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Suppose we have a training data set with variables $a,b$ and $c$ and binary outcome variable $y$. We fit a logistic regression model to this data set:

$$\text{logit}(p) = \hat{\beta_0}+ \hat{\beta_{1}}a + \hat{\beta_{2}}b + \hat{\beta_{3}}c$$

When we get the predicted probabilities from the training data set using the logistic regression model, are we supposed to get perfect classification? Or this depend on the threshold we use?

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Proportion classified correctly is an improper scoring rule, i.e., it is optimized by selecting the wrong features and estimating the wrong coefficients. It is arbitrary and problematic to use any thresholds for continuous quantities.

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Assuming you prediction threshold is 0.5, if your training data are linearly separatable w.r.t. the independent variables (i.e., $a$, $b$, and $c$ in this case) and the optimization underlying the training process converges, yes, you can get a 0 training error.

Getting a perfect classification during training is common when you have a high-dimensional data set. Such data sets are often encountered in text-based classification, bioinformatics, etc.

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