Probabilistic treatment of the uncertainty from the finite size of weighted Monte Carlo data
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The finite size of Monte Carlo samples carries intrinsic uncertainty that can lead to a substantial bias in parameter estimation if it is neglected and the sample size is small. We introduce a probabilistic treatment of this problem by replacing the usual likelihood functions with novel generalized probability distributions that incorporate the finite statistics via suitable marginalization. These new PDFs are analytic, and can be used to replace the Poisson, multinomial, and sample-based unbinned likelihoods, which covers many use cases in high-energy physics. In the limit of infinite statistics, they reduce to the respective standard probability distributions. In the general case of arbitrary Monte Carlo weights, the expressions involve the fourth Lauricella function F_D, for which we find a new representation as a contour integral that allows an exact and efficient calculation. The result also entails a new expression for the probability generating function of the Dirichlet-multinomial distribution with integer parameters. We demonstrate the bias reduction of our approach with a typical toy Monte Carlo problem, estimating the normalization of a peak in a falling energy spectrum, and compare the results with previously published methods from the literature.
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