Quantile

Given a probability distribution $P$ over a real-valued random variable $\theta$, the $(1 - \delta)$ quantile of the distribution, denoted as:

\[\text{Quantile}_{1 - \delta}(P)\]

is the smallest value $q$ such that:

\[\mathbb{P}(\theta \leq q) \geq 1 - \delta.\]

In other words, $q$ is a threshold below which the random variable $\theta$ falls with probability at least $1 - \delta$.

For a Beta distribution $\theta \sim \text{Beta}(\alpha, \beta)$, this quantile can be computed as:

\[\text{Quantile}_{1 - \delta}\left( \text{Beta}(\alpha, \beta) \right) = \texttt{beta.ppf}(1 - \delta, \alpha, \beta)\]

where beta.ppf is the percent-point function (inverse CDF) from standard statistical libraries.¹