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This section presents the zoology of available verification tools and briefly summarizes their benefits and
limitations. First, we define scoring rules and their key properties. Then, we recall univariate scoring rules,
starting with ones derived from scoring functions used in point forecasting. Finally, we provide an overview
of verification tools for multivariate forecasts.
2.1 Calibration, sharpness, and propriety
Gneiting et al. (2007) proposed a paradigm for the evaluation of probabilistic forecasts: "maximizing the
sharpness of the predictive distributions subject to calibration". Calibration is the statistical compatibility
between the forecast and the observations. Sharpness is the concentration of the forecast and is a property
of the forecast itself. In other words, the paradigm aims at minimizing the uncertainty of the forecast given
that the forecast is statistically consistent with the observations. Tsyplakov (2011) states that the notion of
calibration in the paradigm is too vague but it holds if the definition of calibration is refined. This principle
for the evaluation of probabilistic forecasts has reached a consensus in the field of probabilistic forecast
ing (see, e.g., Gneiting and Katzfuss 2014; Thorarinsdottir and Schuhen 2018). The paradigm proposed
in Gneiting et al. (2007) is not the first mention of the link between sharpness and calibration: for exam
ple, Murphy and Winkler (1987) mentioned the relation between refinement (i.e., sharpness) and calibration.
For univariate forecasts, multiple definitions of calibration are available depending on the setting. The
most used definition is probabilistic calibration and, broadly speaking, consists of computing the rank of
observations among samples of the forecast and checking for uniformity with respect to observations. If
the forecast is calibrated, observations should not be distinguishable from forecast samples, and thus, the
distribution of their ranks should be uniform. Probabilistic calibration can be assessed by probability integral
transform (PIT) histograms (Dawid, 1984) or rank histograms (Anderson, 1996; Talagrand et al., 1997)
for ensemble forecasts when observations are stationary (i.e., their distribution is the same across time).
The shape of the PIT or rank histogram gives information about the type of (potential) miscalibration:
a triangular-shaped histogram suggests that the probabilistic forecast has a systematic bias, a ∪-shaped
histogram suggests that the probabilistic forecast is under-dispersed and a ∩-shaped histogram suggests that
the probabilistic forecast is over-dispersed. Moreover, probabilistic calibration implies that rank histograms
should be uniform but uniformity is not sufficient. For example, rank histograms should also be uniform
conditionally on different forecast scenarios (e.g., conditionally on the value of the observations available
when the forecast is issued). Additionally, under certain hypotheses, calibration tools have been developed
to consider real-world limitations such as serial dependence (Bröcker and Ben Bouallègue, 2020). Statistical
tests have been developed to check the uniformity of rank histograms (Jolliffe and Primo, 2008). Readers
interested in a more in-depth understanding of univariate forecast calibration are encouraged to consult
Tsyplakov (2013, 2020).
For multivariate forecasts, a popular approach relies on a similar principle: first, multivariate forecast
samples are transformed into univariate quantities using so-called pre-rank functions and then the calibration
is assessed by techniques used in the univariate case (see, e.g., Gneiting et al. 2008). Pre-rank functions may
be interpretable and allow targeting the calibration of specific aspects of the forecast such as the dependence
structure. Readers interested in the calibration of multivariate forecasts can refer to Allen et al. (2024) for
a comprehensive review of multivariate calibration.
A scoring rule S assigns a real-valued quantity S(F,y) to a forecast-observation pair (F,y), where F ∈ F
is a probabilistic forecast and y ∈ Rd is an observation. In the negative-oriented convention, a scoring rule
S is proper relative to the class F if
EG[S(G,Y )] ≤ EG[S(F,Y )]
(1)
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for all F,G ∈ F, where EG[···] is the expectation with respect to Y ∼ G. In simple terms, a scoring rule
is proper relative to a class of distribution if its expected value is minimal when the true distribution is
predicted, for any distribution within the class. Forecasts minimizing the expected scoring rule are said to
be efficient and the other forecasts are said to be sub-efficient. Moreover, the scoring rule S is strictly proper
relative to the class F if the equality in (1) holds if and only if F = G. This ensures the characterization of
the ideal forecast (i.e., there is a unique efficient forecast and it is the true distribution). Moreover, proper
scoring rules are powerful tools as they allow the assessment of calibration and sharpness simultaneously
(Winkler, 1977; Winkler et al., 1996). Sharpness can be assessed individually using the entropy associated
with proper scoring rules, defined by eS(F) = EF[S(F,Y )]. The sharper the forecast, the smaller its entropy.
Strictly proper scoring rules can also be used to infer the parameters of a parametric probabilistic forecast
(see, e.g., Gneiting et al. 2005; Pacchiardi et al. 2024).
Regardless of all the interesting properties of proper scoring rules, it is worth noting that they have some
limitations. Proper scoring rules may have multiple efficient forecasts (i.e., associated with their minimal
expected value) and, in the general setting, no guarantee is given on their relevance. Moreover, strict propri
ety ensures that the efficient forecast is unique and that it is the ideal forecast (i.e., the true distribution),
however, no guarantee is available for forecasts within the vicinity of the minimum in the general case. This
is particularly problematic since, in practice, the unavailability of the ideal distribution makes it impossible
to know if the minimum expected score is achieved. In the case of calibrated forecasts, the expected scoring
rule is the entropy of the forecast and the ranking of forecasts is thus linked to the information carried by
the forecast (see Corollary 4, Holzmann and Eulert 2014 for the complete result). These limitations may
explain the plurality of scoring rules depending on application fields.
2.2 Univariate scoring rules
We recall classical univariate scoring rules to explain key concepts. Some univariate scoring rules will be
useful for the multivariate scoring rules construction framework proposed in Section 3. Let P(E) denote the
class of Borel probability measures on E. We consider F ∈ F ⊆ P(R) a probabilistic forecast in the form of
its cumulative distribution function (cdf) and y ∈ R an observation. When the probabilistic forecast F has
a probability density function (pdf), it will be denoted f.
The simplest scoring rules can be derived from scoring functions used to assess point forecasts. The
squared error (SE) is the most popular and is known through its averaged value (the mean squared error;
MSE) or the square root of its average (the root mean squared error; RMSE) which has the advantage of
being expressed in the same units as the observations. As a scoring rule, the SE is expressed as
SE(F,y) = (µF −y)2,
(2)
where µF denotes the mean of the predicted distribution F. The SE solely discriminates the mean of the
forecast (see Appendix A); efficient forecasts for SE are the ones matching the mean of the true distribution.
The SE is proper relative to P2(R), the class of Borel probability measures on R with a finite second moment
(i.e., finite variance). Note that the SE cannot be strictly proper as the equality of mean does not imply the
equality of distributions.
Another well-known scoring rule is the absolute error (AE) defined by
AE(F,y) = |med(F)−y|,
(3)
where med(F) is the median of the predicted distribution F. The mean absolute error (MAE), the average
of the absolute error, is the most seen form of the AE and it is also expressed in the same units as the
observations. Efficient forecasts are forecasts that have a median equal to the median of the true distribution.
The AE is proper relative to P1(R) but not strictly proper. Similarly, the quantile score (QS), also known
as the pinball loss, is a scoring rule focusing on quantiles of level α defined by
QSα(F,y) = (1y≤F−1(α) −α)(F−1(α)−y)
(4)
where 0 < α < 1 is a probability level and F−1(α) is the predicted quantile of level α. The case α = 0.5
corresponds to the AE up to a factor 2. The QS of level α is proper relative to P1(R) but not strictly proper
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since efficient forecasts are ones correctly predicting the quantile of level α (see, e.g., Friederichs and Hense
2008).
Another summary statistic of interest is the exceedance of a threshold t ∈ R. The Brier score (BS; Brier
1950) was initially introduced for binary predictions but allows also to discriminate forecasts based on the