# Exact Poisson CI

#### katxt

##### Well-Known Member

Hi. This is given as the "exact" CI for a Poisson rate in many places. k is the observed number, mu is the true rate. My problem is the different df at each end. When alpha gets large and tends to 100% you would expect the limits to converge on k. They don't.
Is there a real "exact" CI for which you can put in large values of alpha and have it work?

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#### hlsmith

##### Less is more. Stay pure. Stay poor.
Not looking at the equation, philosophically I always wondered when younger what is the true population size and how would the function know that since we never have knowledge of the truth and the truth is always changing given all data can't be collected simultaneously and there always seems to be epistemological error. You could simulate but the error comes into play, so hey simulate it deterministically, well that converges prior to having the full sample (likely).

It seems many precision bounds are exact in small sample setting and asymptotic in larger settings. The +2 seems like one of those sample estimate things with d's that has never clicked in my head kind of like prediction intervals.

#### katxt

##### Well-Known Member
Thanks. I'm not much further on, but I appreciate your deep philosophical thoughts about truth, the meaning of life and the pursuit of happiness. kat

#### Dason

Can you provide a link to any source listing that as "exact"?

#### katxt

##### Well-Known Member
Here's the background. I want a way of generating a range of plausible mu's when k = 1 for a monte carlo analysis. I hoped to use the exact CI equation with alpha uniform on 0-1 but that doesn't work.
I have tried using the Poisson likelihood function for k = 1 but the values I get for 95% CI don't match the "exact" formula. I have also tries approximating the Poisson for k = 1 with the binomial for a very large number but that doesn't match either.
Any suggestions?

#### Dason

Could use the posterior distribution you'd get from a bayesian analysis for that I would think.

#### katxt

##### Well-Known Member
Thanks. Good thought. I'm not too hot on Bayesian stuff, but a uniform prior seems to give (I think) more or less the same results as the likelihood function.

#### Dason

As it should. But simulating from the likelihood isn't really defensible from a frequentist perspective. I'm being pedantic but if that's what you want to do it really makes the most sense to at least acknowledge you're taking a bayesian approach.

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#### katxt

##### Well-Known Member
OK. I just thought it was common sense. Perhaps I'm a closet Bayesian.

#### katxt

##### Well-Known Member
Just in case anyone is following this, I was looking for a function to generate Monte Carlo Poisson rates given an observation k.
Given the "exact" CI expression I used mu = 1/2*CHIINV(k,2*(n+k)), where u is Uniform(0,1). This gives about 2k df for small values of u, and 2k+2 df for large u. It seems to work well enough but any further suggestions appreciated. kat

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