Since the 1950s, Item Response Theory (IRT) has been used in educational testing environments. It outputs $P(a)$ (the probability that the user will answer the item correctly) given: 1. $j$ - the item; 2. $\theta$ - the student's ability level; 3. $\beta_j$ - the item's difficulty level. This can be formulated as (when a sigmoid is used as an item response function): $P(a) = \sigma (\theta - \beta_j) = \frac{1}{1 + \exp(-\theta - \beta_j)}$. IRT can also be used to predict $\theta$ or $\beta_j$.