In many classification problems it is desirable to output well-calibrated probabilities on the different classes. We propose a robust, non-parametric method of calibrating probabilities called SplineCalib that utilizes smoothing splines to determine a calibration function. We demonstrate how applying certain transformations as part of the calibration process can improve performance on problems in deep learning and other domains where the scores tend to be "overconfident". We adapt the approach to multi-class problems and find that better calibration can improve accuracy as well as log-loss by better resolving uncertain cases. Finally, we present a cross-validated approach to calibration which conserves data. Significant improvements to log-loss and accuracy are shown on several different problems. We also introduce the ml-insights python package which contains an implementation of the SplineCalib algorithm.
Australian National University（澳大利亚国立大学）；Data61, CSIRO（联邦科学与工业研究组织 Data61）；Google Research（谷歌研究院）
Calibrating neural networks is of utmost importance when employing them in safety-critical applications where the downstream decision making depends on the predicted probabilities. Measuring calibration error amounts to comparing two empirical distributions. In this work, we introduce a binning-free calibration measure inspired by the classical Kolmogorov-Smirnov (KS) statistical test in which the main idea is to compare the respective cumulative probability distributions. From this, by approximating the empirical cumulative distribution using a differentiable function via splines, we obtain a…
Smoothing splines have been used pervasively in nonparametric regressions. However, the computational burden of smoothing splines is significant when the sample size $n$ is large. When the number of predictors $d\geq2$, the computational cost for smoothing splines is at the order of $O(n^3)$ using the standard approach. Many methods have been developed to approximate smoothing spline estimators by using $q$ basis functions instead of $n$ ones, resulting in a computational cost of the order $O(nq^2)$. These methods are called the basis selection methods. Despite algorithmic benefits, most of th…
The graph structure of a Bayesian network (BN) can be learned from data using the well-known score-and-search approach. Previous work has shown that incorporating structured representations of the conditional probability distributions (CPDs) into the score-and-search approach can improve the accuracy of the learned graph. In this paper, we present a novel approach capable of learning the graph of a BN and simultaneously modelling linear and non-linear local probabilistic relationships between variables. We achieve this by a combination of feature selection to reduce the search space for local …
In many applications, accurate class probability estimates are required, but many types of models produce poor quality probability estimates despite achieving acceptable classification accuracy. Even though probability calibration has been a hot topic of research in recent times, the majority of this has investigated non-sequential data. In this paper, we consider calibrating models that produce class probability estimates from sequences of data, focusing on the case where predictions are obtained from incomplete sequences. We show that traditional calibration techniques are not sufficiently e…
University of Waikato（怀卡托大学）；University of Auckland（奥克兰大学）
Obtaining accurate and well calibrated probability estimates from classifiers is useful in many applications, for example, when minimising the expected cost of classifications. Existing methods of calibrating probability estimates are applied globally, ignoring the potential for improvements by applying a more fine-grained model. We propose probability calibration trees, a modification of logistic model trees that identifies regions of the input space in which different probability calibration models are learned to improve performance. We compare probability calibration trees to two widely use…
University of Bristol（布里斯托大学）；University of Tartu（塔尔图大学）
The task of calibration is to retrospectively adjust the outputs from a machine learning model to provide better probability estimates on the target variable. While calibration has been investigated thoroughly in classification, it has not yet been well-established for regression tasks. This paper considers the problem of calibrating a probabilistic regression model to improve the estimated probability densities over the real-valued targets. We propose to calibrate a regression model through the cumulative probability density, which can be derived from calibrating a multi-class classifier. We …
University of Pittsburgh（匹兹堡大学）；Harvard University（哈佛大学）
Discovery of an accurate causal Bayesian network structure from observational data can be useful in many areas of science. Often the discoveries are made under uncertainty, which can be expressed as probabilities. To guide the use of such discoveries, including directing further investigation, it is important that those probabilities be well-calibrated. In this paper, we introduce a novel framework to derive calibrated probabilities of causal relationships from observational data. The framework consists of three components: (1) an approximate method for generating initial probability estimates…
Uppsala University（乌普萨拉大学）；Veoneer Inc.（维宁尔公司）；Linköping University（林雪平大学）
Probabilistic classifiers output a probability distribution on target classes rather than just a class prediction. Besides providing a clear separation of prediction and decision making, the main advantage of probabilistic models is their ability to represent uncertainty about predictions. In safety-critical applications, it is pivotal for a model to possess an adequate sense of uncertainty, which for probabilistic classifiers translates into outputting probability distributions that are consistent with the empirical frequencies observed from realized outcomes. A classifier with such a propert…
University of Oulu（奥卢大学）；Oura Health Ltd.（欧拉开健康有限公司）
Many classification applications require accurate probability estimates in addition to good class separation but often classifiers are designed focusing only on the latter. Calibration is the process of improving probability estimates by post-processing but commonly used calibration algorithms work poorly on small data sets and assume the classification task to be binary. Both of these restrictions limit their real-world applicability. Previously introduced Data Generation and Grouping algorithm alleviates the problem posed by small data sets and in this article, we will demonstrate that its a…
Binary classification is highly used in credit scoring in the estimation of probability of default. The validation of such predictive models is based both on rank ability, and also on calibration (i.e. how accurately the probabilities output by the model map to the observed probabilities). In this study we cover the current best practices regarding calibration for binary classification, and explore how different approaches yield different results on real world credit scoring data. The limitations of evaluating credit scoring models using only rank ability metrics are explored. A benchmark is r…
Neural networks solving real-world problems are often required not only to make accurate predictions but also to provide a confidence level in the forecast. The calibration of a model indicates how close the estimated confidence is to the true probability. This paper presents a survey of confidence calibration problems in the context of neural networks and provides an empirical comparison of calibration methods. We analyze problem statement, calibration definitions, and different approaches to evaluation: visualizations and scalar measures that estimate whether the model is well-calibrated. We…
Measurement of uncertainty of predictions from machine learning methods is important across scientific domains and applications. We present, to our knowledge, the first such technique that quantifies the uncertainty of predictions from a classifier and accounts for both the classifier's belief and performance. We prove that our method provides an accurate estimate of the probability that the outputs of two neural networks are correct by showing an expected calibration error of less than 0.2% on a binary classifier, and less than 3% on a semantic segmentation network with extreme class imbalanc…
Generating confidence calibrated outputs is of utmost importance for the applications of deep neural networks in safety-critical decision-making systems. The output of a neural network is a probability distribution where the scores are estimated confidences of the input belonging to the corresponding classes, and hence they represent a complete estimate of the output likelihood relative to all classes. In this paper, we propose a novel form of label smoothing to improve confidence calibration. Since different classes are of different intrinsic similarities, more similar classes should result i…
University of Pittsburgh（匹兹堡大学）；Carnegie Mellon University（卡内基梅隆大学）；M*Modal（M*Modal公司）
Most classification models work by first predicting a posterior probability distribution over all classes and then selecting that class with the largest estimated probability. In many settings however, the quality of posterior probability itself (e.g., 65% chance having diabetes), gives more reliable information than the final predicted class alone. When these methods are shown to be poorly calibrated, most fixes to date have relied on posterior calibration, which rescales the predicted probabilities but often has little impact on final classifications. Here we propose an end-to-end training p…
