Uncertainty about the rate at which things occur in time or space

Lightning strikes, car accidents, machine failures, political crises, disease outbreaks - are all random events in time that can be thought of as being independent of each other. Daisies on a lawn, bacteria in a liquid, mould in a silo, diamonds in a rock - can all be thought of as random events in either two (surface) or three (volume) dimensional space.

The most common modelling approach in modelling a distribution of how many of these events a might occur in a given amount of time or space t is to assume that the counts are from a Poisson process, in which case the counts will take a Poisson distribution:

Counts a = Poisson(l*t)

where l is the mean (expected) number of events that would occur per unit t. Care needs to be taken with the units of l and t to ensure that they match.  The product l*t is the expected number of events over the period t and is sometimes called the Poisson intensity.

By applying Bayesian inference with Poisson probabilities we arrive at a neat solution to the uncertainty we have about l, when we have observed a events in a time t:

l = Gamma(a, 1/t)

Example

You have observed 12 sporadic (i.e. each occurring independently of the others) cases of disease X in your country in the last 4 years. How many will there be next year, if the underlying risk remains constant? What is the probability that there will be greater than 6 cases next year?

Assuming that the Poisson process applies, we first need to estimate l:

l = VoseGamma(12, 1/t) = VoseGamma(12, 0.25)

 

The graph above shows that with the amount of information we have about l, we believe it is very likely to lie between 1 and 6 expected cases/year.

How many cases will there be next year? The answer is =VosePoisson(l*1). If we wish to model a first order distribution, we write:  =VosePoisson(VoseGamma(12, 0.25)) = VoseNegBin(12, 0.8). If we wish to model a second order distribution (one that separates uncertainty and randomness), the answer comes from taken random samples from the Gamma distribution, and for each of these samples calculate the complete Poisson distribution. The answer is therefore a set of possible probability distributions. The figure below shows the two options. Either is acceptable, depending on management needs, but what is not acceptable is to write =VosePoisson(12/4): in other words to ignore the uncertainty we have about l. This third, incorrect option, is also shown below.

 

What is the probability that there will be greater than 6 cases next year? The Excel function POISSON will calculate the probability of there being less than or equal to six case next year, so we can use that to determine the probability we are actually interested in:

=1-POISSON(6,VoseGamma(12,0.25),1)

where the uncertainty about the intensity is provided by the imbedded Gamma distribution imbedded. Running a simulation for this cell gives the following output:

 

This plot shows that with the level of historical information, we believe with 80% confidence that there is between a 0.4% and 12.7% probability of having more than six cases next year.