What are some ways we can get \(r_{i}\) as in the features that goes into some kind of way of getting probabilities for preferences?

Brady-Terry

\begin{equation} p\qty(i \succ j) = \sigma \qty(z), z_{ij} = s_{i} - s_{j} \end{equation}

factor model

• itemwise: \(z_{ij} = u_{i} v_{j} + \beta_{j}\)
• pairwise: \(z_{i,a,b} = u_{i}\qty(v_{a} - v_{b}) + \qty(\beta_{a}- \beta_{b})\)

see factor model

Assume a bilinear structure of preferences from above

• \(u_{i} = G_{\psi}\) <- some thing about users
• \(v_{x,a} = E_{\phi}\qty(x,a)\) <- BERT or similar
• \(r\qty(i,x,a) = u_{i}^{T} v_{x,a} + b_{\eta}\qty(x,a)\)

where \(u_{i}\) is the user vector, \(x\) the context, \(a\) the document ID, and \(b_{\eta}\) is the baseline document preferences.

such that \(\phi, \eta, \psi\) are the learnable parameters.

Assume no structure whatsoever

\begin{equation} r_{\phi} \qty(i,x,a) = V_{\phi}\qty(E_{\phi}\qty(x,a), i) \end{equation}

and just fucking learn it.