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If a counting experiment gives one observation $x=5$, and if the prior distribution is given as a uniform function, then is the following a correct way of calculating the posterior function?

First, the posterior can be written as
$f(\theta|x) \propto f(x|\theta)\pi(\theta) $

where $\pi(\theta)$ is the prior and $f(x|\theta)$ is the likelihood function.

Given that $\pi(\theta)=constant$, $f(x|\theta)= \frac{e^{-x}x^{\theta}}{\theta!}$,

$f(\theta|x) = k\frac{e^{-x}x^{\theta}}{\theta!} $

Since $\Sigma_{\theta=0}^{\infty} k\frac{e^{-x}x^{\theta}}{\theta!} =1$, $k=1$.

Then

$f(\theta|x=5) = \frac{e^{-5}5^{\theta}}{\theta!} $

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  • $\begingroup$ Neither $\pi$ nor $f$ are probability density functions. $\endgroup$ – whuber Apr 15 at 15:49
  • $\begingroup$ @whuber Do you mean that they don't have to be normalized, or were you trying to correct terminology? $\endgroup$ – Nownuri Apr 15 at 16:03
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    $\begingroup$ Neither is normalized. You explicitly claim $f$ is (but it's not) and it's crucial to (at a minimum) specify the domain of $\pi,$ because it's highly ambiguous. $\endgroup$ – whuber Apr 15 at 17:15
  • $\begingroup$ You are finding the constant of integration by summing over $\theta$ from $0$ to $\infty$. Is $0$ really a valid value for $\theta$? Do you really mean to have the prior on $\theta$ allow for only integer values of $\theta$? Note that your prior, as @whuber observes, doesn't integrate to one. $\endgroup$ – jbowman Apr 16 at 0:09
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Your specified sampling density for the Poisson distribution is incorrect (you have switched the observed value with the parameter). As whuber points out in the comments, you have also been a bit sloppy with specifying the domain and normalisation in some parts. Your model with a single observation $x$ should give likelihood and (improper) prior:

$$L_x(\theta) = \theta^x e^{-\theta} \mathbb{I}(\theta > 0) \quad \quad \quad \quad \quad \pi(\theta) \propto \mathbb{I}(\theta>0).$$

Hence, your posterior should be a gamma distribution:

$$\pi(\theta|x) \propto \theta^x e^{-\theta} \mathbb{I}(\theta>0) \propto \text{Ga}(\theta| x+1,1).$$

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