What prior to use given a Poisson likelihood?

Question

I am trying to incorporate a prior into a model I am working on. From available data, I have found that the likelihood follows a Poisson distribution with $\lambda = 1.5$.

I have then used R to generate random Poisson values with $\lambda = 1.5$.

However, I suspect that the $\lambda$ value is more likely to be $\lambda=2$. How would I turn this into a prior I could use and what would the posterior be?

Thanks in advance, Neil

Answer 1

Common choice of bayesian prior for the Poisson distribution is the Gamma distribution. Which ultimately has a Gamma distribution for the posterior.

If you have a large set of data which you believe comes from a Poisson distribution. The assumption is that $Y_i \text{ iid}\sim \text{Poisson}(\lambda)$. $$f(y|\lambda) = \frac{\lambda^ye^{-\lambda}}{y!}$$

The idea behind the Bayesian approach of estimation / regression is that there is uncertainty in the parameter $\lambda$ for each $Y_i$. And that for each observation, there may be a natural variation of $\lambda$s such that they have their own distribution $g(\lambda| \nu)$ with hyper-parameters $\nu$. As mentioned, a convenient choice of prior for the Poisson distribution is the gamma distribution because with hyper-parameters $\lambda \sim \Gamma(v, r)$: $$f(\lambda|y) \propto f(y|\lambda)\cdot g(\lambda|\nu)$$ $$\propto \big(\frac{\lambda^ye^{-\lambda}}{y!}\big)\cdot g(\lambda|\nu)$$ $$\propto \big(\frac{\lambda^ye^{-\lambda}}{y!}\big)\cdot\big(\frac{v^r}{\Gamma(r)}\lambda^{r-1}e^{-v\lambda}\big)$$ $$\propto \frac{v^r}{\Gamma(r)}\lambda^{r+y-1}e^{-(v+1)\lambda}\sim\Gamma(r' = r + y, v' = v + 1)$$

Now you use the fact that $E[\lambda] = \frac{r'}{v'},\ \text{Var}[\lambda] = \frac{r'}{v'^2}$ you can solve for the regressed maximum likelihood estimate of $\lambda$, where $r$, and $v$ come from a historical set of data to inform the prior, and $y$ is the next observation or $\sum y_i$ the 'next' set of information.