Equations defining probability tree models

Coloured probability tree models are statistical models coding conditional independence between events depicted in a tree graph. They are more general than the very important class of context-specific Bayesian networks. In this paper, we study the algebraic properties of their ideal of model invariants. The generators of this ideal can be easily read from the tree graph and have a straightforward interpretation in terms of the underlying model: they are differences of odds ratios coming from conditional probabilities. One of the key findings in this analysis is that the tree is a convenient tool for understanding the exact algebraic way in which the sum-to-1 conditions on the parameter space translate into the sum-to-one conditions on the joint probabilities of the statistical model. This enables us to identify necessary and sufficient graphical conditions for a staged tree model to be a toric variety intersected with a probability simplex.Comment: 22 pages, 4 figure

Equivalence Classes of Staged Trees

In this paper we give a complete characterization of the statistical equivalence classes of CEGs and of staged trees. We are able to show that all graphical representations of the same model share a common polynomial description. Then, simple transformations on that polynomial enable us to traverse the corresponding class of graphs. We illustrate our results with a real analysis of the implicit dependence relationships within a previously studied dataset.Comment: 18 pages, 4 figure

Sensitivity analysis in multilinear probabilistic models

Sensitivity methods for the analysis of the outputs of discrete Bayesian networks have been extensively studied and implemented in different software packages. These methods usually focus on the study of sensitivity functions and on the impact of a parameter change to the Chan–Darwiche distance. Although not fully recognized, the majority of these results rely heavily on the multilinear structure of atomic probabilities in terms of the conditional probability parameters associated with this type of network. By defining a statistical model through the polynomial expression of its associated defining conditional probabilities, we develop here a unifying approach to sensitivity methods applicable to a large suite of models including extensions of Bayesian networks, for instance context-specific ones. Our algebraic approach enables us to prove that for models whose defining polynomial is multilinear both the Chan–Darwiche distance and any divergence in the family of ϕ-divergences are minimized for a certain class of multi-parameter contemporaneous variations when parameters are proportionally covaried

Discovery of statistical equivalence classes using computer algebra

Discrete statistical models supported on labelled event trees can be specified using so-called interpolating polynomials which are generalizations of generating functions. These admit a nested representation. A new algorithm exploits the primary decomposition of monomial ideals associated with an interpolating polynomial to quickly compute all nested representations of that polynomial. It hereby determines an important subclass of all trees representing the same statistical model. To illustrate this method we analyze the full polynomial equivalence class of a staged tree representing the best fitting model inferred from a real-world dataset.Comment: 26 pages, 9 figure

Staged tree models with toric structure

A staged tree model is a discrete statistical model encoding relationships between events. These models are realised by directed trees with coloured vertices. In algebro-geometric terms, the model consists of points inside a toric variety. For certain trees, called balanced, the model is in fact the intersection of the toric variety and the probability simplex. This gives the model a straightforward description, and has computational advantages. In this paper we show that the class of staged tree models with a toric structure extends far outside of the balanced case, if we allow a change of coordinates. It is an open problem whether all staged tree models have toric structure

Global projections of the soil microbiome in the Anthropocene

Aim: Soil microbes are essential for maintenance of life‐supporting ecosystem services, but projections of how these microbes will be affected by global change scenarios are lacking. Therefore, our aim was to provide projections of future soil microbial distribution using several scenarios of global change. Location: Global. Time period: 1950–2090. Major taxa studied: Bacteria and fungi. Methods: We used a global database of soil microbial communities across six continents to estimate past and future trends of the soil microbiome. To do so, we used structural equation models to include the direct and indirect effects of changes in climate and land use in our predictions, using current climate (temperature and precipitation) and land‐use projections between 1950 and 2090. Results: Local bacterial richness will increase in all scenarios of change in climate and land use considered, although this increase will be followed by a generalized community homogenization process affecting > 85% of terrestrial ecosystems. Changes in the relative abundance of functional genes associated with the increases in bacterial richness are also expected. Based on an ecological cluster analysis, our results suggest that phylotypes such as Geodermatophilus spp. (typical desert bacteria), Mycobacterium sp. (which are known to include important human pathogens), Streptomyces mirabilis (major producers of antibiotic resistance genes) or potential fungal soil‐borne plant pathogens belonging to Ascomycota fungi (Venturia spp., Devriesia spp.) will become more abundant in their communities. Main conclusions: Our results provide evidence that climate change has a stronger influence on soil microbial communities than change in land use (often including deforestation and agricultural expansion), although most of the effects of climate are indirect, through other environmental variables (e.g., changes in soil pH). The same was found for microbial functions such as the prevalence of phosphate transport genes. We provide reliable predictions about the changes in the global distribution of microbial communities, showing an increase in alpha diversity and a homogenization of soil microbial communities in the Anthropocene.This manuscript was developed from discussions within the German Centre of Integrative Biodiversity Research funded by the Deutsche Forschungsgemeinschaft (DFG FZT118). C.A.G. and N.E. acknowledge funding by iDiv (DFG FZT118) Flexpool proposals 34600850 and 34600844. N.E. acknowledges funding by the DFG (FOR 1451) and the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 677232). E.D. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG GRK 2297 –314838170), MathCoRe. M.D.-B. acknowledges support from the Marie Sklodowska-Curie Actions of the Horizon 2020 Framework Program H2020-MSCA-IF-2016 under REA grant agreement number 702057. F.T.M. acknowledges support from the European Research Council grant agreement number 647038 (BIODESERT)

Research-Data Management Planning in the German Mathematical Community

In this paper we discuss the notion of research data for the field of mathematics and report on the status quo of research-data management and planning. A number of decentralized approaches are presented and compared to needs and challenges faced in three use cases from different mathematical subdisciplines. We highlight the importance of tailoring research-data management plans to mathematicians' research processes and discuss their usage all along the data life cycle

Entwicklungsförderung und Gewaltprävention für junge Menschen. Impulse des DFK-Sachverständigenrates für die Auswahl & Durchführung wirksamer Programme. Ein Leitfaden für die Praxis

Die Stiftung Deutsches Forum für Kriminalprävention (DFK) befasst sich kontinuierlich und schwerpunktmäßig mit der Frage, wie Gewaltprävention systematisch und nachhaltig gestaltet und verbreitet werden kann. Ergebnis ist der vorliegende Leitfaden „Entwicklungsförderung und Gewaltprävention für junge Menschen“, der im Rahmen des 18. Deutschen Präventionstages (DPT) in Bielefeld vorgestellt und diskutiert wird. Er knüpft an die Expertise „Gelingensbedingungen für die Prävention von interpersonaler Gewalt im Kindes- und Jugendalter“ an und erweitert die fördernde und präventive Perspektive insbesondere um Aspekte der Effektivität, der Messung von Wirksamkeit und Umsetzungsqualität sowie der Implementierung in Kitas und Schulen. Schließlich werden Fragen des Transfers und einer weitergehenden Verbreitung (Dissemination) von wirksamen und praxistauglichen Präventionsangeboten erörtert. Der Leitfaden richtet sich an professionelle Praktiker, aber auch an Entscheidungsverantwortliche in Institutionen, in Verwaltung und nicht zuletzt in Politik