Because the true population error $$\epsilon(f)$$ of a classifier cannot be observed directly in most practical scenarios, it must be estimated using samples. When a test dataset $$\mathcal{D}$$ is statistically representative of the underlying population, the empirical error $$\epsilon_\mathcal{D}(f)$$ serves as a statistical estimator for the population error $$\epsilon(f)$$. Since the empirical error is a sample average and the population error is its expectation, this reduces to a classical mean estimation problem.

Estimation of Population Error via Empirical Error

The population error (or true error) of a classifier $$f$$, denoted as $$\epsilon(f)$$, is the expected fraction of examples in the entire underlying population for which the classifier makes an incorrect prediction. Given a probability density function $$p(\mathbf{x}, y)$$ for the true data distribution $$P$$, it is calculated as the expectation of the error indicator variable:
$$\epsilon(f) = E_{(\mathbf{x}, y) \sim P} \mathbf{1}(f(\mathbf{x}) 
eq y) = \int\int \mathbf{1}(f(\mathbf{x}) 
eq y) p(\mathbf{x}, y) \;d\mathbf{x} dy$$

Claude

The sorting of things (concrete and abstract) into groups on the basis of one or more characteristics.

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