An unbiased point estimate for the rate $\lambda$ of a Poisson process is $x/t$, with $x$ the number of observed occurrences within $t$. However, for small $\lambda$ or for short observation periods $t$, it can happen that $x$ is *zero*. In fact, $x$ being *zero* is usually desired when observing failures of safety-critical systems. However, declaring $\lambda$ as zero based on such an observation is clearly unsatisfactory for such systems.

Using a Bayesian strategy to quantify the uncertainty about the rate of a Poisson process, the uncertainty about $\lambda$ can be expressed analytically. This allows us to derive upper credible intervals for $\lambda$, even if $x$ is *zero*. These credible intervals are ideally suited for engineering applications, as they are actually on the conservative side.

In the figure above, upper credible intervals and the mean value for lambda are shown as a function of the observation time, assuming $x=0$. For example, if $x$ is still *zero* after observing the Poisson process for $10^4$ hours, the expected value of $\lambda$ is $1.0\times10^{-4}$ events per hour, the $95\%$ upper credible interval for $\lambda$ is $3.0\times10^{-4}$ events per hour and the $99\%$ upper credible interval is $4.6\times 10^{-4}$ events per hour.

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