We refer the readers to ( Peters et al., 2017) for more detailed graphical terminology. We base our proof mostly on ( Kirsch, 2019). The first statement follows directly from the first theorem in ( Haviland, 1936). Without loss of generality, we reorder the variables according to reversed topological ordering, i.e. a Follows directly from Lemma 1. Lemma 4. Recall condition 2) in Causal de Finetti states that 8 i, 8 n 2 N: X The first equality holds by well-defindedness. The fourth equality follow from well-definedness.

There are plenty of previous studies targeting the problem from different aspects. For temporal point process, agreat number of works [3, 13, 15, 16, 28] attempt to model the intensify function from statistic views, and recent studies harness deep recurrent model [24], generative adversarial network [23] and reinforcement learning [19, 18] to learn the temporal process. These researches mainly focus on one-dimension eventsequences where eacheventpossesses thesame marker.

Temporal logics have been used in prior research as a language forexpressing desirable system behaviors, and canimprovetheinterpretability ofspecifications if expressed as compositions of simpler templates (akin to those described by Dwyer et al. [2]).

We propose a constructive algorithm for identifying complete data distributions in graphical models of missing data. The complete data distribution is unrestricted, while the missingness mechanism is assumed to factorize according to a conditional directed acyclic graph. Our approach follows an interventionist perspective in which missingness indicators are treated as variables that can be intervened on. A central challenge in this setting is that sequences of interventions on missingness indicators may induce and propagate selection bias, so that identification can fail even when a propensity score is invariant to available interventions. To address this challenge, we introduce a tree-based identification algorithm that explicitly tracks the creation and propagation of selection bias and determines whether it can be avoided through admissible intervention strategies. The resulting tree provides both a diagnostic and a constructive characterization of identifiability under a given missingness mechanism. Building on these results, we develop recursive inverse probability weighting procedures that mirror the intervention logic of the identification algorithm, yielding valid estimating equations for both the missingness mechanism and functionals of the complete data distribution. Simulation studies and a real-data application illustrate the practical performance of the proposed methods. An accompanying R package, flexMissing, implements all proposed procedures.

Many (if not all) real-world applications require efficient handling of uncertainty and a compact representation of a wide variety of knowledge. Indeed, complex concepts and relationships that typically comprise expert knowledge may be difficult to express in graphical models but can be represented compactly using classical logic.