Does the evaluation stand up to evaluation? A first-principle approach to the evaluation of classifiers

How can one meaningfully make a measurement, if the meter does not conform to any standard and its scale expands or shrinks depending on what is measured? In the present work it is argued that current evaluation practices for machine-learning classifiers are affected by this kind of problem, leading to negative consequences when classifiers are put to real use; consequences that could have been avoided. It is proposed that evaluation be grounded on Decision Theory, and the implications of such foundation are explored. The main result is that every evaluation metric must be a linear combination of confusion-matrix elements, with coefficients - "utilities" - that depend on the specific classification problem. For binary classification, the space of such possible metrics is effectively two-dimensional. It is shown that popular metrics such as precision, balanced accuracy, Matthews Correlation Coefficient, Fowlkes-Mallows index, F1-measure, and Area Under the Curve are never optimal: they always give rise to an in-principle avoidable fraction of incorrect evaluations. This fraction is even larger than would be caused by the use of a decision-theoretic metric with moderately wrong coefficients.

Don't guess what's true: choose what's optimal. A probability transducer for machine-learning classifiers

In fields such as medicine and drug discovery, the ultimate goal of a classification is not to guess a class, but to choose the optimal course of action among a set of possible ones, usually not in one-one correspondence with the set of classes. This decision-theoretic problem requires sensible probabilities for the classes. Probabilities conditional on the features are computationally almost impossible to find in many important cases. The main idea of the present work is to calculate probabilities conditional not on the features, but on the trained classifier's output. This calculation is cheap, needs to be made only once, and provides an output-to-probability "transducer" that can be applied to all future outputs of the classifier. In conjunction with problem-dependent utilities, the probabilities of the transducer allow us to find the optimal choice among the classes or among a set of more general decisions, by means of expected-utility maximization. This idea is demonstrated in a simplified drug-discovery problem with a highly imbalanced dataset. The transducer and utility maximization together always lead to improved results, sometimes close to theoretical maximum, for all sets of problem-dependent utilities. The one-time-only calculation of the transducer also provides, automatically: (i) a quantification of the uncertainty about the transducer itself; (ii) the expected utility of the augmented algorithm (including its uncertainty), which can be used for algorithm selection; (iii) the possibility of using the algorithm in a "generative mode", useful if the training dataset is biased.