Bayesian: Poisson with constant prior

Question

Need some help.

Assume the sampling distribution is Poisson with sample size $n$.

Assume constant prior distribution.

Likelihood:

$L(\lambda | \mathbb{x})=\prod_{i=1}^{n}\dfrac{e^{-\lambda}\lambda^{x_{i}}}{x_{i}!}=\dfrac{e^{-n\lambda}\lambda^{\sum_{i=1}^{n}x_{i}}}{\prod_{i=1}^{n}x_{i}!}$

Prior: $p(\lambda)=\frac{1}{a}$, $a$ is some real constant.

Therefore, posterior:

$\pi(\lambda | \mathbb{x})=\dfrac{e^{-n\lambda}\lambda^{\sum_{i=1}^{n}x_{i}}}{\prod_{i=1}^{n}x_{i}!}\cdot\dfrac{1}{a}$

$\pi(\lambda | \mathbb{x})\propto e^{-n\lambda}\lambda^{\sum_{i=1}^{n}x_{i}}$

Is this correct? And if so, is there no nice posterior for the Poisson/constant?

Answer 1

The posterior with such an improper prior is a gamma distribution and you can pretty much read off the parameter values from what you wrote down. And yes, your derivation seems right.

Answer 2

This looks mostly reasonable to me, but your terminology is a bit loose, and I believe there is also one mathematical oversight. My understanding of the problem would be as follows.

First, applying Bayes theorem to the Poisson parameter $\lambda$ and the data $\mathbb{x}$ gives the posterior formally as $$p(\lambda \mid \mathbb{x})=\frac{p(\mathbb{x} \mid \lambda)p(\lambda)}{p(\mathbb{x})} \,,\, p(\mathbb{x})=\int_0^{\infty}{p(\mathbb{x} \mid \lambda)p(\lambda)d\lambda}$$ where the integral for the normalization factor $p(\mathbb{x})$ is over all possible $\lambda\in\mathbb{R}^+$.

Then, assuming i.i.d. data $\mathbb{x}\in\mathbb{N}^n$, your expression for the data likelihood appears to be correct, and is equivalent to $$p(\mathbb{x} \mid \lambda)=\frac{(e^{-\lambda}\lambda^{\bar{x}})^n}{\Omega}\,,\, \bar{x}=\frac{1}{n}\sum_{i=1}^nx_i\,,\,\Omega=\prod_{i=1}^{n}x_{i}!$$ where $\Omega$ just collects the factors that do not depend on $\lambda$.

Next we must consider the prior $p(\lambda)$, and this is where I believe your mathematical oversight occurs. For a generic Poisson distribution, we know $\lambda\in\mathbb{R}^+$. The prior you give, when interpreted literally, does not give a valid PDF over $\mathbb{R}^+$ (i.e. it integrates to $\infty$). So really the implied prior is a uniform PDF over $[0,a]$, i.e. $$p(\lambda)=\frac{\mathbb{1}_{\lambda\in[0,a]}}{a}$$ where the notation $\mathbb{1}_C$ is an indicator function that is $1$ when condition $C$ holds, and $0$ otherwise.

Combining the above, the posterior would then be $$p(\lambda \mid \mathbb{x})=\frac{(e^{-\lambda}\lambda^{\bar{x}})^n}{\int_0^a{(e^{-\lambda}\lambda^{\bar{x}})^nd\lambda}}\mathbb{1}_{\lambda\in[0,a]}$$

(Warning: I am probably out of my depth, so may be wrong here!)

Answer 3

The likelihood function can be write as:

$L(\mathbb{x}|\lambda) \propto \prod_{i=1}^N e^{\lambda}x^{x_i}$

$\propto$ means you can ignore the normalize constant at this step.

Prior: $p(\lambda) \propto \frac{1}{a} \propto 1$

Therefore, the posterior distribution is:

$\pi(\lambda | \mathbb{x}) \propto L(\mathbb{x}|\lambda) \times p(\lambda)$

$\pi(\lambda | \mathbb{x}) \propto e^{N\lambda}x^{\sum_{i=1}^Nx_i}$

If you are going to just write down the Bayes rule, then it is like this. But if you want to implement MCMC, then you need to consider if a non-informative prior is proper.