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I am learning Bayes on "Applied Bayesian Statistics" by MK Cowles.

The chapter about "Bayesian Hierarchical Models" mentioned an example that we estimate a softball player’s batting average from her number of hits in 30 at bats that occurred during 8 games.

Suppose in Game i (i=1,2,...,8), the player gave $y_i$ hits and $n_i$ of them were at bats. Furthermore, we will assume that the player’s probability $\pi_i$ of getting a hit could be different in different games.

In the first stage, we write down the likelihood:

$p\left(y_{1}, y_{2}, \ldots, y_{8} \mid \pi_{1}, \pi_{2}, \ldots, \pi_{8}\right)=\prod_{i=1}^{8}\left[\left(\begin{array}{c}n_{i} \\ y_{i}\end{array}\right) \pi_{i}^{y_{i}}\left(1-\pi_{i}\right)^{n_{i}-y_{i}}\right]$

In the second stage, we choose beta distribution as the prior distribution:

$\pi_{i} \sim \operatorname{Beta}(\alpha, \beta), \quad i=1, \ldots 8$

In the third stage, we have to specify prior distributions. This is the step that confuses me a lot:

As $\alpha$ and $\beta$ must be strictly positive, we place gamma priors on both $\alpha$ and $\beta$.

$\alpha \sim \operatorname{Exp}(?)$

$\beta \sim \operatorname{Exp}(?)$

How to set the parameters of the distribution of $\alpha$ and $\beta$ so that it could be weakly informative? Or is there a distribution that is better than gamma densities here?

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  • $\begingroup$ From a practical point of view, I'd probably come up with a reasonable guess of the range of values to expect for alpha and beta, and then choose the hyper priors such that this range of values is well covered... $\endgroup$ – jhin Jul 2 '20 at 10:44
  • $\begingroup$ Also, you might want to look into Jeffrey's priors, which are - as I understand it - the canonical solution to choosing non-informative priors. en.wikipedia.org/wiki/Jeffreys_prior $\endgroup$ – jhin Jul 2 '20 at 10:44
  • $\begingroup$ @jhin Thanks. Jeffreys prior is a good one, but its support is not purely positive. $\endgroup$ – CuteCat Jul 2 '20 at 10:51
  • $\begingroup$ @jhin Yeah. Actually what the textbook does is just a reasonable guess. $\endgroup$ – CuteCat Jul 2 '20 at 10:52
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[Expansion of my comment] From a practical point of view, I'd try to come up with a reasonable guess of the range of values to expect for alpha and beta, and then choose the hyper priors such that this range of values is well covered.

I'll try to give an intuitive explanation of why this makes sense: I believe the idea is that the more layers of priors you have, the less the specific values you choose matter. In your example, what you're actually interested in are the $\pi_i$. The importance of the prior you choose for $\pi_i$ diminishes as more data become available, so unless you have extremely little data, what you choose as a prior doesn't matter so much for the final estimates.

And now you're not even choosing the prior itself, but you're even learning that prior from the data, providing even more flexibility! So as long as the correct values of the $\pi_i$ are permitted by your choice of the hyper priors, you should be fine.

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