The evaluation of probabilistic forecasts plays a central role both in the interpretation and in the use of forecast systems and their development. Probabilistic scores (scoring rules) provide statistical measures to assess the quality of probabilistic forecasts. Often, many probabilistic forecast systems are available while evaluations of their performance are not standardized, with different scoring rules being used to measure different aspects of forecast performance. Even when the discussion is restricted to strictly proper scoring rules, there remains considerable variability between them; indeed strictly proper scoring rules need not rank competing forecast systems in the same order when none of these systems are perfect. The locality property is explored to further distinguish scoring rules. The nonlocal strictly proper scoring rules considered are shown to have a property that can produce "unfortunate" evaluations. Particularly the fact that Continuous Rank Probability Score prefers the outcome close to the median of the forecast distribution regardless the probability mass assigned to the value at/near the median raises concern to its use. The only local strictly proper scoring rules, the logarithmic score, has direct interpretations in terms of probabilities and bits of information. The nonlocal strictly proper scoring rules, on the other hand, lack meaningful direct interpretation for decision support. The logarithmic score is also shown to be invariant under smooth transformation of the forecast variable, while the nonlocal strictly proper scoring rules considered may, however, change their preferences due to the transformation. It is therefore suggested that the logarithmic score always be included in the evaluation of probabilistic forecasts.
Scoring rules are used to evaluate the quality of predictions that take the form of probability distributions. A scoring rule is strictly proper if its expected value is uniquely minimized by the true probability distribution. One of the most well-known and widely used strictly proper scoring rules is the logarithmic scoring rule. We propose a version of the logarithmic scoring rule for competing risks data and show that it remains strictly proper under non-informative censoring.
In recent years, probabilistic forecasting is an emerging topic, which is why there is a growing need of suitable methods for the evaluation of multivariate predictions. We analyze the sensitivity of the most common scoring rules, especially regarding quality of the forecasted dependency structures. Additionally, we propose scoring rules based on the copula, which uniquely describes the dependency structure for every probability distribution with continuous marginal distributions. Efficient estimation of the considered scoring rules and evaluation methods such as the Diebold-Mariano test are d…
A Strictly Proper Scoring-Rule Theory for Calibrating Stochastic Car-Following Models
Institute of Systems Engineering, College of Management and Economics, Tianjin University（天津大学管理与经济学部系统工程研究所）；Laboratory of Computation and Analytics of Complex Management Systems (CACMS), Tianjin University（天津大学复杂管理系统计算与分析实验室）；School of Systems Science, Beijing Jiaotong University（北京交通大学系统科学学院）
We propose a holistic framework for constructing sensitivity measures for any elicitable functional $T$ of a response variable. The sensitivity measures, termed score-based sensitivities, are constructed via scoring functions that are (strictly) consistent for $T$. These score-based sensitivities quantify the relative improvement in predictive accuracy when available information, e.g., from explanatory variables, is used ideally. We establish intuitive and desirable properties of these sensitivities and discuss advantageous choices of scoring functions leading to scale-invariant sensitivities.…
We discuss weighted scoring rules for forecast evaluation and their connection to hypothesis testing. First, a general construction principle for strictly locally proper weighted scoring rules based on conditional densities and scoring rules for probability forecasts is proposed. We show how likelihood-based weighted scoring rules from the literature fit into this framework, and also introduce a weighted version of the Hyvärinen score, which is a local scoring rule in the sense that it only depends on the forecast density and its derivatives at the observation, and does not require evaluation …
The construction of objective priors is, at best, challenging for multidimensional parameter spaces. A common practice is to assume independence and set up the joint prior as the product of marginal distributions obtained via "standard" objective methods, such as Jeffreys or reference priors. However, the assumption of independence a priori is not always reasonable, and whether it can be viewed as strictly objective is still open to discussion. In this paper, by extending a previously proposed objective approach based on scoring rules for the one dimensional case, we propose a novel objective …
Multi-dimensional distributions whose marginal distributions are uniform are called copulas. Among them, the one that satisfies given constraints on expectation and is closest to the independent distribution in the sense of Kullback-Leibler divergence is called the minimum information copula. The density function of the minimum information copula contains a set of functions called the normalizing functions, which are often difficult to compute. Although a number of proper scoring rules for probability distributions having normalizing constants such as exponential families are proposed, these s…
This note is a discussion of the article "Bayesian model selection based on proper scoring rules" by A.P. Dawid and M. Musio, to appear in Bayesian Analysis. While appreciating the concepts behind the use of proper scoring rules, including the inclusion of improper priors, we point out here some possible practical difficulties with the advocated approach.
In situations where forecasters are scored on the quality of their probabilistic predictions, it is standard to use `proper' scoring rules to perform such scoring. These rules are desirable because they give forecasters no incentive to lie about their probabilistic beliefs. However, in the real world context of creating a training program designed to help people improve calibration through prediction practice, there are a variety of desirable traits for scoring rules that go beyond properness. These potentially may have a substantial impact on the user experience, usability of the program, or …
We introduce a class of proper scoring rules for evaluating spatial point process forecasts based on summary statistics. These scoring rules rely on Monte-Carlo approximations of expectations and can therefore easily be evaluated for any point process model that can be simulated. In this regard, they are more flexible than the commonly used logarithmic score and other existing proper scores for point process predictions. The scoring rules allow for evaluating the calibration of a model to specific aspects of a point process, such as its spatial distribution or tendency towards clustering. Usin…
Averages of proper scoring rules are often used to rank probabilistic forecasts. In many cases, the individual terms in these averages are based on observations and forecasts from different distributions. We show that some of the most popular proper scoring rules, such as the continuous ranked probability score (CRPS), give more importance to observations with large uncertainty which can lead to unintuitive rankings. To describe this issue, we define the concept of local scale invariance for scoring rules. A new class of generalized proper kernel scoring rules is derived and as a member of thi…
Typically, point forecasting methods are compared and assessed by means of an error measure or scoring function, such as the absolute error or the squared error. The individual scores are then averaged over forecast cases, to result in a summary measure of the predictive performance, such as the mean absolute error or the (root) mean squared error. I demonstrate that this common practice can lead to grossly misguided inferences, unless the scoring function and the forecasting task are carefully matched. Effective point forecasting requires that the scoring function be specified ex ante, or tha…
A plethora of static and dynamic models exist to forecast Value-at-Risk and other quantile-related metrics used in financial risk management. Industry practice tends to favour simpler, static models such as historical simulation or its variants whereas most academic research centres on dynamic models in the GARCH family. While numerous studies examine the accuracy of multivariate models for forecasting risk metrics, there is little research on accurately predicting the entire multivariate distribution. Yet this is an essential element of asset pricing or portfolio optimization problems having …
In this paper we introduce a novel objective prior distribution levering on the connections between information, divergence and scoring rules. In particular, we do so from the starting point of convex functions representing information in density functions. This provides a natural route to proper local scoring rules using Bregman divergence. Specifically, we determine the prior which solves setting the score function to be a constant. While in itself this provides motivation for an objective prior, the prior also minimizes a corresponding information criterion.
A confidence distribution is a distribution for a parameter of interest based on a parametric statistical model. As such, it serves the same purpose for frequentist statisticians as a posterior distribution for Bayesians, since it allows to reach point estimates, to assess their precision, to set up tests along with measures of evidence, to derive confidence intervals, comparing the parameter of interest with other parameters from other studies, etc. A general recipe for deriving confidence distributions is based on classical pivotal quantities and their exact or approximate distributions. How…
Standard Bayesian analyses can be difficult to perform when the full likelihood, and consequently the full posterior distribution, is too complex and difficult to specify or if robustness with respect to data or to model misspecifications is required. In these situations, we suggest to resort to a posterior distribution for the parameter of interest based on proper scoring rules. Scoring rules are loss functions designed to measure the quality of a probability distribution for a random variable, given its observed value. Important examples are the Tsallis score and the Hyvärinen score, which a…
Proper scoring rules are commonly applied to quantify the accuracy of distribution forecasts. Given an observation they assign a scalar score to each distribution forecast, with the the lowest expected score attributed to the true distribution. The energy and variogram scores are two rules that have recently gained some popularity in multivariate settings because their computation does not require a forecast to have parametric density function and so they are broadly applicable. Here we conduct a simulation study to compare the discrimination ability between the energy score and three variogra…
Many forecasts consist not of point predictions but concern the evolution of quantities. For example, a central bank might predict the interest rates during the next quarter, an epidemiologist might predict trajectories of infection rates, a clinician might predict the behaviour of medical markers over the next day, etc. The situation is further complicated since these forecasts sometimes only concern the approximate "shape of the future evolution" or "order of events". Formally, such forecasts can be seen as probability measures on spaces of equivalence classes of paths modulo time-parametriz…
Objective prior distributions represent an important tool that allows one to have the advantages of using the Bayesian framework even when information about the parameters of a model is not available. The usual objective approaches work off the chosen statistical model and in the majority of cases the resulting prior is improper, which can pose limitations to a practical implementation, even when the complexity of the model is moderate. In this paper we propose to take a novel look at the construction of objective prior distributions, where the connection with a chosen sampling distribution mo…
This paper proposes a novel asymmetric continuous probabilistic score (ACPS) for evaluating and comparing density forecasts. It extends the proposed score and defines a weighted version, which emphasizes regions of interest, such as the tails or the center of a variable's range. A test is also introduced to statistically compare the predictive ability of different forecasts. The ACPS is of general use in any situation where the decision maker has asymmetric preferences in the evaluation of the forecasts. In an artificial experiment, the implications of varying the level of asymmetry in the ACP…
We develop new approaches in multi-class settings for constructing proper scoring rules and hinge-like losses and establishing corresponding regret bounds with respect to the zero-one or cost-weighted classification loss. Our construction of losses involves deriving new inverse mappings from a concave generalized entropy to a loss through the use of a convex dissimilarity function related to the multi-distribution $f$-divergence. Moreover, we identify new classes of multi-class proper scoring rules, which also recover and reveal interesting relationships between various composite losses curren…
