Bayes estimator of possion distribution with Pareto prior

Question

Consider a random sample of size $n$ following the possion distribution with parameter $\ln \theta$, that is

$$ f(x|\theta)=\frac{(\ln\theta)^x}{\theta x!}, x=0,1,2,\cdots $$

and the prior of the parameter $\theta$ follows a Pareto distribution:

$$ \mathbb{P}[\theta]= \frac{1}{\beta}\theta^{-\frac{1}{\beta}-1} , \theta > 1 $$

Find the Bayes estimator of $\theta$.

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I have got the expression of the postprior:

$$ \int_1^{\infty} \frac{\ln^x \theta}{\theta x!}\cdot \frac{1}{\beta}\theta^{-\frac1\beta-1}\text{d}\theta $$

I have no idea about this integral. Mathematica shows that this integral equals to $$ -\frac1{\beta x!}\Gamma\left(1+x, \frac{(1+\beta)\ln \theta}{\beta}\right)\left(\frac{1+\beta}{\beta}\right)^{-1-x}\bigg|_{1}^{\infty}=\frac{\beta^x(x+1)}{(1+\beta)^{x+1}} $$

I am not sure about whether my expression of the postprior is correct (there seems no information about the sample). Also I have difficulties about the integral. Any hints will be helpful and appreciated, thank you!

Answer 1

I will simplify the analysis by using the parameter $\alpha = 1/\beta$. Letting $\phi = \ln (\theta)$ we have $d \theta /d \phi = \exp(\phi)$ so we can write the prior for $\phi$ as:

$$\begin{align} \pi (\phi) &= \text{Pareto}(\exp(\phi) | \alpha) \cdot \Bigg| \frac{d \theta}{d \phi} \Bigg| \\[10pt] &= \alpha \exp(\phi)^{-\alpha-1} \cdot \mathbb{I}(\exp(\phi) > 1) \cdot \exp(\phi) \\[16pt] &= \alpha \exp(-\alpha \phi) \cdot \mathbb{I}(\phi > 0). \\[6pt] \end{align}$$

Given a sample $\mathbf{x} = (x_1,...,x_n)$ with sample mean $\bar{x}_n$ the posterior for $\phi$ is then given by:

$$\begin{align} \pi (\phi | \mathbf{x}) &\propto \bigg( \prod_{i=1}^n f(x_i|\phi) \bigg) \cdot \pi (\phi) \\[6pt] &\propto \frac{\phi^{n \bar{x}_n}}{\exp(n\phi)} \cdot \exp(-\alpha \phi) \cdot \mathbb{I}(\phi > 0) \\[12pt] &= \phi^{n \bar{x}_n} \cdot \exp(-(\alpha+n) \phi) \cdot \mathbb{I}(\phi > 0) \\[16pt] &\propto \text{Gamma}(\phi | \text{Shape} = n \bar{x}_n+1, \text{Rate} = \alpha+n). \\[18pt] \end{align}$$

This establishes the posterior $\phi | \mathbf{x} \sim \text{Gamma}(\text{Shape} = {n \bar{x}_n}+1, \text{Rate} = \alpha+n)$ so the posterior for $\theta$ follows the corresponding exponentiated gamma distribution with density function:

$$\begin{align} \pi (\theta | \mathbf{x}) &= \text{Gamma}(\ln(\theta) | \text{Shape} = n \bar{x}_n+1, \text{Rate} = \alpha+n) \cdot \bigg| \frac{d \phi}{d \theta} \bigg| \\[8pt] &= \frac{(\alpha+n)^{n \bar{x}_n+1}}{\Gamma(n \bar{x}_n+1)} \cdot \ln(\theta)^{n \bar{x}_n} \cdot \exp(-(\alpha+n) \ln(\theta)) \cdot \mathbb{I}(\ln(\theta) > 0) \cdot \frac{1}{\theta} \\[6pt] &= \frac{(\alpha+n)^{n \bar{x}_n+1}}{\Gamma(n \bar{x}_n+1)} \cdot \ln(\theta)^{n \bar{x}_n} \cdot \theta^{-(\alpha+n+1)} \cdot \mathbb{I}(\theta > 1). \\[12pt] \end{align}$$