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SUMMARY:A matrix algebra for graphical statistical models - Qingyuan Zhao 
 (Statistical Laboratory)
DTSTART:20241018T143000Z
DTEND:20241018T160000Z
UID:TALK223030@talks.cam.ac.uk
CONTACT:Martina Scauda
DESCRIPTION:Directed mixed graphs permit directed and bidirected edges bet
 ween any two vertices. They were\nfirst considered in the path analysis de
 veloped by Sewall Wright and play an essential role in statistical\nmodeli
 ng. We introduce a matrix algebra for walks on such graphs. Each element o
 f the algebra is\na matrix whose entries are sets of walks on the graph fr
 om the corresponding row to the corresponding column. The matrix algebra i
 s then generated by applying addition (set union)\, multiplication\n(conca
 tenation)\, and transpose to the two basic matrices consisting of directed
  and bidirected edges.\nWe use it to formalize\, in the context of Gaussia
 n linear systems\, the correspondence between important graphical concepts
  such as latent projection and graph separation with important probabilist
 ic\nconcepts such as marginalization and (conditional) independence. In tw
 o further examples regarding\nconfounder adjustment and the augmentation c
 riterion\, we illustrate how the algebra allows us to visualize complex gr
 aphical proofs. A “dictionary” and LATEXmacros for the matrix algebra 
 are provided\nin the Appendix.\n\nPreprint available: https://arxiv.org/pd
 f/2407.15744
LOCATION:MR12\, Centre for Mathematical Sciences\, Wilberforce Road\, Camb
 ridge
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