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I'm trying to understand why a poisson distribution may be preferred over a binomial one when modeling binary cases.

  1. Is there a case where you can't use a binomial distribution to solve a problem that can be solved by a poisson distribution? For example: The average rate for fishing is 2 fish caught per day. What is the probability that 1 fish a day is caught? This can be modeled by a binomial distribution if you define a trial as each hour and the probability of success is approximately 2 fish/24 hour = 0.0833 per hour. Assuming there are 24 trials, the answer calculated from Binomial(24, 0.0833) is approximately the same as the poisson.

  2. When is a binomial distribution unreasonable to calculate computationally? I understand that calculating a probability based on a poisson distribution is less computationally expensive than calculating it based on a binomial distribution. What are the practical applications of Poisson where computational time actually makes large impact?

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    $\begingroup$ Read up on rare events. $\endgroup$ – Scortchi Jan 15 '15 at 17:34
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    $\begingroup$ How you phrase the first question is amusing because it flips the title on its head. You are asking why not use the Binomial to approximate a Poisson distribution! Here we have a situation where you seem to be advocating approximating the correct distribution with something computationally more difficult to evaluate. Why is that? $\endgroup$ – whuber Jan 15 '15 at 17:38
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    $\begingroup$ (When modeling fishing, the Poisson distribution sounds like it would be particularly applicable. Francophones will understand what I mean :-).) $\endgroup$ – whuber Jan 15 '15 at 19:40
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A binomial process is a descretization of the Poisson process, which depends on an artificial binwidth.

Thus, when performing regression on continous (time-series) data such as fish caught or neuronal spiking, the advantage of using Poisson regression (such as w/ a GLM w/ an exponential link function) over bernoulli regression (such as logistic regression) is that once you have the estimated poisson rate function $r(t)$, you can always recreate the bernoulli firing probability in every bin as $P(event-in-\Delta_t)=\int_{\Delta_t} r(t)dt$

tl'dr: you can always transform from Poisson to bernoulli, but you cant transform the other way. Thus Poisson has more information

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In your first fishing example, binomial is a poor choice because it disallows more than one catch per hour! Any subdivision of the time scale will be artificial, since there is no logical upper bound to the number of catches. Therefore Poisson is a more natural model.

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