# Posterior predictive for Gamma distribution with unknown scale and shape

I have a question that needs clarification.

The posterior predictive distribution can be described as the distribution that a new i.i.d. data point $\tilde{x}$ would have, given a set of $N$ existing i.i.d. observations $\mathbf{X} = \{x_1, \dots, x_N\}$

In the Bayesian context this means that we have to calculate the integral:

$p( \tilde{x} | X, \alpha ) = \int_{\theta } p( \tilde{x}| \theta) p(\theta| X, \alpha) d\theta$

Now if I understood correctly how to do that, this means the following: Suppose I have a sequence of i.i.d. datapoints $X_1,...,X_n \sim Poisson(\lambda)$ and I want to find the $\lambda$ that best fits the data.

In the bayesian context I can assign a Gamma prior to the $\lambda$ with hyperparameters $\alpha$ and $\beta$. Now $\lambda \sim Gamma(\alpha, \beta)$ and $\lambda| \textbf{X} = Gamma( \sum x_i + \alpha , n + \beta )$ .

This leads to a posterior predictive of

$p( \tilde{x} | X, \alpha ) = \int_{\theta } p( \tilde{x}| \theta) p(\theta| X, \alpha) d\theta \Rightarrow$

$p( \tilde{x} | X, \alpha ) = \int_0^{\infty} [\frac{\lambda^{x_new} e^{-\lambda}}{x_{new}!}][ \frac{(n+ \beta)^{\sum(x_i + \alpha)}}{ \Gamma ( \sum x_i + \alpha)} \lambda^{\sum x_i + a - 1} e^{-(n+ \beta)\lambda}] d\lambda$..

As soon as I integrate out my parameters, I will end up with the posterior predictive that I need.

I now want to apply the same process on data distributed by a Gamma distribution with unknown shape $\alpha$ and scale $\beta$ parameters. The model is then the following:

$X_1,...,X_n \sim \Gamma(\alpha, \beta)$

$\alpha, \beta \sim exponential(p,q,r,s)$

Where exponential means that the conjugate prior belongs to the exponential family of functions.

My problem is how to calculate the posterior predictive distribution for this model.

I have found a paper that defines(page 25) what the conjugate prior for Gamma with unknown parameters should be. So the model becomes the following

$p( \tilde{x} | X, \alpha ) = \int_0^{\infty} [ \frac{ x^{ \alpha-1}e^ { \frac{-x}{\beta}} }{ \Gamma(\alpha) \beta^{\alpha}} ] [ \frac{1}{K} \frac{ p^{\alpha -1 } e^{\beta q}}{ \Gamma(\alpha) ^ r \beta^{-\alpha s}} ] d\alpha d\beta$

Where K is a normalization factor, and p,q,r,s are hyperparameters.

My problem is that I now have both $\alpha$ and $\beta$ that I need to integrate over both to get my posterior predictive. Is acquiring an analytical closed form posterior predictive plausible in this case?

Cheers, P.

• Generally the answer is no, but you could use a discrete prior and find an analytical form. More importantly it is not clear that you actually want to do what you are describing. If you had $\lambda_i \sim Ga(\alpha,\beta)$, then this would make more sense. – jaradniemi Apr 21 '15 at 15:58
• Thanks for the remark, I changed the notation, I hope it makes more sense. – Panagiotis Chatzichristodoulou Apr 21 '15 at 16:07