International Journal of Approximate Reasoning最新文献

Differential privacy (DP) is a class of mathematical standards for assessing the privacy provided by a data-release mechanism. This work concerns two important flavors of DP that are related yet conceptually distinct: pure ε-differential privacy (ε-DP) and Pufferfish privacy. We restate ε-DP and Pufferfish privacy as Lipschitz continuity conditions and provide their formulations in terms of an object from the imprecise probability literature: the interval of measures. We use these formulations to derive limits on key quantities in frequentist hypothesis testing and in Bayesian inference using data that are sanitised according to either of these two privacy standards. Under very mild conditions, the results in this work are valid for arbitrary parameters, priors and data generating models. These bounds are weaker than those attainable when analysing specific data generating models or data-release mechanisms. However, they provide generally applicable limits on the ability to learn from differentially private data – even when the analyst's knowledge of the model or mechanism is limited. They also shed light on the semantic interpretations of the two DP flavors under examination, a subject of contention in the current literature.¹

Formal concept analysis is a powerful mathematical framework based on mathematical logic and lattice theory for analyzing object-attribute relational systems. Over the decades, Formal concept analysis has evolved from its theoretical foundations into a versatile methodology applied across various disciplines. A heterogeneous formal context provides a feasible generalization of a formal context, enabling diverse truth-degrees of objects, attributes, and fuzzy relations. In our paper, we present extended theoretical results on heterogeneous formal contexts, including bimorphisms, Galois connections, and heterogeneous attribute implications. We recall the basic notions and properties of the heterogeneous formal context and its concept lattice. Moreover, we present extended results on bimorphisms and Galois connections in a heterogeneous formal context, including a self-contained proof of the main result. We include examples of introduced notions in heterogeneous formal contexts and two-valued logic. We propose the extension of attribute implications for heterogeneous formal contexts and explore their validity. By embracing heterogeneity in Formal concept analysis, we enrich its extended theoretical foundations and pave the way for innovative applications across diverse domains, including personal data protection and cybersecurity.

We identify the (filter representation of the) logic behind the recent theory of coherent sets of desirable (sets of) things, which generalise coherent sets of desirable (sets of) gambles as well as coherent choice functions, and show that this identification allows us to establish various representation results for such coherent models in terms of simpler ones.

We give a complete characterisation of the single and double arrow relations of the standard context $K\left(L_n\right)$ of the lattice $L_n$ of partitions of any positive integer n under the dominance order, thereby addressing an open question of Ganter, 2020/2022.

Our study is centered on the aggregation of fuzzy graphs, looking for conditions under which the aggregation process yields another fuzzy graph. We conduct an in-depth analysis of the preservation of several important properties and structures inherent to fuzzy graphs, like paths, cycles, or bridges. In addition we obtain appropriate criteria for when the aggregation of complete fuzzy graphs is again a complete fuzzy graph.

To reduce the estimator's variance and prevent overfitting, regularization techniques have attracted great interest from the statistics and machine learning communities. Most existing regularized methods rely on the sparsity assumption that a model with fewer parameters predicts better than one with many parameters. This assumption works particularly well in high-dimensional problems. However, the sparsity assumption may not be necessary when the number of predictors is relatively small compared to the number of training instances. This paper argues that shrinking the coefficients towards a low-variance data-driven estimate could be a better regularization strategy for such situations. For low-dimensional classification problems, we propose a naïve Bayes regularized logistic regression (NBRLR) that shrinks the logistic regression coefficients toward the naïve Bayes estimate to provide a reduction in variance. Our approach is primarily motivated by the fact that naïve Bayes is functionally equivalent to logistic regression if naïve Bayes' conditional independence assumption holds. Under standard conditions, we prove the consistency of the NBRLR estimator. Extensive simulation and empirical experimental results show that NBRLR is a competitive alternative to various state-of-the-art classifiers, especially on low-dimensional datasets.

We examine the recently proposed language of Logical Credal Networks, a powerful representation formalism that combines probabilities and logic. In particular we investigate the consequences of distinct Markov conditions upon their underlying semantics. We introduce the notion of structure for a Logical Credal Network and show that a structure without directed cycles leads to a well-known factorization result. For networks with directed cycles, we discuss the differences between Markov conditions, factorization results, and specification requirements. We consider several scenarios in causal reasoning that can be tackled by the formalism, in particular looking at partial identifiability and cycles.

In this paper, we study a frequentist approach to belief updating in the framework of Dempster-Shafer Theory (DST). We propose several mechanisms that allow the gathering of possibly ambiguous pieces of evidence over time to obtain a belief mass assignment. We then use our approach to study the impact of ambiguous evidence on the belief distribution of agents in social networks. We illustrate our approach by taking three representative situations. In the first one, we suppose that there is an unknown state of nature, and agents form belief in the set of possible states. Nature constantly sends a signal which reflects the true state with some probability but which can also be ambiguous. In the second situation, there is no ground truth, and agents are against or in favor of some ethical or societal issues. In the third situation, there is no ground state either, but agents have opinions on left, center, and right political parties. We show that our approach can model various phenomena often observed in social networks, such as polarization or bounded confidence effects.