I'm getting confused on estimating the intensity for a Poisson process. My background in the subject is weak.

Suppose I'm interested in modelling the probability of an event occurring given some input x. Then, the event occurs with probability $\lambda(x)$, and the probability of the event occurring between $t$ and $dt$ is $\lambda(x)dt$.

I want to specify a function for $\lambda(x)$, so I assume it is some kind of exponential function with some parameters (Not necessarily an exponential pdf).

When it comes to data, the only data I have is: given x, the amount of time it takes for the event to occur. How do I back out this probability $\lambda(x)$ that I am looking for? (I.e. so that I can equate it to the exponential function and get a fit for the parameters). Is it equal to the amount of time until an event occurs? Is it equal to the average time it takes for an event to occur?

It would be great if someone could tie together all these concepts for me. How do I connect an exponential waiting time between events to some poisson intensity to get a probability of occurrence. Thanks.

Also, as a side point, what is meant by the compensator process $\Lambda_t =\int_0^T\lambda(x_t)dt?$



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