Recalibrating binary probabilistic classifiers

Recalibration of binary probabilistic classifiers to a target prior probability is an important task in areas like credit risk management. We analyse methods for recalibration from a distribution shift perspective. Distribution shift assumptions linked to the area under the curve (AUC) of a probabilistic classifier are found to be useful for the design of meaningful recalibration methods. Two new methods called parametric covariate shift with posterior drift (CSPD) and ROC-based quasi moment matching (QMM) are proposed and tested together with some other methods in an example setting. The outcomes of the test suggest that the QMM methods discussed in the paper can provide appropriately conservative results in evaluations with concave functionals like for instance risk weights functions for credit risk.

Comments on Friedman's Method for Class Distribution Estimation

The purpose of class distribution estimation (also known as quantification) is to determine the values of the prior class probabilities in a test dataset without class label observations. A variety of methods to achieve this have been proposed in the literature, most of them based on the assumption that the distributions of the training and test data are related through prior probability shift (also known as label shift). Among these methods, Friedman's method has recently been found to perform relatively well both for binary and multi-class quantification. We discuss the properties of Friedman's method and another approach mentioned by Friedman (called DeBias method in the literature) in the context of a general framework for designing linear equation systems for class distribution estimation.

Invariance assumptions for class distribution estimation

We study the problem of class distribution estimation under dataset shift. On the training dataset, both features and class labels are observed while on the test dataset only the features can be observed. The task then is the estimation of the distribution of the class labels, i.e. the estimation of the class prior probabilities, in the test dataset. Assumptions of invariance between the training joint distribution of features and labels and the test distribution can considerably facilitate this task. We discuss the assumptions of covariate shift, factorizable joint shift, and sparse joint shift and their implications for class distribution estimation.

Sparse joint shift in multinomial classification

Sparse joint shift (SJS) was recently proposed as a tractable model for general dataset shift which may cause changes to the marginal distributions of features and labels as well as the posterior probabilities and the class-conditional feature distributions. Fitting SJS for a target dataset without label observations may produce valid predictions of labels and estimates of class prior probabilities. We present new results on the transmission of SJS from sets of features to larger sets of features, a conditional correction formula for the class posterior probabilities under the target distribution, identifiability of SJS, and the relationship between SJS and covariate shift. In addition, we point out inconsistencies in the algorithms which were proposed for estimating the characteristics of SJS, as they could hamper the search for optimal solutions.

Factorizable Joint Shift in Multinomial Classification

Factorizable joint shift was recently proposed as a type of dataset shift for which the characteristics can be estimated from observed data. For the multinomial (multi-class) classification setting, we derive a representation of factorizable joint shift in terms of the source (training) distribution, the target (test) prior class probabilities and the target marginal distribution of the features. On the basis of this result, we propose alternatives to joint importance aligning, at the same time pointing out the limitations encountered when making an assumption of factorizable joint shift. Other results of the paper include correction formulae for the posterior class probabilities both under general dataset shift and factorizable joint shift. In addition, we investigate the consequences of assuming factorizable joint shift for the bias caused by sample selection.

Class Prior Estimation under Covariate Shift -- no Problem?

We show that in the context of classification the property of source and target distributions to be related by covariate shift may break down when the information content captured in the covariates is reduced, for instance by discretization of the covariates, dropping some of them, or by any transformation of the covariates even if it is domain-invariant. The consequences of this observation for class prior estimation under covariate shift are discussed. A probing algorithm as alternative approach to class prior estimation under covariate shift is proposed.

Minimising quantifier variance under prior probability shift

For the binary prevalence quantification problem under prior probability shift, we determine the asymptotic variance of the maximum likelihood estimator. We find that it is a function of the Brier score for the regression of the class label against the features under the test data set distribution. This observation suggests that optimising the accuracy of a base classifier on the training data set helps to reduce the variance of the related quantifier on the test data set. Therefore, we also point out training criteria for the base classifier that imply optimisation of both of the Brier scores on the training and the test data sets.

Calibrating sufficiently

When probabilistic classifiers are trained and calibrated, the so-called grouping loss component of the calibration loss can easily be overlooked. Grouping loss refers to the gap between observable information and information actually exploited in the calibration exercise. We investigate the relation between grouping loss and the concept of sufficiency, identifying comonotonicity as a useful criterion for sufficiency. We revisit the probing reduction approach of Langford & Zadrozny (2005) and find that it produces an estimator of probabilistic classifiers that reduces grouping loss. Finally, we discuss Brier curves as tools to support training and 'sufficient' calibration of probabilistic classifiers.

Confidence intervals for class prevalences under prior probability shift

Point estimation of class prevalences in the presence of data set shift has been a popular research topic for more than two decades. Less attention has been paid to the construction of confidence and prediction intervals for estimates of class prevalences. One little considered question is whether or not it is necessary for practical purposes to distinguish confidence and prediction intervals. Another question so far not yet conclusively answered is whether or not the discriminatory power of the classifier or score at the basis of an estimation method matters for the accuracy of the estimates of the class prevalences. This paper presents a simulation study aimed at shedding some light on these and other related questions.

A plug-in approach to maximising precision at the top and recall at the top

For information retrieval and binary classification, we show that precision at the top (or precision at k) and recall at the top (or recall at k) are maximised by thresholding the posterior probability of the positive class. This finding is a consequence of a result on constrained minimisation of the cost-sensitive expected classification error which generalises an earlier related result from the literature.