Confidence region prediction is a practically useful extension to the commonly studied pattern recognition problem. Instead of predicting a single label, the constraint is relaxed to allow prediction of a subset of labels given a desired confidence level 1-delta. Ideally, effective region predictions should be (1) well calibrated - predictive regions at confidence level 1-delta should err with relative frequency at most delta and (2) be as narrow (or certain) as possible. We present a simple technique to generate confidence region predictions from conditional probability estimates (probability forecasts). We use this 'conversion' technique to generate confidence region predictions from probability forecasts output by standard machine learning algorithms when tested on 15 multi-class datasets. Our results show that approximately 44% of experiments demonstrate well-calibrated confidence region predictions, with the K-Nearest Neighbour algorithm tending to perform consistently well across all data. Our results illustrate the practical benefits of effective confidence region prediction with respect to medical diagnostics, where guarantees of capturing the true disease label can be given.

The Universal Similarity Metric (USM) has been demonstrated to give practically useful measures of "similarity" between sequence data. Here we have used the USM as an alternative distance metric in a K-Nearest Neighbours (K-NN) learner to allow effective pattern recognition of variable length sequence data. We compare this USM approach with the commonly used string-to-word vector approach. Our experiments have used two data sets of divergent domains: (1) spam email filtering and (2) protein subcellular localisation. Our results with this data reveal that the USM based K-NN learner (1) gives predictions with higher classification accuracy than those output by techniques that use the string to word vector approach, and (2) can be used to generate reliable probability forecasts.

Probability forecasts for binary outcomes, often referred to as probabilistic classifiers or confidence scores, are ubiquitous in science and society, and methods for evaluating and comparing them are in great demand. We propose and study a triptych of diagnostic graphics that focus on distinct and complementary aspects of forecast performance: The reliability diagram addresses calibration, the receiver operating characteristic (ROC) curve diagnoses discrimination ability, and the Murphy diagram visualizes overall predictive performance and value. A Murphy curve shows a forecast's mean elementary scores, including the widely used misclassification rate, and the area under a Murphy curve equals the mean Brier score. For a calibrated forecast, the reliability curve lies on the diagonal, and for competing calibrated forecasts, the ROC and Murphy curves share the same number of crossing points. We invoke the recently developed CORP (Consistent, Optimally binned, Reproducible, and Pool-Adjacent-Violators (PAV) algorithm based) approach to craft reliability diagrams and decompose a mean score into miscalibration (MCB), discrimination (DSC), and uncertainty (UNC) components. Plots of the DSC measure of discrimination ability versus the calibration metric MCB visualize classifier performance across multiple competitors. The proposed tools are illustrated in empirical examples from astrophysics, economics, and social science.

Forecast combinations have flourished remarkably in the forecasting community and, in recent years, have become part of the mainstream of forecasting research and activities. Combining multiple forecasts produced from single (target) series is now widely used to improve accuracy through the integration of information gleaned from different sources, thereby mitigating the risk of identifying a single "best" forecast. Combination schemes have evolved from simple combination methods without estimation, to sophisticated methods involving time-varying weights, nonlinear combinations, correlations among components, and cross-learning. They include combining point forecasts and combining probabilistic forecasts. This paper provides an up-to-date review of the extensive literature on forecast combinations, together with reference to available open-source software implementations. We discuss the potential and limitations of various methods and highlight how these ideas have developed over time. Some important issues concerning the utility of forecast combinations are also surveyed. Finally, we conclude with current research gaps and potential insights for future research.

We consider two or more forecasters each making a sequence of predictions over time and tackle the problem of how to compare them -- either online or post-hoc. In fields ranging from meteorology to sports, forecasters make predictions on different events or quantities over time, and this work describes how to compare them in a statistically rigorous manner. Specifically, we design a nonasymptotic sequential inference procedure for estimating the time-varying difference in forecast quality when using a relatively large class of scoring rules (bounded scores with a linear equivalent). The resulting confidence intervals can be continuously monitored and yield statistically valid comparisons at arbitrary data-dependent stopping times ("anytime-valid"); this is enabled by adapting recent variance-adaptive confidence sequences (CS) to our setting. In the spirit of Shafer and Vovk's game-theoretic probability, the coverage guarantees for our CSs are also distribution-free, in the sense that they make no distributional assumptions whatsoever on the forecasts or outcomes. Additionally, in contrast to a recent preprint by Henzi and Ziegel, we show how to sequentially test a weak null hypothesis about whether one forecaster outperforms another on average over time, by designing different e-processes that quantify the evidence at any stopping time. We examine the validity of our methods over their fixed-time and asymptotic counterparts in synthetic experiments and demonstrate their effectiveness in real-data settings, including comparing probability forecasts on Major League Baseball (MLB) games and comparing statistical postprocessing methods for ensemble weather forecasts.

The remarkable results of Foster and Vohra was a starting point for a series of papers which show that any sequence of outcomes can be learned (with no prior knowledge) using some universal randomized forecasting algorithm and forecast-dependent checking rules. We show that for the class of all computationally efficient outcome-forecast-based checking rules, this property is violated. Moreover, we present a probabilistic algorithm generating with probability close to one a sequence with a subsequence which simultaneously miscalibrates all partially weakly computable randomized forecasting algorithms. %subsequences non-learnable by each randomized algorithm. According to the Dawid's prequential framework we consider partial recursive randomized algorithms.