Marginal problems naturally arise in a variety of different fields: basically, the question is whether some marginal/partial information is compatible with a joint probability distribution. To this aim, the characterization of marginal sets via quantifier elimination and polyhedral projection algorithms is of primal importance. In this work, before considering specific problems, we review polyhedral projection algorithms with focus on applications in information theory, and, alongside known algorithms, we also present a newly developed geometric algorithm which walks along the face lattice of the polyhedron in the projection space. One important application of this is in the field of quantum non-locality, where marginal problems arise in the computation of Bell inequalities. We apply the discussed algorithms to discover many tight entropic Bell inequalities of the tripartite Bell scenario as well as more complex networks arising in the field of causal inference. Finally, we analyze the usefulness of these inequalities as nonlocality witnesses by searching for violating quantum states.

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<p>Carbon is an essential element and contributor to healthy soil conditions as well as ecological soil function and productivity. Additionally, carbon is a component of all plants and animals on the planet and is a necessary component of life. Natural vegetation serves as a significant but highly dynamic carbon sink. When vegetation is removed quicker than it can regenerate, for example by harvesting crops or timber, soil carbon is depleted. Thus, understanding the environmental effects and dynamics of loss of vegetation is a crucial prerequisite to turning our natural resource management from a carbon emitter to a carbon sink to avoid that and achieve sustainability. At the same time, the spatial distribution of soil organic carbon is also highly heterogeneous, with variations in climate, other soil characteristics, and land use/land cover affecting how our ecosystem reacts to the loss of vegetation. Thus, to effectively improve green metrics and contribute to the creation of future policies, it is required to conduct research on the changes in vegetation and their effect on soil organic carbon and provide regionally appropriate management advice. Here, in this research, our goal is to examine the "individual treatment effects" (ITE), which are a personalized or individualized effect estimation of one variable on the output, and utilize causal inference to address them.  Using the LUCAS dataset, we explore the heterogeneous treatment effect of percent tree coverage (PTC), as a parameter of the density of trees on the ground, on the soil organic carbon content in Germany. We do this by leveraging some parameters, such as climate data, land use/land cover information, and other information from the soil. We thus offer a data-driven viewpoint for focusing on sustainable behaviors and effectively increasing soil organic carbon content levels.</p>

<p>Artificial neural networks (ANN), which are mainly used in pattern and image recognition, have now found a wide range of applications in soil science and geoscience. They have proven to be a useful tool for complex questions that also involve a large amount of data, for example prediction of soil classes or soil properties on various scales. However, we face two main challenges when applying ANN: In their basic form, deep-learning algorithms do not provide interpretable predictive uncertainty. Thus, in geosciences and in particular in soil science, they have been used more as black-box models and properties of a machine learning model such as the certainty and plausibility of the predicted variables, for example soil classes, were interpretation by experts rather than quantified by metrics validating the ANN. In most cases regression coefficients or comparable statistical measure are reported for the overall performance of the model. This leads to the second challenge, that is that these algorithms have high confidence of their predictions in areas far away from the training area or in areas where they receive only little information from a small number of data points. <br />In order to gain a better understanding of these aforementioned properties, we implement in our explorative study on soil classification a Bayesian deep learning approach (i.e., a method to add uncertainty to deep networks) known as last layer Laplace approximation. This is a technique that can be applied as a post-hoc "add-on" without destroying the otherwise good performance of deep classifiers. It helps us to correct the overconfident areas without reducing the accuracy of our prediction, giving us a more realistic uncertainty expression of the model's prediction.  <br />Our predictor variable soil type provides us with a large amount of complex information about soil processes and properties, which is a great advantage since it would take a lot of time and money to collect all this information individually. At the same time, soil maps are in high demand by authorities, construction companies or farmers. In our study area around Tübingen in southern Germany, there are 41 different soil types, determined according to the German soil classification, sub divisible into typical soils of the Neckar and Ammer valleys, the Swabian Jura and Black Forest, and non-area related soil. In addition to the underlying soil map, remotely sensed variables, a digital elevation model and its derivatives are used as input to the ANN, which is designed to learn the relationship between these and the soil type. As a test case, we then explicitly exclude the Swabian Jura and Black Forest in the training area but include them as prediction regions. Both regions are characterized by very different soil types compared to the rest of the study area due to their considerably different geology, climate, and terrain. Our goal is then to enrich soil type maps with a structured uncertainty to better understand the causality of machine learning models in soil science and their transferability to regions other than the training and validation area.</p>

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