6.5 Latent propensity representation

Sometimes you will see the binary outcome problem represented as a latent propensity where \(Y^*_i\) is a continuous variable that represents an unobserved propensity (e.g., to have a dispute, to be a toxic tweet, to participate), where

\[\begin{gather*} Y_i = \begin{cases}1, \; y^*_i > \tau \\ 0,\; y^*_i \leq \tau \end{cases} \end{gather*}\]

and \(\tau\) is some threshold after which a the event (e.g., dispute) occurs.

This becomes particularly relevant when the goal is to classify outcome estimates given certain \(X\) features. This type of threshold will also be relevant when we move into ordinal outcome variables where we want to estimate the probability an outcome belongs to a specific category.