{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T07:07:05Z","timestamp":1774940825370,"version":"3.50.1"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2015,6,11]],"date-time":"2015-06-11T00:00:00Z","timestamp":1433980800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2015,7]]},"DOI":"10.1007\/s11222-015-9558-5","type":"journal-article","created":{"date-parts":[[2015,6,11]],"date-time":"2015-06-11T04:29:01Z","timestamp":1433996941000},"page":"755-766","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Computing functions of random variables via reproducing kernel Hilbert space representations"],"prefix":"10.1007","volume":"25","author":[{"given":"Bernhard","family":"Sch\u00f6lkopf","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Krikamol","family":"Muandet","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kenji","family":"Fukumizu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stefan","family":"Harmeling","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jonas","family":"Peters","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2015,6,11]]},"reference":[{"key":"9558_CR1","first-page":"1","volume":"3","author":"FR Bach","year":"2002","unstructured":"Bach, F.R., Jordan, M.I.: Kernel independent component analysis. J. Mach. Learn. Res. 3, 1\u201348 (2002)","journal-title":"J. Mach. Learn. Res."},{"key":"9558_CR2","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-9096-9","volume-title":"Reproducing Kernel Hilbert Spaces in Probability and Statistics","author":"A Berlinet","year":"2004","unstructured":"Berlinet, A., Agnan, T.C.: Reproducing Kernel Hilbert Spaces in Probability and Statistics. Kluwer Academic Publishers, Boston (2004)"},{"key":"9558_CR3","doi-asserted-by":"crossref","unstructured":"Boser, B.E., Guyon, I.M., Vapnik, V.N.: A training algorithm for optimal margin classifiers. In: Proceedings of the Fifth Annual Workshop on Computational Learning Theory. COLT \u201992, pp. 144\u2013152. ACM, New York (1992)","DOI":"10.1145\/130385.130401"},{"key":"9558_CR4","doi-asserted-by":"crossref","unstructured":"Cassel, J.: Probabilistic programming with stochastic memoization: implementing non-parametric bayesian inference. Math. J. 16 (2014). doi: 10.3888\/tmj.16-1","DOI":"10.3888\/tmj.16-1"},{"issue":"3","key":"9558_CR5","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1214\/aoms\/1177730201","volume":"19","author":"B Epstein","year":"1948","unstructured":"Epstein, B.: Some applications of the Mellin transform in statistics. Ann. Math. Stat. 19(3), 370\u2013379 (1948)","journal-title":"Ann. Math. Stat."},{"issue":"4","key":"9558_CR6","doi-asserted-by":"crossref","first-page":"990","DOI":"10.1080\/10807039609383659","volume":"2","author":"S Ferson","year":"1996","unstructured":"Ferson, S.: What Monte Carlo methods cannot do. Hum. Ecol. Risk Assess.: Int. J. 2(4), 990\u20131007 (1996). doi: 10.1080\/10807039609383659","journal-title":"Hum. Ecol. Risk Assess.: Int. J."},{"issue":"4","key":"9558_CR7","doi-asserted-by":"crossref","first-page":"1871","DOI":"10.1214\/08-AOS637","volume":"37","author":"K Fukumizu","year":"2009","unstructured":"Fukumizu, K., Bach, F., Jordan, M.I.: Kernel dimension reduction in regression. Ann. Stat. 37(4), 1871\u20131905 (2009)","journal-title":"Ann. Stat."},{"key":"9558_CR8","first-page":"489","volume-title":"Advances in Neural Information Processing Systems","author":"K Fukumizu","year":"2008","unstructured":"Fukumizu, K., Gretton, A., Sun, X., Sch\u00f6lkopf, B.: Kernel measures of conditional dependence. In: Platt, J.C., Koller, D., Singer, Y., Roweis, S. (eds.) Advances in Neural Information Processing Systems, vol. 20, pp. 489\u2013496. Curran, Red Hook (2008)"},{"key":"9558_CR9","first-page":"3753","volume":"14","author":"K Fukumizu","year":"2013","unstructured":"Fukumizu, K., Song, L., Gretton, A.: Kernel Bayes\u2019 rule: Bayesian inference with positive definite kernels. J. Mach. Learn. Res. 14, 3753\u20133783 (2013)","journal-title":"J. Mach. Learn. Res."},{"key":"9558_CR10","doi-asserted-by":"crossref","unstructured":"Gordon, A.D., Henzinger, T.A., Nori, A.V., Rajamani, S.K.: Probabilistic programming. In: International Conference on Software Engineering (ICSE, FOSE track) (2014)","DOI":"10.1145\/2593882.2593900"},{"key":"9558_CR11","first-page":"723","volume":"13","author":"A Gretton","year":"2012","unstructured":"Gretton, A., Borgwardt, K.M., Rasch, M.J., Sch\u00f6lkopf, B., Smola, A.: A kernel two-sample test. J. Mach. Learn. Res. 13, 723\u2013773 (2012)","journal-title":"J. Mach. Learn. Res."},{"key":"9558_CR12","doi-asserted-by":"crossref","unstructured":"Gretton, A., Bousquet, O., Smola, A.J., Sch\u00f6lkopf, B.: Measuring statistical dependence with Hilbert-Schmidt norms. In: Jain, S., Simon, H.U., Tomita, E. (eds.) Algorithmic Learning Theory: 16th International Conference, pp. 63\u201378. Springer, Berlin (2005)","DOI":"10.1007\/11564089_7"},{"key":"9558_CR13","first-page":"131","volume-title":"Dataset Shift in Machine Learning","author":"A Gretton","year":"2009","unstructured":"Gretton, A., Smola, A.J., Huang, J., Schmittfull, M., Borgwardt, K., Sch\u00f6lkopf, B.: Covariate shift by kernel mean matching. In: Candela, J.Q., Sugiyama, M., Schwaighofer, A., Lawrence, N.D. (eds.) Dataset Shift in Machine Learning, pp. 131\u2013160. MIT Press, Cambridge (2009)"},{"key":"9558_CR14","doi-asserted-by":"crossref","unstructured":"Harmeling, S., Hirsch, M., Schlkopf, B.: On a link between kernel mean maps and Fraunhofer diffraction, with an application to super-resolution beyond the diffraction limit. In: CVPR, pp. 1083\u20131090. IEEE (2013)","DOI":"10.1109\/CVPR.2013.144"},{"key":"9558_CR15","doi-asserted-by":"crossref","unstructured":"Hofmann, T., Sch\u00f6lkopf, B., Smola, A.J.: Kernel methods in machine learning. Ann. Stat. 36(3), 1171\u20131220 (2008)","DOI":"10.1214\/009053607000000677"},{"key":"9558_CR16","first-page":"601","volume-title":"Advances in Neural Information Processing Systems (NIPS)","author":"J Huang","year":"2007","unstructured":"Huang, J., Smola, A., Gretton, A., Borgwardt, K., Sch\u00f6lkopf, B.: Correcting sample selection bias by unlabeled data. In: Sch\u00f6lkopf, B., Platt, J., Hoffman, T. (eds.) Advances in Neural Information Processing Systems (NIPS), pp. 601\u2013608. MIT Press, Cambridge (2007)"},{"key":"9558_CR17","unstructured":"James, W., Stein, C.: Estimation with quadratic loss. Proc. Fourth Berkeley Symp. Math. Statist. Prob. 361\u2013379 (1961)"},{"issue":"3","key":"9558_CR18","doi-asserted-by":"crossref","first-page":"A1241","DOI":"10.1137\/110839680","volume":"34","author":"S Jaroszewicz","year":"2012","unstructured":"Jaroszewicz, S., Korzen, M.: Arithmetic operations on independent random variables: a numerical approach. SIAM J. Sci. Comput. 34(3), A1241\u2013A1265 (2012)","journal-title":"SIAM J. Sci. Comput."},{"key":"9558_CR19","unstructured":"Kanagawa, M., Fukumizu, K.: Recovering distributions from Gaussian RKHS embeddings. In: JMLR W&CP 33 (Proc. AISTATS 2014), pp. 457\u2013465. (2014)"},{"key":"9558_CR20","unstructured":"Kpotufe, S., Sgouritsa, E., Janzing, D., Sch\u00f6lkopf, B.: Consistency of causal inference under the additive noise model. In: ICML (2014)"},{"key":"9558_CR21","unstructured":"Lopez-Paz, D., Muandet, K., Sch\u00f6lkopf, B., Tolstikhin, I.: Towards a learning theory of cause-effect inference. In: Proceedings of the 32nd International Conference on Machine Learning, JMLR: W&CP, Lille, France, in press (2015)"},{"key":"9558_CR22","unstructured":"Milios, D.: Probability Distributions as Program Variables. Master\u2019s thesis, School of Informatics, University of Edinburgh (2009)"},{"key":"9558_CR23","unstructured":"Mooij, J.M., Peters, J., Janzing, D., Zscheischler, J., Sch\u00f6lkopf, B.: Distinguishing cause from effect using observational data: methods and benchmarks. arXiv:1412.3773 (2014). Accessed 18 Jan 2015"},{"key":"9558_CR24","unstructured":"Muandet, K., Balduzzi, D., Sch\u00f6lkopf, B.: Domain generalization via invariant feature representation. In: ICML, pp. 10\u201318 (2013)"},{"key":"9558_CR25","first-page":"10","volume-title":"Advances in Neural Information Processing Systems 25","author":"K Muandet","year":"2012","unstructured":"Muandet, K., Fukumizu, K., Dinuzzo, F., Sch\u00f6lkopf, B.: Learning from distributions via support measure machines. In: Bartlett, P., Pereira, F., Burges, C., Bottou, L., Weinberger, K. (eds.) Advances in Neural Information Processing Systems 25, pp. 10\u201318. Curran Associates, Inc., Red Hook (2012)"},{"key":"9558_CR26","unstructured":"Muandet, K., Fukumizu, K., Sriperumbudur, B., Gretton, A., Sch\u00f6lkopf, B.: Kernel mean estimation and Stein effect. In: Xing, E.P., Jebara, T. (eds.). Proceedings of the 31st International Conference on Machine Learning, JMLR W&CP, vol. 32. (2014a)"},{"key":"9558_CR27","unstructured":"Muandet, K., Sriperumbudur, B., Sch\u00f6lkopf, B.: Kernel mean estimation via spectral filtering. In: Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N., Weinberger, K. (eds.) Advances in Neural Information Processing Systems, vol. 27. Curran Associates Inc. (2014b)"},{"key":"9558_CR28","unstructured":"Paige, B., Wood, F.: A compilation target for probabilistic programming languages. In: Journal of Machine Learning Research; ICML 2014, pp. 1935\u20131943 (2014)"},{"key":"9558_CR29","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511803161","volume-title":"Causality: Models, Reasoning, and Inference","author":"J Pearl","year":"2009","unstructured":"Pearl, J.: Causality: Models, Reasoning, and Inference, 2nd edn. Cambridge University Press, New York (2009)","edition":"2"},{"key":"9558_CR30","first-page":"2009","volume":"15","author":"J Peters","year":"2014","unstructured":"Peters, J., Mooij, J., Janzing, D., Sch\u00f6lkopf, B.: Causal discovery with continuous additive noise models. J. Mach. Learn. Res. 15, 2009\u20132053 (2014)","journal-title":"J. Mach. Learn. Res."},{"issue":"3","key":"9558_CR31","doi-asserted-by":"crossref","first-page":"614","DOI":"10.1137\/0118054","volume":"18","author":"RD Prasad","year":"1970","unstructured":"Prasad, R.D.: Probability distributions of algebraic functions of independent random variables. SIAM J. Appl. Math. 18(3), 614\u2013626 (1970)","journal-title":"SIAM J. Appl. Math."},{"key":"9558_CR32","unstructured":"Rahimi, A., Recht, B.: Random features for large-scale kernel machines. In: Platt, J., Koller, D., Singer, Y., Roweis, S. (eds.) Advances in Neural Information Processing Systems, vol 20, pp. 1177\u20131184. Curran Associates, Inc. (2007)"},{"issue":"7","key":"9558_CR33","doi-asserted-by":"crossref","first-page":"1443","DOI":"10.1162\/089976601750264965","volume":"13","author":"B Sch\u00f6lkopf","year":"2001","unstructured":"Sch\u00f6lkopf, B., Platt, J.C., Shawe-Taylor, J.C., Smola, A.J., Williamson, R.C.: Estimating the support of a high-dimensional distribution. Neural Comput. 13(7), 1443\u20131471 (2001)","journal-title":"Neural Comput."},{"key":"9558_CR34","doi-asserted-by":"crossref","unstructured":"Sch\u00f6lkopf, B., Smola, A.J.: Learning with Kernels. MIT Press, Cambridge, MA, USA (2002)","DOI":"10.7551\/mitpress\/4175.001.0001"},{"key":"9558_CR35","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511809682","volume-title":"Kernel Methods for Pattern Analysis","author":"J Shawe-Taylor","year":"2004","unstructured":"Shawe-Taylor, J., Cristianini, N.: Kernel Methods for Pattern Analysis. Cambridge University Press, Cambridge (2004)"},{"key":"9558_CR36","doi-asserted-by":"crossref","unstructured":"Smola, A., Gretton, A., Song, L., Sch\u00f6lkopf, B.: A Hilbert space embedding for distributions. In: Proc. Algorithmic Learning Theory, pp. 13\u201331. Springer-Verlag (2007)","DOI":"10.1007\/978-3-540-75225-7_5"},{"key":"9558_CR37","unstructured":"Song, L.: Learning via Hilbert space embedding of distributions. Ph.D. thesis, The School of Information Technologies, The University of Sydney (2008)"},{"key":"9558_CR38","unstructured":"Song, L., Boots, B., Siddiqi, S.M., Gordon, G., Smola, A.J.: Hilbert space embeddings of hidden Markov models. In: Proceedings of the 27th International Conference on Machine Learning (ICML) (2010)"},{"key":"9558_CR39","unstructured":"Song, L., Gretton, A., Bickson, D., Low, Y., Guestrin, C.: Kernel belief propagation. In: Proceedings of the 14th International Conference on Artificial Intelligence and Statistics (AISTATS) (2011)"},{"key":"9558_CR40","doi-asserted-by":"crossref","unstructured":"Song, L., Huang, J., Smola, A., Fukumizu, K.: Hilbert space embeddings of conditional distributions with applications to dynamical systems. In: ICML (2009)","DOI":"10.1145\/1553374.1553497"},{"key":"9558_CR41","doi-asserted-by":"crossref","unstructured":"Song, L., Zhang, X., Smola, A.J., Gretton, A., Sch\u00f6lkopf, B.: Tailoring density estimation via reproducing kernel moment matching. In: Cohen, W.W., McCallum, A., Roweis, S. (eds.). In: Proceedings of the 25th International Conference on Machine Learning, pp. 992\u2013999. ACM Press, New York (2008)","DOI":"10.1145\/1390156.1390281"},{"key":"9558_CR42","volume-title":"The Algebra of Random Variables, Wiley Series in Probability and Mathematical Statistics","author":"M Springer","year":"1979","unstructured":"Springer, M.: The Algebra of Random Variables, Wiley Series in Probability and Mathematical Statistics. Wiley, Hoboken (1979)"},{"issue":"3","key":"9558_CR43","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1137\/0114046","volume":"14","author":"MD Springer","year":"1966","unstructured":"Springer, M.D., Thompson, W.E.: The distribution of products of independent random variables. SIAM J. Appl. Math. 14(3), 511\u2013526 (1966)","journal-title":"SIAM J. Appl. Math."},{"key":"9558_CR44","unstructured":"Sriperumbudur, B.K., Gretton, A., Fukumizu, K., Lanckriet, G., Sch\u00f6lkopf, B.: Injective Hilbert space embeddings of probability measures. In: The 21st Annual Conference on Learning Theory (COLT) (2008)"},{"key":"9558_CR45","first-page":"1517","volume":"99","author":"BK Sriperumbudur","year":"2010","unstructured":"Sriperumbudur, B.K., Gretton, A., Fukumizu, K., Sch\u00f6lkopf, B., Lanckriet, G.: Hilbert space embeddings and metrics on probability measures. J. Mach. Learn. Res. 99, 1517\u20131561 (2010)","journal-title":"J. Mach. Learn. Res."},{"key":"9558_CR46","first-page":"67","volume":"2","author":"I Steinwart","year":"2002","unstructured":"Steinwart, I.: On the influence of the kernel on the consistency of support vector machines. J. Mach. Learn. Res. 2, 67\u201393 (2002)","journal-title":"J. Mach. Learn. Res."},{"key":"9558_CR47","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-77242-4","volume-title":"Support Vector Machines","author":"I Steinwart","year":"2008","unstructured":"Steinwart, I., Christmann, A.: Support Vector Machines. Springer, New York (2008)"},{"key":"9558_CR48","unstructured":"Szab\u00f3, Z., Gretton, A., P\u00f3czos, B., Sriperumbudur, B.: Two-stage sampled learning theory on distributions. arXiv:1402.1754 (2014). Accessed 18 Jan 2015"},{"key":"9558_CR49","unstructured":"Williamson, R.C.: Probabilistic arithmetic. Ph.D. thesis, Department of Electrical Engineering, University of Queensland, St. Lucia, Queensland, Australia (1989)"},{"key":"9558_CR50","unstructured":"Wood, F., van de Meent, J.W., Mansinghka, V.: A new approach to probabilistic programming inference. In: Proceedings of the 17th International Conference on Artificial Intelligence and Statistics (AISTATS), JMLR: W&CP, vol. 33. Reykjavik, Iceland (2014)"},{"key":"9558_CR51","unstructured":"Zhang, K., Peters, J., Janzing, D., Sch\u00f6lkopf, B.: Kernel-based Conditional Independence Test and Application in Causal Discovery. In: Cozman, F., Pfeffer, A. (eds.). In: 27th Conference on Uncertainty in Artificial Intelligence (UAI 2011), pp. 804\u2013813. AUAI Press, Corvallis, OR, USA (2011)"}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-015-9558-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11222-015-9558-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-015-9558-5","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,28]],"date-time":"2025-05-28T13:54:20Z","timestamp":1748440460000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11222-015-9558-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,6,11]]},"references-count":51,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2015,7]]}},"alternative-id":["9558"],"URL":"https:\/\/doi.org\/10.1007\/s11222-015-9558-5","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,6,11]]}}}