{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:54:35Z","timestamp":1760241275126,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,12,29]],"date-time":"2019-12-29T00:00:00Z","timestamp":1577577600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We address the problem of two-variable causal inference without intervention. This task is to infer an existing causal relation between two random variables, i.e.,     X \u2192 Y     or     Y \u2192 X    , from purely observational data. As the option to modify a potential cause is not given in many situations, only structural properties of the data can be used to solve this ill-posed problem. We briefly review a number of state-of-the-art methods for this, including very recent ones. A novel inference method is introduced, Bayesian Causal Inference (BCI) which assumes a generative Bayesian hierarchical model to pursue the strategy of Bayesian model selection. In the adopted model, the distribution of the cause variable is given by a Poisson lognormal distribution, which allows to explicitly regard the discrete nature of datasets, correlations in the parameter spaces, as well as the variance of probability densities on logarithmic scales. We assume Fourier diagonal Field covariance operators. The model itself is restricted to use cases where a direct causal relation     X \u2192 Y     has to be decided against a relation     Y \u2192 X    , therefore we compare it other methods for this exact problem setting. The generative model assumed provides synthetic causal data for benchmarking our model in comparison to existing state-of-the-art models, namely LiNGAM, ANM-HSIC, ANM-MML, IGCI, and CGNN. We explore how well the above methods perform in case of high noise settings, strongly discretized data, and very sparse data. BCI performs generally reliably with synthetic data as well as with the real world TCEP benchmark set, with an accuracy comparable to state-of-the-art algorithms. We discuss directions for the future development of BCI.<\/jats:p>","DOI":"10.3390\/e22010046","type":"journal-article","created":{"date-parts":[[2019,12,30]],"date-time":"2019-12-30T05:49:41Z","timestamp":1577684981000},"page":"46","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Bayesian Model for Bivariate Causal Inference"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1270-3702","authenticated-orcid":false,"given":"Maximilian","family":"Kurthen","sequence":"first","affiliation":[{"name":"Max-Planck-Institut f\u00fcr Astrophysik, Karl-Schwarzschildstr. 1, 85748 Garching, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5246-1624","authenticated-orcid":false,"given":"Torsten","family":"En\u00dflin","sequence":"additional","affiliation":[{"name":"Max-Planck-Institut f\u00fcr Astrophysik, Karl-Schwarzschildstr. 1, 85748 Garching, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,29]]},"reference":[{"key":"ref_1","unstructured":"Kurthen, M. 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