Is the beta distribution the only conjugate prior distribution to the binomial distribution?
I understand that the beta distribution is a conjugate prior however I can find very little information online which confirms if this is the only conjugate prior distribution and so I have posted this question.
Many thanks
It is slightly misleading to say the beta distribution is the conjugate prior for a binomially distributed likelihood, as there is more than one beta distribution. It would be better to say that beta distributions are a conjugate family of distributions for binomially distributed likelihoods.
The reason they are conjugate is that their densities are proportional to $\theta^\alpha (1-\theta)^\beta$ and when multiplied by a binomial likelihood you get a posterior density from the same beta distribution family though with different $\alpha$ and $\beta$.
There are other families for which this would be true: indeed with a well-behaved function $f(x)$ on $[0,1]$ (non-negative and integrable to a finite positive value), you could have a family of distributions with density proportional to $f(\theta)\theta^\alpha (1-\theta)^\beta$ so that when multiplied by a binomial likelihood you get a posterior density from the same distribution family though with different $\alpha$ and $\beta$.
For example you could get strange wavy priors such as $2\cos^2(10\pi \theta)$ leading to a conjugate family with densities proportional to $\cos^2(10\pi \theta)\theta^\alpha (1-\theta)^\beta$. These particular distributions are not well-known (I just invented them), not as easy to handle and not as useful, but they make up a conjugate family with some curious properties: if say you observed $7000$ successes and $13000$ failures, then your posterior distribution would be bimodal.