It is essential to approach the interpretation of our algorithm's results with caution and subject them to critical evaluation. In this section, we provide the definition of partial ancestral graphs (P AGs). A P AG shares the same adjacencies as any MAG in the observational equivalence class of MAGs. Section 2. For any v W, let G In this section, we derive the causal effect for the SMCM in Figure 3(top), i.e., (6), as well as prove D.1 Proof of (6) First, using the law of total probability, we have P(y |do (t = t)) = null Rule 3a, (c) follows from Rule 1, and (g) follows from Rule 2. D.2 Proof of Theorem 3.1 Lemma 1. Suppose Assumptions 1 to 3 hold. Given this claim, Theorem 3.1 follows from Tian and Pearl [2002, Theorem 4].

Granger causality (GC) [15] is a time series causal discovery framework that uses predictive modeling to identify the underlying causal structure of a time series system. Relying on the assumption that cause precedes effect, GC assesses whether including the lagged information from one time series in the autoregressive model of a second time series enhances its predictions. This improvement indicates a predictive relationship between the time series variables, where one time series provides supplemental information about the future of another time series, thereby signifying the presence of a (Granger) causal relationship. GC requires only observational data, and has been used for time series causal discovery across diverse domains, including climate science [33], political and social sciences [17], econometrics [4], and biological systems studies [13]. The original formulation of GC requires several assumptions to be satisfied for causal identifiability. In regards to the candidate time series system, it is assumed that the time series variables are stationary, and that all variables are observed (absence of latent confounders). GC was initially proposed for bivariate time series systems, but was generalised for the multivariate setting to accommodate the assumption that all relevant variables are included in the analysis [15]. Additional assumptions are made with regard to the types of causal relationships that can be identified within the time series system. GC cannot estimate a causal relationship between time series at an instantaneous time point, relying on the relationship between the lags and predicted values to determine a GC relationship.

From a particle smasher encircling the moon to an "impossible" laser, five scientists reveal the experiments they would run in a world powered purely by imagination In physics, breakthroughs are rare. Experiments are slow, expensive and often end up refining, rather than rewriting, our understanding of the universe. But what if the only constraint on scientific ambition were imagination? We asked five physicists to describe the kind of experiment they would do if they didn't have to worry about budgets, engineering limitations or political realities. Not because we expect any of it to happen soon - though in a few cases, momentum is building - but because it is revealing to see where their minds go when the usual boundaries are stripped away. One researcher wants to launch radio telescopes deep into space to probe dark matter with cosmic energy flashes. Others are dreaming of completely new kinds of particle accelerator or lasers that push the at bounds of the possible.

Explaining a complex system through their cause and e ff ect relations is one of the fundamental challenges in science.

There can be no loops or directed cycles. See Figure 1A for an example. The results also hold for Maximal Ancestral Graphs (MAG) [Richardson and Spirtes, 2002] without selection variables. Kinships are defined as usual: parents Our approach does not involve modified graph constructions as in van der Zander et al. [2019] and other works. A node is an ancestor and descendant of itself, but not a parent/child/spouse of itself.

Shpitser et al. (2010) extended the BD and

Identifying causal effects is one prominent task throughout empirical sciences.

In this paper, we investigate the task of structural learning in non-Markovian systems (i.e., when latent variables a ect more than one observable) from a combination of observational and soft experimental data when the interventional targets are unknown.