(Model) General Bayesian model of pedagogical reasoning, after Shafto et al. (2014)

@2014shafto formalize rational pedagogical reasoning—that is, reasoning in a setting in which one agent chooses information to transmit to another agent in order to teach a concept—using a recursive Bayesian model that captures two problems:

• The teacher needs to choose helpful examples that serve as data for the learner;
• The learner needs to infer the correct concept from the data, under the assumption that the teacher is choosing helpful examples.

The teacher and learner problems form a mutually dependent system of equations given by:

$$P_{\text{teacher}}(d|h)\propto {\left(\frac{P_{\text{teacher}}(d|h)P(h)}{{\sum }_{h^{\prime }}{P}_{\text{teacher}}(d|h^{\prime })P(h^{\prime })}\right)}^{\alpha },$$

where $d$ is data chosen by the teacher and $h$ is the correct hypothesis.

The parameter $\alpha$ captures a “soft maximization” of the posterior probability, or the degree to which the teacher tends towards maximizing the posterior instead of choosing the most representative samples. As $\alpha \to 0$, the teacher chooses uniformly among data consistent with the hypothesis, with $\alpha =0$ being the case of random sampling (i.e., not helpful) and $\alpha =1$ representing selection in direct proportion to the posterior probability they give to the target hypothesis.

Under this model, teachers compute the utility of selecting an example using theory-of-mind. @2022chen extend this model to capture uncertainty during adaptive and interactive pedagogical setting.