We consider the problem of inferring the directed, causal graph from observational data, assuming no hidden confounders. We take an information theoretic approach, and make three main contributions. First, we show how through algorithmic information theory we can obtain SCI, a highly robust, effective and computationally efficient test for conditional independence---and show it outperforms the state of the art when applied in constraint-based inference methods such as stable PC. Second, building upon on SCI, we show how to tell apart the parents and children of a given node based on the algorithmic Markov condition. We give the Climb algorithm to efficiently discover the directed, causal Markov blanket---and show it is at least as accurate as inferring the global network, while being much more efficient. Last, but not least, we detail how we can use the Climb score to direct those edges that state of the art causal discovery algorithms based on PC or GES leave undirected---and show this improves their precision, recall and F1 scores by up to 20 .
Many methods for discovering Markov blankets and causal networks rely on conditional independence tests @cite_15 @cite_4 ---which, by conditioning on the empty set, can also be used to measure association. For discrete data, the state of the art commonly relies on either the @math test or conditional mutual information (CMI) @cite_15 @cite_29 @cite_14 . Both measures, however, have drawbacks. The @math test is biased to independence when only limited data is available, whereas CMI has the opposite problem. In particular, it holds that @math for any @math , even if @math . For both @math and CMI it is necessary to define a significance threshold or cut-off, which can be arbitrary. Our proposed method based on algorithmic independence overcomes these limitations.
@cite_14 @cite_15 @cite_29 @cite_4
2951039901 2601307951 2126653228 609741286
Conditional independence testing is an important problem, especially in Bayesian network learning and causal discovery. Due to the curse of dimensionality, testing for conditional independence of continuous variables is particularly challenging. We propose a Kernel-based Conditional Independence test (KCI-test), by constructing an appropriate test statistic and deriving its asymptotic distribution under the null hypothesis of conditional independence. The proposed method is computationally efficient and easy to implement. Experimental results show that it outperforms other methods, especially when the conditioning set is large or the sample size is not very large, in which case other methods encounter difficulties. 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. A new method is proposed for exploiting causal independencies in exact Bayesian network inference. A Bayesian network can be viewed as representing a factorization of a joint probability into the multiplication of a set of conditional probabilities. We present a notion of causal independence that enables one to further factorize the conditional probabilities into a combination of even smaller factors and consequently obtain a finer-grain factorization of the joint probability. The new formulation of causal independence lets us specify the conditional probability of a variable given its parents in terms of an associative and commutative operator, such as "or", "sum" or "max", on the contribution of each parent. We start with a simple algorithm VE for Bayesian network inference that, given evidence and a query variable, uses the factorization to find the posterior distribution of the query. We show how this algorithm can be extended to exploit causal independence. Empirical studies, based on the CPCS networks for medical diagnosis, show that this method is more efficient than previous methods and allows for inference in larger networks than previous algorithms. Determining conditional independence (CI) relationships between random variables is a challenging but important task for problems such as Bayesian network learning and causal discovery. We propose a new kernel CI test that uses a single, learned permutation to convert the CI test problem into an easier two-sample test problem. The learned permutation leaves the joint distribution unchanged if and only if the null hypothesis of CI holds. Then, a kernel two-sample test, which has been studied extensively in prior work, can be applied to a permuted and an unpermuted sample to test for CI. We demonstrate that the test (1) easily allows the incorporation of prior knowledge during the permutation step, (2) has power competitive with state-of-the-art kernel CI tests, and (3) accurately estimates the null distribution of the test statistic, even as the dimensionality of the conditioning variable grows.