Counting statistics and Poisson likelihoods

Let r be a nonnegative source rate before detection, epsilon a known detection probability, b a known detected background rate, and T the live acquisition time. Independent thinning of a constant-rate Poisson source gives detected signal rate epsilon times r. Independent Poisson background adds its mean. The observed random count N then has mean mu, which is dimensionless; r and b have inverse-time units. Exposure refers to live time, so unmodeled dead time cannot simply be ignored.

The Poisson assumption requires more than rare individual events. Independent increments and a stable intensity are central. Afterpulsing, pileup, bursty sources, or fluctuating efficiencies can produce correlations or overdispersion. A fixed number of trials with a success probability instead leads to a binomial model. Conditioning independent Poisson bins on their total yields a multinomial shape model, discarding information about the total rate. These are different sampling experiments even when their histograms look similar.

After observing n, hold it fixed and vary r in the likelihood. The factorial can be omitted when maximizing because it has no parameter dependence. The expected-count normalization term cannot: it penalizes predicting many events when few appeared. Differentiating the log likelihood gives an unconstrained estimate from the observed total rate minus background, divided by efficiency. If this is negative, the maximum over the physical parameter space lies at zero; that boundary changes its sampling behavior.

For an interior solution, the variance of the unconstrained rate estimator follows directly from the Poisson variance through its linear transformation. Substituting n for the unknown mu gives an estimated standard error, not an exact confidence interval. At zero observed counts this plug-in error vanishes even though positive rates remain compatible with the observation. Small-count inference therefore needs an interval construction or explicit posterior rather than a symmetric error bar justified only by a local curvature approximation.

Suppose a second acquisition measures background alone. Its count fluctuates independently if it uses a separate Poisson process and the detector conditions are stable. Subtracting a time-scaled background count removes its expectation, while adding its variance to that of the signal-region count. The resulting difference need not be nonnegative and is not itself a Poisson count. Modeling the two original observations jointly preserves their discrete information and allows the common background rate to become a nuisance parameter.

Efficiency uncertainty, exposure uncertainty, and contamination from the source in the control region require further terms. A shared calibration can correlate acquisitions that would otherwise have independent counting noise. Before adding variances, state whether independence concerns events, runs, or calibration information. Inspecting count-versus-time and control-region distributions helps test the assumptions. Increasing exposure reduces independent counting error, but it does not automatically remove a common efficiency bias or repair a wrong background model.