Why a beta distribution with the parameters $\alpha=0$ and $\beta=0$ as a prior is bad

Question

what happened if I define a beta distribution with $\alpha=0$ and $\beta=0$ as a prior? in other words if $p(\theta) \varpropto \frac{1}{\theta(1-\theta)}$.

Thanks

Answer 1

A prior of the form of a Beta distribution with parameters $\alpha=\beta=0$ is 'bad' in the sense that it is what is referred to as an improper prior. In some circumstances, it leads to an improper posterior distribution, which of course is undesirable. For example, the Beta distribution is a conjugate prior to a bernoulli likelihood, where the posterior is a Beta($\alpha + \sum_{i=1}^n x_i, \beta + n - \sum_{i=1}^nx_i$) distribution. Note that if you have $\alpha=\beta=0$ and either only successes ($x_i=1\forall i$) or failures ($x_i=0\forall i$), one of these two parameters in the posterior is going to be $0$, and hence you will have a posterior that is improper. Improper priors also severely complicate model comparisons.