Brier score

About: Brier score is a research topic. Over the lifetime, 590 publications have been published within this topic receiving 18964 citations.

Strictly Proper Scoring Rules, Prediction, and Estimation

Abstract: Scoring rules assess the quality of probabilistic forecasts, by assigning a numerical score based on the predictive distribution and on the event or value that materializes. A scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distributionF if he or she issues the probabilistic forecast F, rather than G ≠ F. It is strictly proper if the maximum is unique. In prediction problems, proper scoring rules encourage the forecaster to make careful assessments and to be honest. In estimation problems, strictly proper scoring rules provide attractive loss and utility functions that can be tailored to the problem at hand. This article reviews and develops the theory of proper scoring rules on general probability spaces, and proposes and discusses examples thereof. Proper scoring rules derive from convex functions and relate to information measures, entropy functions, and Bregman divergences. In the case of categorical variables, we prove a rigorous version of the ...

Assessing the performance of prediction models: a framework for traditional and novel measures.

Abstract: The performance of prediction models can be assessed using a variety of methods and metrics. Traditional measures for binary and survival outcomes include the Brier score to indicate overall model performance, the concordance (or c) statistic for discriminative ability (or area under the receiver operating characteristic [ROC] curve), and goodness-of-fit statistics for calibration.Several new measures have recently been proposed that can be seen as refinements of discrimination measures, including variants of the c statistic for survival, reclassification tables, net reclassification improvement (NRI), and integrated discrimination improvement (IDI). Moreover, decision-analytic measures have been proposed, including decision curves to plot the net benefit achieved by making decisions based on model predictions.We aimed to define the role of these relatively novel approaches in the evaluation of the performance of prediction models. For illustration, we present a case study of predicting the presence of residual tumor versus benign tissue in patients with testicular cancer (n = 544 for model development, n = 273 for external validation).We suggest that reporting discrimination and calibration will always be important for a prediction model. Decision-analytic measures should be reported if the predictive model is to be used for clinical decisions. Other measures of performance may be warranted in specific applications, such as reclassification metrics to gain insight into the value of adding a novel predictor to an established model.

Decomposition of the Continuous Ranked Probability Score for Ensemble Prediction Systems

Abstract: Some time ago, the continuous ranked probability score (CRPS) was proposed as a new verification tool for (probabilistic) forecast systems. Its focus is on the entire permissible range of a certain (weather) parameter. The CRPS can be seen as a ranked probability score with an infinite number of classes, each of zero width. Alternatively, it can be interpreted as the integral of the Brier score over all possible threshold values for the parameter under consideration. For a deterministic forecast system the CRPS reduces to the mean absolute error. In this paper it is shown that for an ensemble prediction system the CRPS can be decomposed into a reliability part and a resolution/uncertainty part, in a way that is similar to the decomposition of the Brier score. The reliability part of the CRPS is closely connected to the rank histogram of the ensemble, while the resolution/ uncertainty part can be related to the average spread within the ensemble and the behavior of its outliers. The usefulness of such a decomposition is illustrated for the ensemble prediction system running at the European Centre for Medium-Range Weather Forecasts. The evaluation of the CRPS and its decomposition proposed in this paper can be extended to systems issuing continuous probability forecasts, by realizing that these can be interpreted as the limit of ensemble forecasts with an infinite number of members.

A boosted decision tree approach using Bayesian hyper-parameter optimization for credit scoring

Abstract: Credit scoring is an effective tool for banks to properly guide decision profitably on granting loans. Ensemble methods, which according to their structures can be divided into parallel and sequential ensembles, have been recently developed in the credit scoring domain. These methods have proven their superiority in discriminating borrowers accurately. However, among the ensemble models, little consideration has been provided to the following: (1) highlighting the hyper-parameter tuning of base learner despite being critical to well-performed ensemble models; (2) building sequential models (i.e., boosting, as most have focused on developing the same or different algorithms in parallel); and (3) focusing on the comprehensibility of models. This paper aims to propose a sequential ensemble credit scoring model based on a variant of gradient boosting machine (i.e., extreme gradient boosting (XGBoost)). The model mainly comprises three steps. First, data pre-processing is employed to scale the data and handle missing values. Second, a model-based feature selection system based on the relative feature importance scores is utilized to remove redundant variables. Third, the hyper-parameters of XGBoost are adaptively tuned with Bayesian hyper-parameter optimization and used to train the model with selected feature subset. Several hyper-parameter optimization methods and baseline classifiers are considered as reference points in the experiment. Results demonstrate that Bayesian hyper-parameter optimization performs better than random search, grid search, and manual search. Moreover, the proposed model outperforms baseline models on average over four evaluation measures: accuracy, error rate, the area under the curve (AUC) H measure (AUC-H measure), and Brier score. The proposed model also provides feature importance scores and decision chart, which enhance the interpretability of credit scoring model.

Random forest versus logistic regression: a large-scale benchmark experiment.

Abstract: The Random Forest (RF) algorithm for regression and classification has considerably gained popularity since its introduction in 2001. Meanwhile, it has grown to a standard classification approach competing with logistic regression in many innovation-friendly scientific fields. In this context, we present a large scale benchmarking experiment based on 243 real datasets comparing the prediction performance of the original version of RF with default parameters and LR as binary classification tools. Most importantly, the design of our benchmark experiment is inspired from clinical trial methodology, thus avoiding common pitfalls and major sources of biases. RF performed better than LR according to the considered accuracy measured in approximately 69% of the datasets. The mean difference between RF and LR was 0.029 (95%-CI =[0.022,0.038]) for the accuracy, 0.041 (95%-CI =[0.031,0.053]) for the Area Under the Curve, and − 0.027 (95%-CI =[−0.034,−0.021]) for the Brier score, all measures thus suggesting a significantly better performance of RF. As a side-result of our benchmarking experiment, we observed that the results were noticeably dependent on the inclusion criteria used to select the example datasets, thus emphasizing the importance of clear statements regarding this dataset selection process. We also stress that neutral studies similar to ours, based on a high number of datasets and carefully designed, will be necessary in the future to evaluate further variants, implementations or parameters of random forests which may yield improved accuracy compared to the original version with default values.