Machine learning methods have been remarkably successful for a wide range of application areas in the extraction of essential information from data. An exciting and relatively recent development is the uptake of machine learning in the natural sciences, where the major goal is to obtain novel scientific insights and discoveries from observational or simulated data. A prerequisite for obtaining a scientific outcome is domain knowledge, which is needed to gain explainability, but also to enhance scientific consistency. In this article we review explainable machine learning in view of applications in the natural sciences and discuss three core elements which we identified as relevant in this context: transparency, interpretability, and explainability. With respect to these core elements, we provide a survey of recent scientific works that incorporate machine learning and the way that explainable machine learning is used in combination with domain knowledge from the application areas. * R. Roscher and J. Garcke contributed equally to this work arXiv:1905.08883v3 [cs.LG]

and safety-critical real world applications remains limited. The main factors responsible for this limitation are• the lack of expressiveness and transparency of a deep neural network's inference model, which makes it difficult to trust their outcomes [2],• the inability to distinguish between in-domain and out-ofdomain samples [11], [12] and the sensitivity to domain shifts [13],• the inability to provide reliable uncertainty estimates for a deep neural network's decision [14] and frequently occurring overconfident predictions [15], [16],• the sensitivity to adversarial attacks that make deep neural networks vulnerable for sabotage [17], [18], [19].

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The berry size is one of the most important fruit traits in grapevine breeding. Non-invasive, image-based phenotyping promises a fast and precise method for the monitoring of the grapevine berry size. In the present study an automated image analyzing framework was developed in order to estimate the size of grapevine berries from images in a high-throughput manner. The framework includes (i) the detection of circular structures which are potentially berries and (ii) the classification of these into the class 'berry' or 'non-berry' by utilizing a conditional random field. The approach used the concept of a one-class classification, since only the target class 'berry' is of interest and needs to be modeled. Moreover, the classification was carried out by using an automated active learning approach, i.e. no user interaction is required during the classification process and in addition, the process adapts automatically to changing image conditions, e.g. illumination or berry color. The framework was tested on three datasets consisting in total of 139 images. The images were taken in an experimental vineyard at different stages of grapevine growth according to the BBCH scale. The mean berry size of a plant estimated by the framework correlates with the manually measured berry size by 0.88. 1

Yield estimation and forecasting are of special interest in the field of grapevine breeding and viticulture. The number of harvested berries per plant is strongly correlated with the resulting quality. Therefore, early yield forecasting can enable a focused thinning of berries to ensure a high quality end product. Traditionally yield estimation is done by extrapolating from a small sample size and by utilizing historic data. Moreover, it needs to be carried out by skilled experts with much experience in this field. Berry detection in images offers a cheap, fast and non-invasive alternative to the otherwise time-consuming and subjective on-site analysis by experts. We apply fully convolutional neural networks on images acquired with the Phenoliner, a field phenotyping platform. We count single berries in images to avoid the error-prone detection of grapevine clusters. Clusters are often overlapping and can vary a lot in the size which makes the reliable detection of them difficult. We address especially the detection of white grapes directly in the vineyard. The detection of single berries is formulated as a classification task with three classes, namely 'berry', 'edge' and 'background'. A connected component algorithm is applied to determine the number of berries in one image. We compare the automatically counted number of berries with the manually detected berries in 60 images showing Riesling plants in vertical shoot positioned trellis (VSP) and semi minimal pruned hedges (SMPH). We are able to detect berries correctly within the VSP system with an accuracy of 94.0 % and for the SMPH system with 85.6 %.

Over the last decade, neural networks have reached almost every field of science and become a crucial part of various real world applications. Due to the increasing spread, confidence in neural network predictions has become more and more important. However, basic neural networks do not deliver certainty estimates or suffer from over- or under-confidence, i.e. are badly calibrated. To overcome this, many researchers have been working on understanding and quantifying uncertainty in a neural network’s prediction. As a result, different types and sources of uncertainty have been identified and various approaches to measure and quantify uncertainty in neural networks have been proposed. This work gives a comprehensive overview of uncertainty estimation in neural networks, reviews recent advances in the field, highlights current challenges, and identifies potential research opportunities. It is intended to give anyone interested in uncertainty estimation in neural networks a broad overview and introduction, without presupposing prior knowledge in this field. For that, a comprehensive introduction to the most crucial sources of uncertainty is given and their separation into reducible model uncertainty and irreducible data uncertainty is presented. The modeling of these uncertainties based on deterministic neural networks, Bayesian neural networks (BNNs), ensemble of neural networks, and test-time data augmentation approaches is introduced and different branches of these fields as well as the latest developments are discussed. For a practical application, we discuss different measures of uncertainty, approaches for calibrating neural networks, and give an overview of existing baselines and available implementations. Different examples from the wide spectrum of challenges in the fields of medical image analysis, robotics, and earth observation give an idea of the needs and challenges regarding uncertainties in the practical applications of neural networks. Additionally, the practical limitations of uncertainty quantification methods in neural networks for mission- and safety-critical real world applications are discussed and an outlook on the next steps towards a broader usage of such methods is given.

SummaryThe need for new methods to deal with big data is a common theme in most scientific fields, although its definition tends to vary with the context. Statistical ideas are an essential part of this, and as a partial response, a thematic program on statistical inference, learning and models in big data was held in 2015 in Canada, under the general direction of the Canadian Statistical Sciences Institute, with major funding from, and most activities located at, the Fields Institute for Research in Mathematical Sciences. This paper gives an overview of the topics covered, describing challenges and strategies that seem common to many different areas of application and including some examples of applications to make these challenges and strategies more concrete.

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