Let $X_{1},..,X_{n}$ be a sample from a poisson$({\lambda})$ distribution. Let the prior be ${\pi}({\lambda})=1/{\sqrt{\lambda}}$. Find the posterior distribution.
My work: We have $f(x|{\lambda})=\frac{{\lambda}^{x_{1}+..+x_{n}-0.5}e^{-{\lambda}n}}{x_{1}!....x_{n}!}$. To find the marginal density I believe I need to sum this expression through ${\lambda}=0,1,2,......$. How to do this?
Thanks
$p(\lambda|X)=\frac{p(X|\lambda)\pi(\lambda)}{p(X)}=\frac{L_\lambda(X)\pi(\lambda)}{\int_0^\infty p(X,\lambda)d\lambda}$
$L_\lambda(X)=\frac{e^{-n\lambda }\lambda^{\sum x_i}}{\Pi(x_i!)}$
$\int_0^\infty p(X,\lambda)d\lambda=constant$
$p(\lambda|X)\propto \pi(\lambda)L_\lambda(X)=\frac{e^{-n\lambda }\lambda^{\sum x_i-0.5}}{\Pi(x_i!)}$
$\int_0^\infty p(X,\lambda)d\lambda=\int_0^\infty p(\lambda)(X|\lambda)d\lambda$, since we believe $Cov(\lambda,x_i)=0$
$\int_0^\infty p(\lambda)(X|\lambda)d\lambda=\int_0^\infty\frac{e^{-n\lambda }\lambda^{\sum x_i-0.5}}{\Pi(x_i!)}d\lambda=\int_0^\infty\frac{1}{n^{\sum x_i+0.5}\Pi(x_i!)}\times e^{-n\lambda }(n\lambda)^{\sum x_i-0.5}d(n\lambda)$
The result of the integral is a Gamma function
$\int_0^\infty\frac{1}{n^{\sum x_i+0.5}\Pi(x_i!)}\times e^{-n\lambda }(n\lambda)^{\sum x_i-0.5}d(n\lambda)=\frac{\Gamma(\sum x_i+0.5)}{n^{\sum x_i+0.5}\Pi(x_i!)}$
$p(\lambda|X)=\frac{e^{-n\lambda }\lambda^{\sum x_i-0.5}}{\Pi(x_i!)}\div\frac{\Gamma(\sum x_i+0.5)}{n^{\sum x_i+0.5}\Pi(x_i!)}=\frac{exp\{-n\lambda\}\lambda^{\sum x_i-0.5}n^{\sum x_i+0.5}}{\Gamma(\sum x_i+0.5)}$
$f(\lambda|X)$ is a $\gamma$ distribution as $f(\lambda|X;\alpha,\beta)$, where $alpha=\sum x_i+0.5$ and $\beta=n$
$\Gamma(\sum x_i+0.5)=\frac{(2\sum x_i)!\sqrt\pi}{(\sum x_i)!4^{\sum x_i}}$
$p(\lambda|X)=\frac{exp\{-n\lambda\}\lambda^{\sum x_i-0.5}n^{\sum x_i+0.5}(\sum x_i)!4^{\sum x_i}}{(2\sum x_i)!\sqrt\pi}$
And from the Bayesian Analysis, we have much confidence to believe that the probability of $\lambda$ of a Poisson distribution to be $\gamma$ distribution, almost regardless of the prior $\pi(\lambda)$.
That's much the reason, I believe, that we usually assume $\pi(\lambda)=\gamma(\lambda;\alpha,\beta)$, a $\gamma$ function for the prior of the parameter $\lambda$ of the Possion distrubtion Pois($\lambda$).
And we usually use flat prior for $\alpha,\beta$, namely $\alpha\sim U,\beta\sim U$, to acknowledge that we have no information of $\alpha$ and $\beta$.
In this case we have a hierarchical prior.
A gamma prior $\gamma(\lambda;\alpha,\beta)$ for $\lambda$ of the Poisson distribution Pois($\lambda$).
Flat hyperpriors $\alpha\sim U, \beta\sim U$ for $\alpha$ and $\beta$ of the $\gamma(\lambda;\alpha,\beta)$ prior.
The above discussion, I think, is pretty common in standard text books regarding priors.
