# Learning operating probabilities from interval data

Suppose I have a machine. When the machine is active (operating), it runs for at least $$\mu > 0$$ time. I know that at some point in the time interval $$[l, h]$$ ($$l, h \in \mathbb R_{\ge 0}, l < h$$) the machine was active, but I do not know when, or for how long (other than that each operation took at least $$\mu$$ time). It might have been active multiple times.

I have data on (possibly overlapping) subintervals $$[l_i, h_i] \subseteq [l, h]$$ for $$i = 1,\dots,n$$ during which the machine was observed to be active. The machine was not observed to be active during the entire subinterval, but it is known it was active at least at some point in this interval. [for example: you have a printer in the office, and saw it printing in the morning. When you recall this a few hours later, you might not remember exactly when that was, but you know it was between 9:00 at the earliest and 11:00 at the latest.]

What I am interested in are (approximate) probabilities $$\Pr (t)$$ that the machine was active at any point $$t \in [l, h]$$. I suspect such an approximation can somehow be learned from the $$n$$ subintervals I have. Are there additional assumptions I need to impose for the problem to be well-defined? Is there literature investigating this, or a similar problem?

• Do you observe the output in each interval? Feb 2, 2022 at 19:51
• @dimitriy of the machine, after it finishes operating? I unfortunately do not have that data because it has not been observed/collected in the past. Feb 2, 2022 at 19:58
• Not having machine output for each period makes the problem a lot harder. Feb 2, 2022 at 21:47