Many causal discovery algorithms, including the celebrated FCI algorithm, output a Partial Ancestral Graph (PAG). PAGs serve as an abstract graphical representation of the underlying causal structure, modeled by directed acyclic graphs with latent and selection variables. This paper develops a characterization of the set of extended-type conditional independence relations that are invariant across all causal models represented by a PAG. This theory allows us to formulate a general measure-theoretic version of Pearl's causal calculus and a sound and complete identification algorithm for PAGs under selection bias. Our results also apply when PAGs are learned by certain algorithms that integrate observational data with experimental data and incorporate background knowledge.

Learning causal relations from observational data is a fundamental problem with wide-ranging applications across many fields. Constraint-based methods infer the underlying causal structure by performing conditional independence tests. However, existing algorithms such as the prominent PC algorithm need to perform a large number of independence tests, which in the worst case is exponential in the maximum degree of the causal graph. Despite extensive research, it remains unclear if there exist algorithms with better complexity without additional assumptions. Here, we establish an algorithm that achieves a better complexity of $p^{\mathcal{O}(s)}$ tests, where $p$ is the number of nodes in the graph and $s$ denotes the maximum undirected clique size of the underlying essential graph. Complementing this result, we prove that any constraint-based algorithm must perform at least $2^{Ω(s)}$ conditional independence tests, establishing that our proposed algorithm achieves exponent-optimality up to a logarithmic factor in terms of the number of conditional independence tests needed. Finally, we validate our theoretical findings through simulations, on semi-synthetic gene-expression data, and real-world data, demonstrating the efficiency of our algorithm compared to existing methods in terms of number of conditional independence tests needed.

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].

Thistask is challenging because the causal graph is unknown and even there may exist latent confounders.

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.