Comparison of classical and Bayesian estimates of intensity l in a Poisson process

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The classical statistics estimate of l is compared with the Bayesian estimate with a p(l) = 1/l prior, and with just the likelihood function (equivalent to a Bayesian analysis with a Uniform prior). Comparisons are made with t = 1 because the time unit is just a scaling factor.

You can see that the Bayesian analysis consistently offers a lower estimate than the classical method, but the difference diminishes with greater a. This is in part due to the uninformed prior that has so much density at low values of l. Note that a Bayesian analysis with a Uniform prior would fall between the two.