At lunch today, Adrian Price-Whelan (Flatiron) challenged me to explain the form of the variable-rate Poisson process likelihood function. I waved my hands! But I think the argument goes like this (don't quote me on this!): You boxelize your space (time or space or whatever you are working in) in small-enough boxels that every boxel contains exactly one or zero data points. Then you take the limit as the boxel sizes go to zero (and become infinitely numerous). The occupied boxels deliver a sum (in the log likelihood) of log densities at the locations of the observed data. The unoccupied boxels deliver an integral of the density over all of space. Something like that?
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