Ranking methods or models based on their performance is of prime importance but is tricky because performance is fundamentally multidimensional. In the case of classification, precision and recall are scores with probabilistic interpretations that are both important to consider and complementary. The rankings induced by these two scores are often in partial contradiction. In practice, therefore, it is extremely useful to establish a compromise between the two views to obtain a single, global ranking. Over the last fifty years or so,it has been proposed to take a weighted harmonic mean, known as the F-score, F-measure, or $F_β$. Generally speaking, by averaging basic scores, we obtain a score that is intermediate in terms of values. However, there is no guarantee that these scores lead to meaningful rankings and no guarantee that the rankings are good tradeoffs between these base scores. Given the ubiquity of $F_β$ scores in the literature, some clarification is in order. Concretely: (1) We establish that $F_β$-induced rankings are meaningful and define a shortest path between precision- and recall-induced rankings. (2) We frame the problem of finding a tradeoff between two scores as an optimization problem expressed with Kendall rank correlations. We show that $F_1$ and its skew-insensitive version are far from being optimal in that regard. (3) We provide theoretical tools and a closed-form expression to find the optimal value for $β$ for any distribution or set of performances, and we illustrate their use on six case studies.

The ROC curve is widely used to assess binary classification performance. Yet for some applications such as alert systems for hospitalized patient monitoring, conventional ROC analysis cannot capture crucial factors that impact deployment, such as enforcing a minimum precision constraint to avoid false alarm fatigue or imposing an upper bound on the number of predicted positives to represent the capacity of hospital staff. The usual area under the curve metric also does not reflect asymmetric costs for false positives and false negatives. In this paper we address all three of these issues. First, we show how the subset of classifiers that meet given precision and capacity constraints can be represented as a feasible region in ROC space. We establish the geometry of this feasible region. We then define the partial area of lesser classifiers, a performance metric that is monotonic with cost and only accounts for the feasible portion of ROC space. Averaging this area over a desired range of cost parameters results in the partial volume over the ROC surface, or partial VOROS. In experiments predicting mortality risk using vital sign history on the MIMIC-IV dataset, we show this cost-aware metric is better than alternatives for ranking classifiers in hospital alert applications.

The area under the ROC curve is a common measure that is often used to rank the relative performance of different binary classifiers. However, as has been also previously noted, it can be a measure that ill-captures the benefits of different classifiers when either the true class values or misclassification costs are highly unbalanced between the two classes. We introduce a third dimension to capture these costs, and lift the ROC curve to a ROC surface in a natural way. We study both this surface and introduce the VOROS, the volume over this ROC surface, as a 3D generalization of the 2D area under the ROC curve. For problems where there are only bounds on the expected costs or class imbalances, we restrict consideration to the volume of the appropriate subregion of the ROC surface. We show how the VOROS can better capture the costs of different classifiers on both a classical and a modern example dataset.

Machine learning is about machines making decisions and, as we have already discussed, we can produce multiple models for any given problem and measure their accuracy. It is intuitively obvious that we would elect to use the most accurate model and most of the time, of course, we do. But there are times when we will actually elect to use one of the less accurate ones. The underlying reason is that the estimates we make of accuracy, whilst very useful, take no account of the cost of being right and being wrong. We might be trying to identify which of our customers on a clothing website are women and which men so that our recommendation engine makes the appropriate clothing suggestions.

Machine learning (ML) is all about getting machines to learn but how do we know how well they are doing? Suppose we have existing data about people who buy books from our website – age, amount spent, preferred author and so on. These columns of data are known, in ML terms, as the "Predictors". For some of the customers we also know their gender, for others we don't, but we want to know it for all customers. We start with the customers where we do know the gender and we feed both the Predictor and Response data into a machine learning algorithm and get it to build a model that can (hopefully) forecast the Response from the Predictors.

Deep learning research in medicine is a bit like the Wild West at the moment; sometimes you find gold, sometimes a giant steampunk spider-bot causes a ruckus. This has derailed my series on whether AI will be replacing doctors soon, as I have felt the need to focus a bit more on how to assess the quality of medical AI research. I wanted to start closing out my series on the role of AI in medicine. What has happened instead is that several papers have claimed to beat doctors, and have failed to justify these claims. Despite this, and despite not going through peer review, the groups involved have issued press releases about their achievements, marketing the results direct to the public and the media. I don't think this is malicious. I think there is a cultural divide between the machine learning and the medical communities, a different way of doing research, and a different level of evidence required for making strong claims. If I have to be honest, I think the machine learning community has a fair bit to learn from medical research in this regard. Last time I made a set of three rules about how to assess medical AI research, and the third was a glib recommendation to "actually read the paper".

The business world is full of streams of items that need to be filtered or evaluated: parts on an assembly line, resumés in an application pile, emails in a delivery queue, transactions awaiting processing. Machine learning techniques are increasingly being used to make such processes more efficient: image processing to flag bad parts, text analysis to surface good candidates, spam filtering to sort email, fraud detection to lower transaction costs etc. In this article, I show how you can take business factors into account when using machine learning to solve these kinds of problems with binary classifiers. Specifically, I show how the concept of expected utility from the field of economics maps onto the Receiver Operating Characteristic (ROC) space often used by machine learning practitioners to compare and evaluate models for binary classification. I begin with a parable illustrating the dangers of not taking such factors into account. This concrete story is followed by a more formal mathematical look at the use of indifference curves in ROC space to avoid this kind of problem and guide model development. I wrap up with some recommendations for successfully using binary classifiers to solve business problems.

The business world is full of streams of items that need to be filtered or evaluated: parts on an assembly line, resumés in an application pile, emails in a delivery queue, transactions awaiting processing. Machine learning techniques are increasingly being used to make such processes more efficient: image processing to flag bad parts, text analysis to surface good candidates, spam filtering to sort email, fraud detection to lower transaction costs etc. In this article, I show how you can take business factors into account when using machine learning to solve these kinds of problems with binary classifiers. Specifically, I show how the concept of expected utility from the field of economics maps onto the Receiver Operating Characteristic (ROC) space often used by machine learning practitioners to compare and evaluate models for binary classification. I begin with a parable illustrating the dangers of not taking such factors into account. This concrete story is followed by a more formal mathematical look at the use of indifference curves in ROC space to avoid this kind of problem and guide model development. I wrap up with some recommendations for successfully using binary classifiers to solve business problems.

The business world is full of streams of items that need to be filtered or evaluated: parts on an assembly line, resumés in an application pile, emails in a delivery queue, transactions awaiting processing. Machine learning techniques are increasingly being used to make such processes more efficient: image processing to flag bad parts, text analysis to surface good candidates, spam filtering to sort email, fraud detection to lower transaction costs etc. In this article, I show how you can take business factors into account when using machine learning to solve these kinds of problems with binary classifiers. Specifically, I show how the concept of expected utility from the field of economics maps onto the Receiver Operating Characteristic (ROC) space often used by machine learning practitioners to compare and evaluate models for binary classification. I begin with a parable illustrating the dangers of not taking such factors into account. This concrete story is followed by a more formal mathematical look at the use of indifference curves in ROC space to avoid this kind of problem and guide model development. I wrap up with some recommendations for successfully using binary classifiers to solve business problems.

ROC curves and cost curves are two popular ways of visualising classifier performance, finding appropriate thresholds according to the operating condition, and deriving useful aggregated measures such as the area under the ROC curve (AUC) or the area under the optimal cost curve. In this note we present some new findings and connections between ROC space and cost space, by using the expected loss over a range of operating conditions. In particular, we show that ROC curves can be transferred to cost space by means of a very natural way of understanding how thresholds should be chosen, by selecting the threshold such that the proportion of positive predictions equals the operating condition (either in the form of cost proportion or skew). We call these new curves {ROC Cost Curves}, and we demonstrate that the expected loss as measured by the area under these curves is linearly related to AUC. This opens up a series of new possibilities and clarifies the notion of cost curve and its relation to ROC analysis. In addition, we show that for a classifier that assigns the scores in an evenly-spaced way, these curves are equal to the Brier Curves. As a result, this establishes the first clear connection between AUC and the Brier score.