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Consider the following:

  1. We have a training dataset ($y_i$, $\mathbf{x}_i$) $i \in [1,n]$ where $y_i \in \{-1, 1\}$.
  2. We can build any model (logistic / SVM / anything else) to predict $y_i$ given $\mathbf{x}_i$ using this training data.
  3. We are then given a new set of predictors $\mathbf{x}'_j$ where $j \in [1, m]$ and we need to predict the corresponding $y'_j$ for each $\mathbf{x}'_j$ using our model.
  4. Each $y'_j \in \{-1, 1, 0\}$.
  5. For each $y'_j \in \{-1, 1\}$, if correct we are rewarded with $1$ dollar and if wrong we are penalized $1$ dollar.
  6. For each $y'_j = 0$, there is no reward or penalty (this is a 'don't care' state).
  7. We want to maximize our payoff.

What's the best approach here? I think building a logistic regression model where we set $y'_j = 0$ if the probability is close to $0.5$ is the best way to go but I haven't been able to flesh this out yet.

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