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We consider the problem of inferring causal relationships between two or more passively ob-served variables. While the problem of such causal discovery has been extensively studied,especially in the bivariate setting, the majority of current methods assume a linear causal relationship, and the few methods which consider non-linear relations usually make the assumption of additive noise. Here, we propose a framework through which we can perform causal discovery in the presence of general nonlinear relationships. The proposed method is based on recent progress in non-linear in-dependent component analysis (ICA) and exploits the nonstationarity of observations in order to recover the underlying sources. We show rigorously that in the case of bivariate causal discovery, such non-linear ICA can be used to infer causal direction via a series of in-dependence tests. We further propose an alternative measure for inferring causal direction based on asymptotic approximations to the likelihood ratio, as well as an extension to multivariate causal discovery. We demonstrate the capabilities of the proposed method via a series of simulation studies and conclude with an application to neuroimaging data.
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ON AN INTEGRAL OF -BESSEL FUNCTIONS AND ITS APPLICATION TO MAHLER MEASURE
Abstract Cogdell et al. [‘Evaluating the Mahler measure of linear forms via Kronecker limit formulas on complex projective space’, Trans. Amer. Math. Soc. (2021), to appear] developed infinite series representations for the logarithmic Mahler measure of a complex linear form with four or more variables. We establish the case of three variables by bounding an integral with integrand involving the random walk probability density $$a\int _0^\infty tJ_0(at) \prod _{m=0}^2 J_0(r_m t)\,dt$$ , where $$J_0$$ is the order-zero Bessel function of the first kind and a and $$r_m$$ are positive real numbers. To facilitate our proof we develop an alternative description of the integral’s asymptotic behaviour at its known points of divergence. As a computational aid for numerical experiments, an algorithm to calculate these series is presented in the appendix.
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- Award ID(s):
- 1820731
- PAR ID:
- 10281300
- Date Published:
- Journal Name:
- Bulletin of the Australian Mathematical Society
- ISSN:
- 0004-9727
- Page Range / eLocation ID:
- 1 to 13
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/S0004972721000484
