Updating a beta-binomial

Question

Suppose I'm modeling a set of processes using a beta-binomial prior. I can build parameterized beta-binomial models that average over large groups of the processes to give reasonable, although coarse, priors.

$p_i \sim \beta B(n, \alpha_i, \beta_i)$ (roughly)

I know how to update those priors using observed partial data via Bayes' rule. However, for a subset of the priors, I actually have a little more historical data that I'd like to incorporate into the prior, call it $h_j$, where $j \in h$ is a subset of the $i$s. So the result would be an updated distribution, call it $p'_i$. That additional data is a scalar. For example, if I've got a beta-binomial with $n=9$, $\alpha=2$ and $\beta=3$ (see the examples for the dbetabin.ab function in the VGAM R package), it has a mode of 3, but I might have additional prior information that suggests the mode should be closer to 6. I happen to know that this additional information is only modestly predictive ($r$ of .4, say). But it's still better than nothing, and for this particular process, it's known to be a better predictor than the expected value of my existing beta-binomial prior ($r$ of around .3).

So, what I'm looking for, is a way to update the beta-binomial, using this scalar, so that the result is also a beta-binomial, which I can then update like any of my other process models as data comes in. (That is, I need a closed-form expression.) $(\alpha'_i, \beta'_i) = f(\alpha_i, \beta_i, h_i, \theta)$, where $\theta$ has something to do with the relative estimated predictiveness of the original beta-binomial and the scalar $h$.

What's a reasonable approach here? Is there a way to adjust the $\alpha$ and $\beta$ parameters so that the central tendency is pulled an appropriate amount towards my modestly-predictive scalar? I'm happy to use cross-validation or something to identify a weighting parameter, if that's the right way to go about this.

Thanks!

Answer 1

Lets see if I understand Harlan's (and Srikant's) formulation correctly.

$$\pi_1 \sim beta(\alpha_1,\beta_1)$$ $$\pi_2 \sim beta(\alpha_2,\beta_2)$$

Say, $\pi_1$ corresponds to the set of data for which you have less information apriori and $\pi_2$ is for the more precise data set.

Using Srikant's formulation:

$$\pi(p) \propto \pi_1(p) \alpha + \pi_2(p) (1-\alpha)$$

Therefore, the complete hierarchical formulation will be: $$\alpha \sim beta(\alpha_0,\beta_0)$$ $$p|\alpha \sim \pi(p)$$ $$y_i | p \sim B(n_i,p) $$

I assume here that $y_i|p$ are iid. I don't know if this is a valid assumption in your case. Am I correct?

You can choose $\alpha_0$ and $\beta_0$ in such a way that mean of this beta distribution is 0.8 (or 0.2) acc. to your formulation.

Now the MCMC sampling can be done, by using OpenBUGS or JAGS (untested).

I will add more to this (and recheck formulation) as soon as I get more time. But please point out if you see a fallacy in my argument.

Answer 2

Assume that prior2 is a beta random variable and set $\alpha$ and $\beta$ as needed subject to the constraint that $\frac{\alpha-1}{\alpha + \beta -2} = 6$.

In response to your comment:

1. Getting to prior2:

   • Fix either $\alpha$ or $\beta$ at the same value as prior1 and tweak the other to match the desired mode.

   • If the above does not work then you can use whatever constraints you want to impose (e.g., same variance) and use some sort of routine (e.g., optimization) to get to your desired mode (e.g., Min abs($\frac{\alpha-1}{\alpha + \beta -2} - 6)$ subject to constraints) or simply play around till your prior2 parameters are consistent with your constraints.

2. Accommodating the fact that you do not fully believe in prior2:

   A principled way to approach the issue of 20% trust in prior2 is to assume mixture priors.

   Thus, your prior is: $f(\alpha_1,\beta_1|-) 0.8 + f(\alpha_2,\beta_2|-) 0.2$. You could multiply your likelihood with the above mixture priors to get a beta-binomial model.