I'm trying to figure out how to incorporate this formula from this paper into the Weng-Lin update rule algorithms:
\begin{align} \mathcal{P}[i \text{ beats } j] &= \frac{\exp{\lambda_i}}{\exp{\lambda_i} + \exp{\lambda_j}} \\ \lambda_i &= \sum_k^N X_{i,k} \beta_k \\ y_i &\sim \mathrm{Bernoulli}(\mathcal{P}[i \text{ beats } j]) \end{align}
Priors:
$$\beta_k \sim \mathcal{N}(0, \sigma_\beta^2)$$
The chief assumption here being that score margins are player specific predictors. Then given that from equation (57) of Weng's paper:
$$ \lambda_i = \frac{\mu_i}{c_{iq}} $$
Since $\lambda_i$ has different values, how do I merge these two concepts? If it's impossible to add a score margin with this method, what other method can be used?
