Discovering causal relationships from observational data is a crucial problem and it has applications in many research areas. The PC algorithm is the state-of-the-art constraint based method for causal discovery. However, runtime of the PC algorithm, in the worst-case, is exponential to the number of nodes (variables), and thus it is inefficient when being applied to high dimensional data, e.g. gene expression datasets. On another note, the advancement of computer hardware in the last decade has resulted in the widespread availability of multi-core personal computers. There is a significant motivation for designing a parallelised PC algorithm that is suitable for personal computers and does not require end users’ parallel computing knowledge beyond their competency in using the PC algorithm. In this paper, we develop parallel-PC, a fast and memory efficient PC algorithm using the parallel computing technique. We apply our method to a range of synthetic and real-world high dimensional datasets. Experimental results on a dataset from the DREAM 5 challenge show that the original PC algorithm could not produce any results after running more than 24 hours; meanwhile, our parallel-PC algorithm managed to finish within around 12 hours with a 4-core CPU computer, and less than 6 hours with a 8-core CPU computer. Furthermore, we integrate parallel-PC into a causal inference method for inferring miRNA-mRNA regulatory relationships. The experimental results show that parallel-PC helps improve both the efficiency and accuracy of the causal inference algorithm.
Meanwhile, the other constraint based methods @cite_26 infer causal relationships by performing CI tests and searching for specific causal structures. For instance, Cooper @cite_56 proposed the LCD method to learn only a small portion of edges rather than the complete network. Specifically, the LCD algorithm searches for the CCC structures which consist of three pairs of correlated variables: ( @math , @math ), ( @math , @math ) and ( @math , @math ). If @math and @math become independent when conditioned on @math , we may infer that one of the following causal relations exists between @math , @math , and @math : @math ; @math ; and @math . @cite_28 proposed the similar method that focuses on the CCU structures. The CCU method searches for two correlated pairs @math and @math and an independent pair @math . When conditioning on @math , if @math becomes dependent then the causal relationships @math are concluded. Compared to the PC algorithm, such specific structure finding methods are more efficient, but the causal discoveries are incomplete.
@cite_28 @cite_26 @cite_56
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Constraint-based causal discovery (CCD) algorithms require fast and accurate conditional independence (CI) testing. The Kernel Conditional Independence Test (KCIT) is currently one of the most popular CI tests in the non-parametric setting, but many investigators cannot use KCIT with large datasets because the test scales cubicly with sample size. We therefore devise two relaxations called the Randomized Conditional Independence Test (RCIT) and the Randomized conditional Correlation Test (RCoT) which both approximate KCIT by utilizing random Fourier features. In practice, both of the proposed tests scale linearly with sample size and return accurate p-values much faster than KCIT in the large sample size context. CCD algorithms run with RCIT or RCoT also return graphs at least as accurate as the same algorithms run with KCIT but with large reductions in run time. Discovering causal relationships in large databases of observational data is challenging. The pioneering work in this area was rooted in the theory of Bayesian network (BN) learning, which however, is a NP-complete problem. Hence several constraint-based algorithms have been developed to efficiently discover causations in large databases. These methods usually use the idea of BN learning, directly or indirectly, and are focused on causal relationships with single cause variables. In this paper, we propose an approach to mine causal rules in large databases of binary variables. Our method expands the scope of causality discovery to causal relationships with multiple cause variables, and we utilise partial association tests to exclude noncausal associations, to ensure the high reliability of discovered causal rules. Furthermore an efficient algorithm is designed for the tests in large databases. We assess the method with a set of real-world diagnostic data. The results show that our method can effectively discover interesting causal rules in large databases. The discovery of causal relationships from purely observational data is a fundamental problem in science. The most elementary form of such a causal discovery problem is to decide whether X causes Y or, alternatively, Y causes X, given joint observations of two variables X,Y. An example is to decide whether altitude causes temperature, or vice versa, given only joint measurements of both variables. Even under the simplifying assumptions of no confounding, no feedback loops, and no selection bias, such bivariate causal discovery problems are challenging. Nevertheless, several approaches for addressing those problems have been proposed in recent years. We review two families of such methods: methods based on Additive Noise Models (ANMs) and Information Geometric Causal Inference (IGCI). We present the benchmark CAUSEEFFECTPAIRS that consists of data for 100 different causee ffect pairs selected from 37 data sets from various domains (e.g., meteorology, biology, medicine, engineering, economy, etc.) and motivate our decisions regarding the "ground truth" causal directions of all pairs. We evaluate the performance of several bivariate causal discovery methods on these real-world benchmark data and in addition on artificially simulated data. Our empirical results on real-world data indicate that certain methods are indeed able to distinguish cause from effect using only purely observational data, although more benchmark data would be needed to obtain statistically significant conclusions. One of the best performing methods overall is the method based on Additive Noise Models that has originally been proposed by (2009), which obtains an accuracy of 63 ± 10 and an AUC of 0.74 ± 0.05 on the real-world benchmark. As the main theoretical contribution of this work we prove the consistency of that method. There are several existing algorithms that under appropriate assumptions can reliably identify a subset of the underlying causal relationships from observational data. This paper introduces the first computationally feasible score-based algorithm that can reliably identify causal relationships in the large sample limit for discrete models, while allowing for the possibility that there are unobserved common causes. In doing so, the algorithm does not ever need to assign scores to causal structures with unobserved common causes. The algorithm is based on the identification of so called Y substructures within Bayesian network structures that can be learned from observational data. An example of a Y substructure is A -> C, B -> C, C -> D. After providing background on causal discovery, the paper proves the conditions under which the algorithm is reliable in the large sample limit.