Beta-binomial vs updating a prior beta distribution

Bear with me, as I've just recently been learning about conjugate priors, prior and posterior distributions, and such material. My understanding of the beta-binomial distribution is that it basically is used to update a prior beta distribution with a binomial likelihood (data that we have observed that follows a binomial distribution). Together, the prior beta and binomial likelihood are compounded to form a posterior beta distribution. If that is the correct understanding (if not please correct me) then my question is: it sounds very similar to updating a prior beta distribution with observed data by manipulating the prior a & b params, so is there a difference?

Bonus question: is there such a thing as a beta likelihood, and if so what would be an example of data that follows that distribution?

• My understanding is beta distributions are conjugate to binomial likelihoods, as you describe. That leaves you with a beta posterior distribution for the parameter $p$, so the predicted distribution of the next experiment's results is not binomial ($p$ is still uncertain) but "beta-binomial", which has a wider dispersion. When you next update your posterior distribution for $p$, you will still treat the likelihood as binomially based. Mar 3 '21 at 23:01
• @Henry I think theres something critical that i'm misunderstanding here. Some explanations for the betabinomial make it seem as though we're updating the prior beta distribution with the binomial likelihood, is this the wrong conception? Mar 3 '21 at 23:26
• I thought that was what I tried to say. Mar 3 '21 at 23:39
• would you happen to have a basic example of this in action? I'm having trouble understanding how the posterior beta distribution feeds into the p of the binomial function Mar 4 '21 at 0:58
• You can use the beta-binomial distribution directly as a likelihood, some examples: stats.stackexchange.com/questions/188916/…, stats.stackexchange.com/questions/167688/…, stats.stackexchange.com/questions/444750/… Mar 4 '21 at 14:01

You begin with a prior distribution. You combine this with the likelihood to create a posterior distribution. From the posterior distribution, you can create a posterior predictive distribution. A prior predictive distribution also exists if you do not collect data.

In your case, you are beginning with the beta distribution as your prior distribution and a binomial likelihood function. Your posterior is a beta distribution. Your posterior predictive distribution is the beta-binomial distribution. The prior predictive distribution would also be the beta-binomial distribution.

The prior distribution carries all the information you have about the parameter from outside the data. The support for distribution is $$\theta\in[0,1].$$

The likelihood carries the information from the sample. The support for the distribution is $$0\dots{n}$$ over the natural numbers, where $$n$$ is your sample size.

The posterior distribution has the same support as the prior $$\theta\in[0,1]$$.

The predictive distribution has support in $$0\dots{m}$$, where $$m$$ is the size of the future sample being predicted.

Because it is conjugate, updating happens in an analytic manner. For example, if your prior was $$\beta(5,12)$$ and you observed seven successes and eighteen failures, then your posterior would be $$\beta(5+7,12+18)=\beta(12,30)$$.

If your posterior parameters were $$\alpha$$ and $$\beta$$ and $$\gamma$$ is your number of future success and $$\delta$$ is your number of future failures, then your posterior predictive distribution is $${\gamma+\delta\choose{\delta}}{\frac{\mathbf{B}(\alpha+\gamma,\alpha+\beta+\gamma+\delta)}{\mathbf{B}(\alpha,\beta)}},\gamma+\delta=m,$$ where $$\mathbf{B}$$ is the beta function and not the beta distribution.

Finally, yes, the beta distribution could be a likelihood function, but I cannot imagine a real world process that where it would be used.

• Thank you, this was really helpful. Mar 8 '21 at 16:20