shows some events we are interested in.
An event could be detecting a certain type of particle.
Or, to make it more demonstrative, the event may be overheating of our contraption. (We know it will automatically recover.)
We start our experiment, knowing that it should not be able to overheat at all.
Let's consider various possible observations after running it for one day:
We started with the assumption that it does not overheat at all.
- It just runs smoothly, and it never overheated.
- That tells us we were probably right.
- It overheats at random times about once per minute.
- We now expect the next overheat event to occur in roughly a minute
- It overheated once.
- What do we know based on this?
We learned that it can overheat. Should we assume that it will overheat again? The probability of overheating is not 0 as we expected. Anything else?
One may argue the method of statistics just does not apply here, because we do not have a set of samples to base on. But there seems to be some information we get.
Do we just update our estimate when an event occurs to be the number of events up to now per time passed since start? Could we say anything about how sure we are about it?