Abstraction is a powerful idea widely used in science, to model, reason and explain the behavior of systems in a more tractable search space, by omitting irrelevant details. While notions of abstraction have matured for deterministic systems, the case for abstracting probabilistic models is not yet fully understood.
In this paper, we provide a semantical framework for analyzing such abstractions from first principles. We develop the framework in a general way, allowing for expressive languages, including logic-based ones that admit relational and hierarchical constructs with stochastic primitives. We motivate a definition of consistency between a high-level model and its low-level counterpart, but also treat the case when the high-level model is missing critical information present in the low-level model. We prove properties of abstractions, both at the level of the parameter as well as the structure of the models. We conclude with some observations about how abstractions can be derived automatically.
RWTH Aachen University（亚琛工业大学）；University of Edinburgh（爱丁堡大学）
Abstraction is a commonly used process to represent some low-level system by a more coarse specification with the goal to omit unnecessary details while preserving important aspects. While recent work on abstraction in the situation calculus has focused on non-probabilistic domains, we describe an approach to abstraction of probabilistic and dynamic systems. Based on a variant of the situation calculus with probabilistic belief, we define a notion of bisimulation that allows to abstract a detailed probabilistic basic action theory with noisy actuators and sensors by a possibly non-stochastic b…
We consider a sequence of successively more restrictive definitions of abstraction for causal models, starting with a notion introduced by Rubenstein et al. (2017) called exact transformation that applies to probabilistic causal models, moving to a notion of uniform transformation that applies to deterministic causal models and does not allow differences to be hidden by the "right" choice of distribution, and then to abstraction, where the interventions of interest are determined by the map from low-level states to high-level states, and strong abstraction, which takes more seriously all poten…
SRI International（SRI国际公司）；George Washington Univ.（乔治华盛顿大学）
In spite of the rapidly increasing number of applications of machine learning in various domains, a principled and systematic approach to the incorporation of domain knowledge in the engineering process is still lacking and ad hoc solutions that are difficult to validate are still the norm in practice, which is of growing concern not only in mission-critical applications. In this note, we introduce Probabilistic Approximate Logic (PALO) as a logic based on the notion of mean approximate probability to overcome conceptual and computational difficulties inherent to strictly probabilistic logics.…
Understanding the behavior of learned classifiers is an important task, and various black-box explanations, logical reasoning approaches, and model-specific methods have been proposed. In this paper, we introduce probabilistic sufficient explanations, which formulate explaining an instance of classification as choosing the "simplest" subset of features such that only observing those features is "sufficient" to explain the classification. That is, sufficient to give us strong probabilistic guarantees that the model will behave similarly when all features are observed under the data distribution…
Frankfurt School of Finance & Management（法兰克福金融与管理学院）；University of Bologna（博洛尼亚大学）
There are two reasons why uncertainty may not be adequately described by Probability Theory. The first one is due to unique or nearly-unique events, that either never realized or occurred too seldom for frequencies to be reliably measured. The second one arises when one fears that something may happen, that one is not even able to figure out, e.g., if one asks: "Climate change, financial crises, pandemic, war, what next?" In both cases, simple one-to-one cognitive maps between available alternatives and possible consequences eventually melt down. However, such destructions reflect into the cha…
ANDRE: An Attention-based Neuro-symbolic Differentiable Rule Extractor for Inductive Logic Programming
Dept. of Mechanical and Aerospace Engineering, George Washington University, USA（机械与航空航天工程系，乔治华盛顿大学）；Dept. of Mechanical Engineering Sciences, University of Surrey, UK（机械工程科学系，萨里大学）
Safe Reinforcement learning (Safe RL) aims at learning optimal policies while staying safe. A popular solution to Safe RL is shielding, which uses a logical safety specification to prevent an RL agent from taking unsafe actions. However, traditional shielding techniques are difficult to integrate with continuous, end-to-end deep RL methods. To this end, we introduce Probabilistic Logic Policy Gradient (PLPG). PLPG is a model-based Safe RL technique that uses probabilistic logic programming to model logical safety constraints as differentiable functions. Therefore, PLPG can be seamlessly applie…
University of California, Los Angeles（加州大学洛杉矶分校）；University of Trento（特伦托大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Weighted model integration (WMI) is a very appealing framework for probabilistic inference: it allows to express the complex dependencies of real-world problems where variables are both continuous and discrete, via the language of Satisfiability Modulo Theories (SMT), as well as to compute probabilistic queries with complex logical and arithmetic constraints. Yet, existing WMI solvers are not ready to scale to these problems. They either ignore the intrinsic dependency structure of the problem at all, or they are limited to too restrictive structures. To narrow this gap, we derive a factorized…
University of California, LA（加利福尼亚大学洛杉矶分校）；AMSS（数学与系统科学研究院）；Chinese Academy of Sciences（中国科学院）；DISI（信息工程与计算机科学系）；University of Trento（特伦托大学）
Weighted model integration (WMI) is a very appealing framework for probabilistic inference: it allows to express the complex dependencies of real-world hybrid scenarios where variables are heterogeneous in nature (both continuous and discrete) via the language of Satisfiability Modulo Theories (SMT); as well as computing probabilistic queries with arbitrarily complex logical constraints. Recent work has shown WMI inference to be reducible to a model integration (MI) problem, under some assumptions, thus effectively allowing hybrid probabilistic reasoning by volume computations. In this paper, …
Istituto Dalle Molle di Studi per l’Intelligenza Artificiale (IDSIA)（达莱莫勒人工智能研究所（IDSIA））
Probabilistic sentential decision diagrams are a class of structured-decomposable probabilistic circuits especially designed to embed logical constraints. To adapt the classical LearnSPN scheme to learn the structure of these models, we propose a new scheme based on a partial closed-world assumption: data implicitly provide the logical base of the circuit. Sum nodes are thus learned by recursively clustering batches in the initial data base, while the partitioning of the variables obeys a given input vtree. Preliminary experiments show that the proposed approach might properly fit training dat…
Dalhousie University（达尔豪斯大学）；Vector Institute for Artificial Intelligence（向量人工智能研究所）
The choice of activation functions and their motivation is a long-standing issue within the neural network community. Neuronal representations within artificial neural networks are commonly understood as logits, representing the log-odds score of presence of features within the stimulus. We derive logit-space operators equivalent to probabilistic Boolean logic-gates AND, OR, and XNOR for independent probabilities. Such theories are important to formalize more complex dendritic operations in real neurons, and these operations can be used as activation functions within a neural network, introduc…
University of California, Berkeley（加州大学伯克利分校）；Stanford University（斯坦福大学）
Qualitative and quantitative approaches to reasoning about uncertainty can lead to different logical systems for formalizing such reasoning, even when the language for expressing uncertainty is the same. In the case of reasoning about relative likelihood, with statements of the form $φ\succsimψ$ expressing that $φ$ is at least as likely as $ψ$, a standard qualitative approach using preordered preferential structures yields a dramatically different logical system than a quantitative approach using probability measures. In fact, the standard preferential approach validates principles of reasonin…
TU Darmstadt（达姆施塔特工业大学）；Hessian Center for AI (hessian.AI)（黑森人工智能中心 (hessian.AI)）
The goal of combining the robustness of neural networks and the expressivity of symbolic methods has rekindled the interest in neuro-symbolic AI. Recent advancements in neuro-symbolic AI often consider specifically-tailored architectures consisting of disjoint neural and symbolic components, and thus do not exhibit desired gains that can be achieved by integrating them into a unifying framework. We introduce SLASH -- a novel deep probabilistic programming language (DPPL). At its core, SLASH consists of Neural-Probabilistic Predicates (NPPs) and logical programs which are united via answer set …
The Bitcoin transaction graph is a public data structure organized as transactions between addresses, each associated with a logical entity. In this work, we introduce a complete probabilistic model of the Bitcoin Blockchain. We first formulate a set of conditional dependencies induced by the Bitcoin protocol at the block level and derive a corresponding fully observed graphical model of a Bitcoin block. We then extend the model to include hidden entity attributes such as the functional category of the associated logical agent and derive asymptotic bounds on the privacy properties implied by t…
Uncertain information is being taken into account in an increasing number of application fields. In the meantime, abduction has been proved a powerful tool for handling hypothetical reasoning and incomplete knowledge. Probabilistic logical models are a suitable framework to handle uncertain information, and in the last decade many probabilistic logical languages have been proposed, as well as inference and learning systems for them. In the realm of Abductive Logic Programming (ALP), a variety of proof procedures have been defined as well. In this paper, we consider a richer logic language, cop…
