  We provide a framework for probabilistic reasoning in Vadalog-based Knowledge
Graphs (KGs), satisfying the requirements of ontological reasoning: full
recursion, powerful existential quantification, expression of inductive
definitions. Vadalog is a Knowledge Representation and Reasoning (KRR) language
based on Warded Datalog+/-, a logical core language of existential rules, with
a good balance between computational complexity and expressive power. Handling
uncertainty is essential for reasoning with KGs. Yet Vadalog and Warded
Datalog+/- are not covered by the existing probabilistic logic programming and
statistical relational learning approaches for several reasons, including
insufficient support for recursion with existential quantification, and the
impossibility to express inductive definitions. In this work, we introduce Soft
Vadalog, a probabilistic extension to Vadalog, satisfying these desiderata. A
Soft Vadalog program induces what we call a Probabilistic Knowledge Graph
(PKG), which consists of a probability distribution on a network of chase
instances, structures obtained by grounding the rules over a database using the
chase procedure. We exploit PKGs for probabilistic marginal inference. We
discuss the theory and present MCMC-chase, a Monte Carlo method to use Soft
Vadalog in practice. We apply our framework to solve data management and
industrial problems, and experimentally evaluate it in the Vadalog system.
  Under consideration in Theory and Practice of Logic Programming (TPLP).
