Understanding a parameter in a bayesian Poisson model ($\beta$)

Question

I would like to know the meaning or signification of the parameter $\beta$ in this Bayesian model. I have a Poisson model :

$ s_{i} \mid \lambda_{i} \sim Poisson(\lambda_{i}t_{i})$

Where

$\lambda_i\mid\beta\sim\mathcal{G}a(\lambda_i\mid 1.8,\beta)$

and

$\beta\sim\mathcal{G}a(\beta\mid 0.01,1)$

I have as data the next table (which can be found in an article called Robust Empirical Bayes Analyses of Event Rates by O'Muircheartaigh & Gaver ):

\begin{array}{|c|c|c|c|} \hline system & fails (s_{i}) & time (t_{i}) & rates (r_{i} = \frac{s_{i}}{t_{i}}) \\ \hline 1&5&94.32&5.3\times 10^{-2}\\ 2&1&15.72&6.4\times 10^{-2}\\ 3&5&62.88&8.0\times 10^{-2}\\ 4&14&125.76&11.1\times 10^{-2}\\ 5&3&5.24&57.3\times 10^{-2}\\ 6&19&31.44&60.4\times 10^{-2}\\ 7&1&1.05&95.4\times 10^{-2}\\ 8&1&1.05&95.4\times 10^{-2}\\ 9&4&2.10&191\times 10^{-2}\\ 10&22&10.48&209.90\times 10^{-2}\\ \hline \end{array}

I tought that it was a kind of priori knowledge but it seems an interpretation too much simplistic. Is something hide that I am not taking into account? Is something that is going to be revealed to me afterwards when I try to find the posterior distribution?

Answer 1

This is a hyperparameter arising in a mixture representation of the prior

For hierarchical Bayesian models of this kind, the parameter $\beta$ is what we usually call a hyperparameter. A hyperparameter is a parameter that arises in a mixture representation of the prior distribution. In your stipulated hierarchical model, the prior distribution for $\lambda_i$ is a gamma mixture of gamma distributions, and it can be written in marginal form as:

$$\pi(\lambda_i) = \int \limits_0^\infty \text{Ga}(\lambda_i| 1.8, \beta) \cdot \text{Ga}(\beta|0.01, 1) \ d \beta.$$

Importantly, this prior distribution is not a function of the hyperparameter $\beta$. The latter only arises as a quantity appearing in one particular mixture representation of the prior distribution. (You could write the prior distribution in an infinite number of other mixture forms involving different parameters if you prefer.)

Now, in some cases, the hierarchical model is formed in this way because there is some objectively useful reason to frame the prior distribution as a mixture form. In such a case there might be a useful interpretation for the hyperparameter $\beta$, pertaining to the reasoning behind this mixture representation of the prior. However, this is not always the case. Sometimes a mixture-based prior in a hierarchical Bayesian model is formed solely as a way of making the prior more diffuse, and the hyperparameter used in the mixture form may not have any particularly illuminating interpretation. If you would like to obtain an interpretation for $\beta$, the best thing to do would be to look into the mathematical interpretation of a mixing parameter in a gamma mixture of gamma distributions, since this is (mathematically) what it is.