Efficient maximum likelihood estimator fitting of histograms

To the Editor: Scientists commonly form histograms of counted events from their data, and extract parameters by fitting to a known model. Anytime a scientist counts photons, molecules, cells or data for individuals in histogram bins and fits that to a distribution, he or she is fitting an event-counting histogram. Here we aim to convince the scientific community to use the maximum likelihood estimator (MLE) for Poisson deviates when fitting event-counting histograms rather than the typically used least-squares measure. We describe how to use the MLE to fit data efficiently and robustly, and provide example code (Supplementary Software). The least-squares measure is the MLE for Gaussian-distributed data. However, event counts in histogram bins are distributed according to the Poisson distribution1. The application of leastsquares procedures to Poisson-distributed data is known to lead to biases, exemplified by results in fitting exponential decays2. The proper procedure with event-counting histograms is to adjust fitting parameters to minimize the MLE for the Poisson distribution. For a dataset x = (x1,x2,...,xN) fitted to a model function f = (f1,f2,... ,fN), the MLE (mle) for Poisson deviates is

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