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?
 A: 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.
