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Counting experiments often rely on Monte Carlo simulations for predictions ofPoisson expectations. The accompanying uncertainty from the finite Monte Carlosample size can be incorporated into parameter estimation by modifying thePoisson likelihood. We first review previous Frequentist methods of this typeby Barlow et al, Bohm et al, and Chirkin, as well as recently proposedprobabilistic methods by the author and Arg\"uelles et al. We show that allthese approaches can be understood in a unified way: they all approximate theunderlying probability distribution of the sum of weights in a given bin, thecompound Poisson distribution (CPD). The Probabilistic methods marginalize thePoisson mean with a distribution that approximates the CPD, while theFrequentist counterparts optimize the same integrand treating the mean as anuisance parameter. With this viewpoint we can motivate three new probabilisticlikelihoods based on generalized gamma-Poisson mixture distributions which wederive in analytic form. Afterwards, we test old and new formulas in differentparameter estimation settings consisting of a "background" and "signal"dataset. The probablistic counterpart to the Ansatz by Barlow et al.outperforms all other existing approaches in various scenarios. We further finda surprising outcome: usage of the exact CPD is actually bad for parameterestimation. A continuous approximation performs much better and in principleallows to perform bias-free inference at any level of simulated livetime if thefirst two moments of the CPD of each dataset are known exactly. Finally, wealso discuss the situation where new Monte Carlo simulation is produced for agiven parameter choice which leads to fluctuations in the computed likelihoodvalue. Two of the new formulas allow to include this Poisson uncertaintydirectly into the likelihood which substantially decreases these fluctuations.

A unified perspective on modified Poisson likelihoods for limited Monte Carlo data