Causal models are crucial for understanding complex systems and identifying causal relationships among variables. Even though causal models are extremely popular, conditional probability calculation of formulas involving interventions pose significant challenges. In case of Causal Bayesian Networks (CBNs), Pearl assumes autonomy of mechanisms that determine interventions to calculate a range of probabilities. We show that by making simple yet often realistic independence assumptions, it is possible to uniquely estimate the probability of an interventional formula (including the well-studied notions of probability of sufficiency and necessity). We discuss when these assumptions are appropriate. Importantly, in many cases of interest, when the assumptions are appropriate, these probability estimates can be evaluated using observational data, which carries immense significance in scenarios where conducting experiments is impractical or unfeasible.

Causal models are crucial for understanding complex systems and identifying causal relationships among variables. Even though causal models are extremely popular, conditional probability calculation of formulas involving interventions pose significant challenges. In case of Causal Bayesian Networks (CBNs), Pearl assumes autonomy of mechanisms that determine interventions to calculate a range of probabilities. We show that by making simple yet often realistic independence assumptions, it is possible to uniquely estimate the probability of an interventional formula (including the well-studied notions of probability of sufficiency and necessity). We discuss when these assumptions are appropriate. Importantly, in many cases of interest, when the assumptions are appropriate, these probability estimates can be evaluated using observational data, which carries immense significance in scenarios where conducting experiments is impractical or unfeasible.

Causal models are crucial for understanding complex systems and identifying causal relationships among variables. Even though causal models are extremely popular, conditional probability calculation of formulas involving interventions pose significant challenges. In case of Causal Bayesian Networks (CBNs), Pearl assumes autonomy of mechanisms that determine interventions to calculate a range of probabilities. We show that by making simple yet often realistic independence assumptions, it is possible to uniquely estimate the probability of an interventional formula (including the well-studied notions of probability of sufficiency and necessity). We discuss when these assumptions are appropriate. Importantly, in many cases of interest, when the assumptions are appropriate, these probability estimates can be evaluated using observational data, which carries immense significance in scenarios where conducting experiments is impractical or unfeasible.

It is well know that basic conditionals satisfy some desirable basic logical and probabilistic properties, such as the compound probability theorem, but checking the validity of these becomes trickier when we switch to compound and iterated conditionals. We consider de Finetti's notion of conditional as a three-valued object and as a conditional random quantity in the betting framework. We recall the notions of conjunction and disjunction among conditionals in selected trivalent logics. First, in the framework of specific three-valued logics we analyze the notions of iterated conditioning introduced by Cooper-Calabrese, de Finetti and Farrell, respectively. We show that the compound probability theorem and other basic properties are not preserved by these objects, by also computing some probability propagation rules. Then, for each trivalent logic we introduce an iterated conditional as a suitable random quantity which satisfies the compound prevision theorem and some of the desirable properties. We also check the validity of two generalized versions of Bayes' Rule for iterated conditionals. We study the p-validity of generalized versions of Modus Ponens and two-premise centering for iterated conditionals. Finally, we observe that all the basic properties are satisfied only by the iterated conditional mainly developed in recent papers by Gilio and Sanfilippo in the setting of conditional random quantities.

In this paper we recall some results for conditional events, compound conditionals, conditional random quantities, p-consistency, and p-entailment. Then, we show the equivalence between bets on conditionals and conditional bets, by reviewing de Finetti's trivalent analysis of conditionals. But our approach goes beyond de Finetti's early trivalent logical analysis and is based on his later ideas, aiming to take his proposals to a higher level. We examine two recent articles that explore trivalent logics for conditionals and their definitions of logical validity and compare them with our approach to compound conditionals. We prove a Probabilistic Deduction Theorem for conditional events. After that, we study some probabilistic deduction theorems, by presenting several examples. We focus on iterated conditionals and the invalidity of the Import-Export principle in the light of our Probabilistic Deduction Theorem. We use the inference from a disjunction, "$A$ or $B$", to the conditional,"if not-$A$ then $B$", as an example to show the invalidity of the Import-Export principle. We also introduce a General Import-Export principle and we illustrate it by examining some p-valid inference rules of System P. Finally, we briefly discuss some related work relevant to AI.

This chapter presents probability logic as a rationality framework for human reasoning under uncertainty. Selected formal-normative aspects of probability logic are discussed in the light of experimental evidence. Specifically, probability logic is characterized as a generalization of bivalent truth-functional propositional logic ( short "logic"), as being connexive, and as being nonmonotonic. The chapter discusses selected argument forms and associated uncertainty propagation rules. Probability logic is a generalization of logicProbability logic as a rationality framework combines probabilistic reasoning with logical rule-based reasoning and studies formal properties of uncertain argument forms. Among various approaches to probability logic ( for overviews see, e.g., Hailperin, 1996; Adams, 1975, 1998; Coletti and Scozzafava, 2002; Haenni, Romeijn, Wheeler, and Williamson, 2011; Demey, Kooi, and Sack, 2017), this chapter reviews selected formal-normative aspects of probability logic in the light of experimental evidence. The focus is on probability logic as a generalization of the classical propositional calculus ( short: logic; for probabilistic generalizations of quantified statements see, e.g., Hailperin, 2011; Pfeifer & Sanfilippo, 2017, 2019).

We generalize, by a progressive procedure, the notions of conjunction and disjunction of two conditional events to the case of $n$ conditional events. In our coherence-based approach, conjunctions and disjunctions are suitable conditional random quantities. We define the notion of negation, by verifying De Morgan's Laws. We also show that conjunction and disjunction satisfy the associative and commutative properties, and a monotonicity property. Then, we give some results on coherence of prevision assessments for some families of compounded conditionals; in particular we examine the Fr\'echet-Hoeffding bounds. Moreover, we study the reverse probabilistic inference from the conjunction $\mathcal{C}_{n+1}$ of $n+1$ conditional events to the family $\{\mathcal{C}_{n},E_{n+1}|H_{n+1}\}$. We consider the relation with the notion of quasi-conjunction and we examine in detail the coherence of the prevision assessments related with the conjunction of three conditional events. Based on conjunction, we also give a characterization of p-consistency and of p-entailment, with applications to several inference rules in probabilistic nonmonotonic reasoning. Finally, we examine some non p-valid inference rules; then, we illustrate by an example two methods which allow to suitably modify non p-valid inference rules in order to get inferences which are p-valid.

We make a probabilistic analysis related to some inference rules which play an important role in nonmonotonic reasoning. In a coherence-based setting, we study the extensions of a probability assessment defined on $n$ conditional events to their quasi conjunction, and by exploiting duality, to their quasi disjunction. The lower and upper bounds coincide with some well known t-norms and t-conorms: minimum, product, Lukasiewicz, and Hamacher t-norms and their dual t-conorms. On this basis we obtain Quasi And and Quasi Or rules. These are rules for which any finite family of conditional events p-entails the associated quasi conjunction and quasi disjunction. We examine some cases of logical dependencies, and we study the relations among coherence, inclusion for conditional events, and p-entailment. We also consider the Or rule, where quasi conjunction and quasi disjunction of premises coincide with the conclusion. We analyze further aspects of quasi conjunction and quasi disjunction, by computing probabilistic bounds on premises from bounds on conclusions. Finally, we consider biconditional events, and we introduce the notion of an $n$-conditional event. Then we give a probabilistic interpretation for a generalized Loop rule. In an appendix we provide explicit expressions for the Hamacher t-norm and t-conorm in the unitary hypercube.

In this paper, we address the problem of merging multiple imprecise probabilistic beliefs represented as Probabilistic Logic Programs (PLPs) obtained from multiple sources. Beliefs in each PLP are modeled as conditional events attached with probability bounds. The major task of syntax-based merging is to obtain the most rational probability bound for each conditional event from the original PLPs to form a new PLP. We require the minimal change principle to be followed so that each source gives up its beliefs as little as possible. Some instantiated merging operators are derived from our merging framework. Furthermore, we propose a set of postulates for merging PLPs, some of which extend the postulates for merging classical knowledge bases, whilst others are specific to the merging of probabilistic beliefs.