Partially directed random fields, for some kind of definition thereof? The classic chain graph of the 80s allows you to have cycling sets of mutually influencing variables, connected by directed acyclic influence. It looks like the CPDAGs of Maathuis et al, and I’m curious what the precise relationship is.

- BoSt95
- Bouckaert, R. R., & Studený, M. (1995) Chain graphs: Semantics and expressiveness. In C. Froidevaux & J. Kohlas (Eds.), Symbolic and Quantitative Approaches to Reasoning and Uncertainty (pp. 69–76). Springer Berlin Heidelberg
- Bunt96
- Buntine, W. L.(1996) A guide to the literature on learning probabilistic networks from data.
*IEEE Transactions on Knowledge and Data Engineering*, 8(2), 195–210. DOI. - Bunt13
- Buntine, W. L.(2013) Chain Graphs for Learning.
*arXiv:1302.4933 [Cs]*. - DaLa93
- Dawid, A. P., & Lauritzen, S. L.(1993) Hyper Markov Laws in the Statistical Analysis of Decomposable Graphical Models.
*The Annals of Statistics*, 21(3), 1272–1317. DOI. - Drto09
- Drton, M. (2009) Discrete chain graph models.
*Bernoulli*, 15(3), 736–753. DOI. - Laur92
- Lauritzen, S. L.(1992) Propagation of Probabilities, Means, and Variances in Mixed Graphical Association Models.
*Journal of the American Statistical Association*, 87(420), 1098–1108. DOI. - LaRi02
- Lauritzen, S. L., & Richardson, T. S.(2002) Chain graph models and their causal interpretations.
*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, 64(3), 321–348. DOI. - LaWe89
- Lauritzen, S. L., & Wermuth, N. (1989) Graphical Models for Associations between Variables, some of which are Qualitative and some Quantitative.
*The Annals of Statistics*, 17(1), 31–57. DOI. - Stud97a
- Studený, M. (1997a) A recovery algorithm for chain graphs.
*International Journal of Approximate Reasoning*, 17(2–3), 265–293. DOI. - Stud97b
- Studený, M. (1997b) On Recovery Algorithm for Chain Graphs.
- StBo98
- Studený, M., & Bouckaert, R. R.(1998) On chain graph models for description of conditional independence structures.
*The Annals of Statistics*, 26(4), 1434–1495. DOI. - WeLa90
- Wermuth, N., & Lauritzen, S. L.(1990) On Substantive Research Hypotheses, Conditional Independence Graphs and Graphical Chain Models.
*Journal of the Royal Statistical Society. Series B (Methodological)*, 52(1), 21–50. - XiJR12
- Xing, E. P., Jordan, M. I., & Russell, S. (2012) A Generalized Mean Field Algorithm for Variational Inference in Exponential Families.
*arXiv:1212.2512 [Cs, Stat]*.