Predictability estimates of ensemble prediction systems are uncertain due to limited numbers of past forecasts and observations. To account for such uncertainty, this paper proposes a Bayesian inferential framework that provides a simple 6-parameter representation of ensemble forecasting systems and the corresponding observations. The framework is probabilistic, and thus allows for quantifying uncertainty in predictability measures such as correlation skill and signal-to-noise ratios. It also provides a natural way to produce recalibrated probabilistic predictions from uncalibrated ensembles forecasts. The framework is used to address important questions concerning the skill of winter hindcasts of the North Atlantic Oscillation for 1992-2011 issued by the Met Office GloSea5 climate prediction system. Although there is much uncertainty in the correlation between ensemble mean and observations, there is strong evidence of skill: the 95% credible interval of the correlation coefficient of [0.19, 0.68] does not overlap zero. There is also strong evidence that the forecasts are not exchangeable with the observations: With over 99% certainty, the signalto-noise ratio of the forecasts is smaller than the signal-to-noise ratio of the observations, which suggests that raw forecasts should not be taken as representative scenarios of the observations. Forecast recalibration is thus required, which can be coherently addressed within the proposed framework.

The rapid emergence of SARS-CoV-2 mutants with new phenotypic properties is a critical challenge to the control of the ongoing pandemic. B.1.1.7 was monitored in the UK through routine testing and S-gene target failures (SGTF), comprising over 90% of cases by March 2021. Now, the reverse is occurring: SGTF cases are being replaced by an S-gene positive variant, which we associate with B.1.617.2. Evidence from the characteristics of S-gene positive cases demonstrates that, following importation, B.1.617.2 is transmitted locally, growing at a rate higher than B.1.1.7 and a doubling time between 5-14 days. S-gene positive cases should be prioritised for sequencing and aggressive control in any countries in which this variant is newly detected.One-Sentence SummaryThe B.1.617.2 variant of SARS-CoV-2 is replacing B.1.1.7 and emerging as the dominant variant in England, evidenced by sustained local transmission.

The skill of weather and climate forecast systems is often assessed by calculating the correlation coefficient between past forecasts and their verifying observations. Improvements in forecast skill can thus be quantified by correlation differences. The uncertainty in the correlation difference needs to be assessed to judge whether the observed difference constitutes a genuine improvement, or is compatible with random sampling variations. A widely used statistical test for correlation difference is known to be unsuitable, because it assumes that the competing forecasting systems are independent. In this paper, appropriate statistical methods are reviewed to assess correlation differences when the competing forecasting systems are strongly correlated with one another. The methods are used to compare correlation skill between seasonal temperature forecasts that differ in initialization scheme and model resolution. A simple power analysis framework is proposed to estimate the probability of correctly detecting skill improvements, and to determine the minimum number of samples required to reliably detect improvements. The proposed statistical test has a higher power of detecting improvements than the traditional test. The main examples suggest that sample sizes of climate hindcasts should be increased to about 40 years to ensure sufficiently high power. It is found that seasonal temperature forecasts are significantly improved by using realistic land surface initial conditions.

Seasonal forecasts of variables like near-surface temperature or precipitation are becoming increasingly important for a wide range of stakeholders. Due to the many possibilities of recalibrating, combining, and verifying ensemble forecasts, there are ambiguities of which methods are most suitable. To address this we compare approaches how to process and verify multimodel seasonal forecasts based on a scientific assessment performed within the framework of the EU Copernicus Climate Change Service (C3S) Quality Assurance for Multi-model Seasonal Forecast Products (QA4Seas) contract C3S 51 lot 3. Our results underpin the importance of processing raw ensemble forecasts differently depending on the final forecast product needed. While ensemble forecasts benefit a lot from bias correction using climate conserving recalibration, this is not the case for the intrinsically bias adjusted multi-category probability forecasts. The same applies for multi-model combination. In this paper, we apply simple, but effective, approaches for multi-model combination of both forecast formats. Further, based on existing literature we recommend to use proper scoring rules like a sample version of the continuous ranked probability score and the ranked probability score for the verification of ensemble forecasts and multi-category probability forecasts, respectively. For a detailed global visualization of calibration as well as bias and dispersion errors, using the Chi-square decomposition of rank histograms proved to be appropriate for the analysis performed within QA4Seas.

With modern high-dimensional data, complex statistical models are necessary, requiring computationally feasible inference schemes. We introduce Max-and-Smooth, an approximate Bayesian inference scheme for a flexible class of latent Gaussian models (LGMs) where one or more of the likelihood parameters are modeled by latent additive Gaussian processes. Our proposed inference scheme is a two-step approach. In the first step (Max), the likelihood function is approximated by a Gaussian density with mean and covariance equal to either (a) the maximum likelihood estimate and the inverse observed information, respectively, or (b) the mean and covariance of the normalized likelihood function. In the second step (Smooth), the latent parameters and hyperparameters are inferred and smoothed with the approximated likelihood function. The proposed method ensures that the uncertainty from the first step is correctly propagated to the second step. Because the prior density for the latent parameters is assumed to be Gaussian and the approximated likelihood function is Gaussian, the approximate posterior density of the latent parameters (conditional on the hyperparameters) is also Gaussian, thus facilitating efficient posterior inference in high dimensions. Furthermore, the approximate marginal posterior distribution of the hyperparameters is tractable, and as a result, the hyperparameters can be sampled independently of the latent parameters. We show that the computational cost of Max-and-Smooth is close to being insensitive to the number of independent data replicates, and that it scales well with increased dimension of the latent parameter vector provided that its Gaussian prior density is specified with a sparse precision matrix. In the case of a large number of independent data replicates, sparse precision matrices, and high-dimensional latent vectors, the speedup is substantial in comparison to an MCMC scheme that infers the posterior density from the exact likelihood function. The accuracy of the Gaussian approximation to the likelihood function increases with the number of data replicates per latent model parameter. The proposed inference scheme is demonstrated on one spatially referenced real dataset and on simulated data mimicking spatial, temporal, and spatio-temporal inference problems. Our results show that Max-and-Smooth is accurate and fast.