A Bradley-Terry Artificial Neural Network Model for Individual ...

8 Joshua Menke, Tony Martinez
rate is low enough. This is because the delta rule is using gradient descent
on an error surface that is strictly convex. Notice that the formulation of
the delta rule in (5) does not include the derivative of the sigmoid. The
sigmoid’s derivative, namely Ouput(1 − Output), is also the inverse of the
derivative of a different objective function called cross-entropy. Therefore,
this version of the delta rule is minimizing the cross-entropy instead of the
squared-error. The cross-entropy of a model is also known as the negative
log-likelihood. This is appealing because the resulting fit will produce the
maximum likelihood estimator. Despite not strictly minimizing the squarederror,
this method will still converge to the global minimum because the
direction of the gradient for minimizing either function is always the same. In
summary, the standard Bradley-Terry model can be fit by reparameterizing
it as a single-layer ANN, and then training it with the delta rule.
3.2 Individial Ratings from Groups
In order to extend the ANN model given in 3.1 to learn individual ratings,
the weights for each group are obtained by averaging the ratings of the individuals
in each group. Huang et. al (Huang et al., 2005) used the sum of
each group which in the ANN model is analagous to having each individual
have an input of 1 if they are on the winning team, −1 if they are on
the losing team, and 0 if they are absent. This model assumes that when
there are an uneven number of individuals on each team, the effect of one
individual is the same in all situations. Here the average is used instead so