Bayesian credible interval construction for Poisson statistics

The construction of the Bayesian credible (confidence) interval for a Poisson observable including both the signal and background with and without systematic uncertainties is presented. Introducing the conditional probability satisfying the requirement of the background not larger than the observed events to construct the Bayesian credible interval is also discussed. A Fortran routine, BPOCI, has been developed to implement the calculation.

💡 Research Summary

The paper presents a comprehensive Bayesian framework for constructing credible intervals for a Poisson‑distributed observable that contains both signal and background contributions, with explicit treatment of systematic uncertainties on the background. Traditional frequentist confidence intervals often become problematic when the observed count is low, the background expectation is comparable to or larger than the data, or when systematic errors are sizable. In such regimes frequentist methods can yield asymmetric intervals, negative lower limits for the signal strength, or overly conservative bounds that do not respect the physical constraint that the background cannot exceed the total observed events.

To overcome these issues the authors adopt a Bayesian approach that incorporates two key ideas. First, they impose a physically motivated constraint by using a conditional probability that enforces n ≥ b, where n is the observed total count and b the (unknown) background count. This is achieved by restricting the integration over the background parameter to the interval 0 ≤ b ≤ n when forming the posterior for the signal strength s. The likelihood remains the standard Poisson form L(n|s,b)=e^{-(s+b)}(s+b)^{n}/n!, but the prior for b is defined only on the non‑negative domain, and the prior for s is likewise non‑negative (often taken as flat or weakly informative). The resulting posterior is

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