causal discovery from interventions
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Causal Discovery from Interventions Brady Neal causalcourse.com - PowerPoint PPT Presentation

Causal Discovery from Interventions Brady Neal causalcourse.com Structural Interventions Single-Node Interventions Multi-Node Interventions Parametric Interventions Interventional Markov Equivalence Miscellaneous Other Settings Brady Neal


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Intervention in Two-Variable Setting or A B A B Same Markov equivalence class / essential graph A B Brady Neal Single-Node Interventions 4 / 28

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Intervention in Two-Variable Setting <latexit 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A := N A or B := f ( A, N B ) A B A B Same Markov equivalence class / essential graph A B Brady Neal Single-Node Interventions 4 / 28

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Intervention in Two-Variable Setting Intervention <latexit 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A := N A or B := f ( A, N B ) A B A B Same Markov equivalence class / essential graph A B Brady Neal Single-Node Interventions 4 / 28

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Intervention in Two-Variable Setting Intervention <latexit 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A := N A or B := f ( A, N B ) A B A B Same Markov equivalence class / essential graph A B Brady Neal Single-Node Interventions 4 / 28

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Intervention in Two-Variable Setting Intervention <latexit 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A := N A or B := f ( A, N B ) A B A B Manipulated graph Same Markov equivalence class / essential graph A B Brady Neal Single-Node Interventions 4 / 28

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Intervention in Two-Variable Setting Intervention <latexit 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A := N A or B := f ( A, N B ) A B A B B := N 0 Manipulated graph B Same Markov equivalence class / essential graph A B Brady Neal Single-Node Interventions 4 / 28

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Intervention in Two-Variable Setting Intervention Intervention <latexit 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A := N A or B := f ( A, N B ) A B A B B := N 0 Manipulated graph B Same Markov equivalence class / essential graph A B Brady Neal Single-Node Interventions 4 / 28

  8. <latexit 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<latexit 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<latexit 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<latexit sha1_base64="5cxwm212S7zL9/sTDKDeLhr3iw=">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</latexit> Intervention in Two-Variable Setting Intervention Intervention <latexit sha1_base64="YoTFQme1yoczA6rgQ3GQmijZ9p8=">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</latexit> A := N A or B := f ( A, N B ) A B A B B := N 0 Manipulated graph Manipulated graph B Same Markov equivalence class / essential graph A B Brady Neal Single-Node Interventions 4 / 28

  9. <latexit 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Intervention in Two-Variable Setting Intervention Intervention <latexit sha1_base64="YoTFQme1yoczA6rgQ3GQmijZ9p8=">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</latexit> A := N A <latexit 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B := N B or B := f ( A, N B ) A := f ( B, N A ) A B A B B := N 0 Manipulated graph Manipulated graph B Same Markov equivalence class / essential graph A B Brady Neal Single-Node Interventions 4 / 28

  10. <latexit 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<latexit sha1_base64="Z9WmOs6HNVD16+uKnqwq1YeS0=">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</latexit> <latexit sha1_base64="wsqcWLVObrqn7zuAI3jdVBSxwg=">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</latexit> <latexit 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<latexit sha1_base64="WOFVOe1GkhAd+LMRe9L9kH/qKo=">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</latexit> Intervention in Two-Variable Setting Intervention Intervention B := N 0 B <latexit sha1_base64="YoTFQme1yoczA6rgQ3GQmijZ9p8=">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</latexit> A := N A <latexit 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B := N B or B := f ( A, N B ) A := f ( B, N A ) A B A B B := N 0 Manipulated graph Manipulated graph B Same Markov equivalence class / essential graph A B Brady Neal Single-Node Interventions 4 / 28

  11. <latexit 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<latexit 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<latexit 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Two Variables: Interventional Essential Graphs True graph A B A B A B Brady Neal Single-Node Interventions 5 / 28

  12. <latexit 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<latexit 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<latexit 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Two Variables: Interventional Essential Graphs Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit 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I = {} A B A B A B Brady Neal Single-Node Interventions 5 / 28

  13. <latexit 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Two Variables: Interventional Essential Graphs A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B Brady Neal Single-Node Interventions 5 / 28

  14. <latexit 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<latexit 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<latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { A } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B Brady Neal Single-Node Interventions 5 / 28

  15. <latexit 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<latexit 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<latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { A } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B Brady Neal Single-Node Interventions 5 / 28

  16. <latexit 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<latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { A } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  17. <latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { A } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  18. <latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { A } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit 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I = {} A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  19. <latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { A } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit 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I = {} A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  20. <latexit 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<latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { A } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  21. <latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { A } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit 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I = {} A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  22. <latexit 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<latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { A } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit 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I = {} A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  23. <latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { A } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  24. <latexit 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<latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { B } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit 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I = {} A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  25. <latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { B } A B Interventional essential graphs True graph <latexit sha1_base64="IT2+fM0h2Pd6il5NqdMJ5oS+U=">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</latexit> I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  26. <latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { B } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  27. <latexit 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<latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { B } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  28. <latexit 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<latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { B } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  29. <latexit 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<latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { B } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">AUenicrVjbtGEN2kt8S9OYnf+sJaKdCisiq5ad0XAQHsBG3RA5gO27jIJAoyiJEixJWXUI/URf2/o/Rf+tAzZ5eiqBulwBJErmZnzszOblsh54bJ/X6v7duv/Pue+9/cOfu1ocfzJp9v37p/FwTCynVM78ILovN2KHc8dOKeJm3jOeRg5Lb/tOS/a/UOZf3HlRLEbDE6S69B5bcuB27XtVsJSOc/WU3rIr0Yv96u1Gt1fqz5QcMKsp8joN7d/9RF6qjAmWrofKVowYqwdhTLRXj+1I1VF2FoL1SKWgRi7nHTVW5AdgsBRwvUPq6X+PfSUAf4L5gxpW1o8fCLIGmpLwxPB+Muqfou+q0p3mU6UmKLjde4tw2mD2qieqCWyWc68u18fVLVp1gDT9wtS5WEpIifrALa+7i7uF/ghXK9RqcDkYdSEUY2aB5oGqK6Ihw154X3/QYiRb5HIzE6lV2r7Ja5gN8+8BqYRzTUrHVUk9NXAbU7NBW4fEY0+WIf2CF2r5VWN3JGlzGXa9CeE9A6as3GBWRV+mczq7lXIlBHi/QJfwJOQbgjkEJmPku/OeCo5irNjBFR4uRvORaQmZQzSD/zPksm0PEc4/2xIyaxXxzqSc0lZXbndnp4d4mdgT5FHM96hFfVGPQ+yI2Zt5sUruCP9G/GfTRnuGXmNlSjyrWJNkVxW4AWbcCZb2hPZnZpG2LgXNMt89Qv1O4h7lV6RvK5CNqCFOotj6hqY2DfRV8QvPq5CFc9mlCp1+LSjx2zpg5br0i7xusImb9eAY1xd2j10KiVKld/DPi2OeaJKJD6h5h3KEekbQ62rqwNgxLmgV37jsTvNaLeaf1K7ujLNaRW7ASrVMVjapqa4emRVLBMTua/ZphvqnhxzdeMZq9q0P+8akqNF+0Qqj08xVh1mY49SRVzx4uJ1LlrLPukyliK7y/B0aRU19Bj1mZoNG1N1d+hqZKA8bB5901O6HoXbaOZmR03jKA5+uc9mTZOZ3YLaYUxf03vT8KsQnRj7bDaTjp6SuljtjZRVlI4zSycxqeZcjh/7RI4m4xgvp65Br+Xzys6Zm1sW+GbzBHKJkbGKQ1kXvMDcWr/xEVdALKsiqoj8l0kKJ+HQSYvxwvkQuMLbp5YBKFLh6WR+XBqDI1DGxmNdzmV1mNvgsvN1yHlhMk9z6IpKZjikqldrlbxclDeavlq2M8n3oqymr5YVyhVz0zX9ICbWeYlcwjiIhD+VpSclUgH3BE3XNv+6hn0D7rVDkxeRsfBFiaTPHqNXFZiqfFZqX8IuGBmfiMxvb+HBK/a/MZ+ANvVjLpts5M1c7vqtfJrLjzb0bC7pb+jfXPJNiaTDvcGd0ETmeWlNCdxKZfOp7I9wcFukj2/6Gqr8tqGF2XP7pjOZJn9RrJC745N9oDqWv15ft+RSm6U1vGQ3bfNPhRc2dF7ZzeGFoeweEa64uZ7/n5cp36WiS3ft4cMg6LumM2o59A8qeQ7PQT+1PLvHzk0XKp8CR2cmy58E9ZoCO4Y/ohc/wjDWmpm/xfNfA9Wlh1sXVc5ZQ9Mvp3H3gXyA/eqI+9rmuKE5De0VzkxFHU+gIfuNzeky5/bMGWAZ0iZ6FyPrOvMnUVmNemnzR649AnsugQxz+PFeKvjuxhZvykJeALIz2E3gTx9xlwPL8+Z5W8Eyt5TLD89h7h3cfXY0WPDe8Ss89jZHfQYfcLXT0+Hc+8qJH/3URWSu+nu/Rl9RfzZ0+8lT69DcOQ8+hSTUFrTphGXnaNDyiTmaTHmTM/WdcKJ2WN2tbPJVXvsFrmn7kmN1i6/V2pTH79m1+cLZfa3xXqz/frz+3ryZu6M+U7vqS/jpQD1GZR7Dr+LjP9Vf6u8H/+3s7ny187VmvX3LyDxQhc/Oo/8BhbX2Yg=</latexit> I = {} A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  30. <latexit 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<latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { B } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  31. <latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { B } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit 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I = {} A B A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  32. <latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { B } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit 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I = {} A B A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  33. <latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { B } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit 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I = {} A B A B A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  34. <latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = { B } A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  35. <latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = {} A B Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit 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I = {} A B A B A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  36. <latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = {} Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  37. <latexit 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<latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = {} Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  38. <latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = {} Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B A B A B A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  39. <latexit 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Two Variables: Interventional Essential Graphs True graph under intervention <latexit 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I = {} Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B A B A B A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  40. <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B A B A B A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  41. <latexit 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<latexit 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Two Variables: Interventional Essential Graphs Need more than one intervention to identify the graph Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit 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I = {} A B A B A B A B A B A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 5 / 28

  42. <latexit 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<latexit 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<latexit 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Two Variables: Two Interventions Identify the Graph Two interventions are sufficient and necessary to identify the graph. Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit 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I = {} A B A B A B A B A B A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 6 / 28

  43. <latexit 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Two Variables: Two Interventions Identify the Graph Two interventions are sufficient and necessary to identify the graph. Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">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</latexit> I = {} A B A B A B A B A B A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 6 / 28

  44. <latexit 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<latexit 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Two Variables: Two Interventions Identify the Graph Two interventions are sufficient and necessary to identify the graph. Interventional essential graphs True graph <latexit 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I = { B } <latexit 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I = { A } <latexit sha1_base64="hD4hMxQA2UtFZ03WDckODdM5J1Q=">AUenicrVjbtGEN2kt8S9OYnf+sJaKdCisiq5ad0XAQHsBG3RA5gO27jIJAoyiJEixJWXUI/URf2/o/Rf+tAzZ5eiqBulwBJErmZnzszOblsh54bJ/X6v7duv/Pue+9/cOfu1ocfzJp9v37p/FwTCynVM78ILovN2KHc8dOKeJm3jOeRg5Lb/tOS/a/UOZf3HlRLEbDE6S69B5bcuB27XtVsJSOc/WU3rIr0Yv96u1Gt1fqz5QcMKsp8joN7d/9RF6qjAmWrofKVowYqwdhTLRXj+1I1VF2FoL1SKWgRi7nHTVW5AdgsBRwvUPq6X+PfSUAf4L5gxpW1o8fCLIGmpLwxPB+Muqfou+q0p3mU6UmKLjde4tw2mD2qieqCWyWc68u18fVLVp1gDT9wtS5WEpIifrALa+7i7uF/ghXK9RqcDkYdSEUY2aB5oGqK6Ihw154X3/QYiRb5HIzE6lV2r7Ja5gN8+8BqYRzTUrHVUk9NXAbU7NBW4fEY0+WIf2CF2r5VWN3JGlzGXa9CeE9A6as3GBWRV+mczq7lXIlBHi/QJfwJOQbgjkEJmPku/OeCo5irNjBFR4uRvORaQmZQzSD/zPksm0PEc4/2xIyaxXxzqSc0lZXbndnp4d4mdgT5FHM96hFfVGPQ+yI2Zt5sUruCP9G/GfTRnuGXmNlSjyrWJNkVxW4AWbcCZb2hPZnZpG2LgXNMt89Qv1O4h7lV6RvK5CNqCFOotj6hqY2DfRV8QvPq5CFc9mlCp1+LSjx2zpg5br0i7xusImb9eAY1xd2j10KiVKld/DPi2OeaJKJD6h5h3KEekbQ62rqwNgxLmgV37jsTvNaLeaf1K7ujLNaRW7ASrVMVjapqa4emRVLBMTua/ZphvqnhxzdeMZq9q0P+8akqNF+0Qqj08xVh1mY49SRVzx4uJ1LlrLPukyliK7y/B0aRU19Bj1mZoNG1N1d+hqZKA8bB5901O6HoXbaOZmR03jKA5+uc9mTZOZ3YLaYUxf03vT8KsQnRj7bDaTjp6SuljtjZRVlI4zSycxqeZcjh/7RI4m4xgvp65Br+Xzys6Zm1sW+GbzBHKJkbGKQ1kXvMDcWr/xEVdALKsiqoj8l0kKJ+HQSYvxwvkQuMLbp5YBKFLh6WR+XBqDI1DGxmNdzmV1mNvgsvN1yHlhMk9z6IpKZjikqldrlbxclDeavlq2M8n3oqymr5YVyhVz0zX9ICbWeYlcwjiIhD+VpSclUgH3BE3XNv+6hn0D7rVDkxeRsfBFiaTPHqNXFZiqfFZqX8IuGBmfiMxvb+HBK/a/MZ+ANvVjLpts5M1c7vqtfJrLjzb0bC7pb+jfXPJNiaTDvcGd0ETmeWlNCdxKZfOp7I9wcFukj2/6Gqr8tqGF2XP7pjOZJn9RrJC745N9oDqWv15ft+RSm6U1vGQ3bfNPhRc2dF7ZzeGFoeweEa64uZ7/n5cp36WiS3ft4cMg6LumM2o59A8qeQ7PQT+1PLvHzk0XKp8CR2cmy58E9ZoCO4Y/ohc/wjDWmpm/xfNfA9Wlh1sXVc5ZQ9Mvp3H3gXyA/eqI+9rmuKE5De0VzkxFHU+gIfuNzeky5/bMGWAZ0iZ6FyPrOvMnUVmNemnzR649AnsugQxz+PFeKvjuxhZvykJeALIz2E3gTx9xlwPL8+Z5W8Eyt5TLD89h7h3cfXY0WPDe8Ss89jZHfQYfcLXT0+Hc+8qJH/3URWSu+nu/Rl9RfzZ0+8lT69DcOQ8+hSTUFrTphGXnaNDyiTmaTHmTM/WdcKJ2WN2tbPJVXvsFrmn7kmN1i6/V2pTH79m1+cLZfa3xXqz/frz+3ryZu6M+U7vqS/jpQD1GZR7Dr+LjP9Vf6u8H/+3s7ny187VmvX3LyDxQhc/Oo/8BhbX2Yg=</latexit> I = {} A B A B A B A B A B A B A B A B A B A B A B A B Brady Neal Single-Node Interventions 6 / 28

  45. Complete Graphs Are the Worst Case Brady Neal Single-Node Interventions 7 / 28

  46. Complete Graphs Are the Worst Case Theorem: Two graphs are Markov equivalent if and only if they have the same skeleton and same immoralities (Verma & Pearl, 1990; Frydenburg, 1990). Brady Neal Single-Node Interventions 7 / 28

  47. Complete Graphs Are the Worst Case Theorem: Two graphs are Markov equivalent if and only if they have the same skeleton and same immoralities (Verma & Pearl, 1990; Frydenburg, 1990). In complete graphs, there are no immoralities, so we can only get the complete skeleton graph (e.g. from PC) as the essential graph in the worse case Brady Neal Single-Node Interventions 7 / 28

  48. <latexit sha1_base64="ZiceQALX4eRkdDF5AOJvDwn4WyU=">AVpnicrVjpbtGEF6lV+JeSQr/6h82dgAHkFXJPVKgMJBESVAUTZAdpzUMlIeK4kQr5KUVYfQe/Qp+gj9275C36Yz3y5FUhflIBJELmdnvpmda7myIs9N0nb7v8aV97/4MOPrl7b+viTz/7/PqNmy+ScBzb8tgOvTB+aZmJ9NxAHqdu6smXUSxN3/LkiTXq8vzJuYwTNwyO0otInvnmIHD7rm2mRHp9o9HpWXLgBlnqjt5Erp2OYzndMvTntvE0dGQye+4F9HiapGYqz4y9+3cMzX2s32HSPbvb87/XEFY/eOcWpJL5wYsTsYpsZh2Dfun5FMd7XMA5IxrfBclmS6LPOAZUoWPnIGJQtv9yIzHcI2SROMU9IwN9dnHtQnsNkTwZOxTuvr+0W218jMVBRw92hP48C29c+0v0hCNCYux8IUgUhp7AlTJPQ9FR3RFhHRzkRGtJhGLualmIotkh0TlyQOk6gjug7o6VRTA3pmzATSNmnx6BeTpCFuax6Hxn1Q1Z31GyXeVToyYLONF3S3NKZP1FQMiVonl3NuLmfR169ZdUpr+AGrdWklESjsB7uy5j7dPXpOaYV8vSBOSOHpGIa2UTziKorCOmu/I8+2aISJjgkzRiq9fZvc5qng/pOyIsk8YJLGVbDfFYxyWAZglbmcdDTFcj/kErVPatw+rP1uAi7moVzHtElJF4Q6Mq8jqd5exazZVq5OkSXcyfgiMg7oQoITLfJf+5xFHNVZswWYeJSA6wlgZ1NLIP2M+z+aI4rkPexJEzUC+udAT6coq7M7t9OhuATsm+YzmhtDvmiSPRLYMbI392IT3DE9TfBkw0Z7jt5CZXI8m7Qmzq4m4Y0486wlCeUP3OLlHUZ0Qz93Re/QL+kuDfhFc7rJsmGsFBlcQJdgY79IfUV9otPV6ayZ3NKEzp82DFEtoyIVuhTSKzva1pFgvrxNGpGdw9ei4DShHb2zwRjH2viI6he0JjB3pY2qBe1xJ3tR3Tilb2jYvutKjVQP5x7arOK+V5QJUqGz8hCa2uJbvWKOANtd9pqtu6HqyQlWN52zyoL9RdfgHK3ax1JFfKqxcpCNQ0hVcdmLy9e5bC0HqJMmYsm+HxDHIaT6mp6gNiOtatUf1dJSHiYePu65xQ9c7aJnMzGdF97SmP+FSd857EM78Tpomc6sF75fl1iI+0fL4bcMfPQF0v9wKVZWNaZTNZtbLuxhJ+EeNOIKL4KvI6zlq9nPKM1siv1u8IFRM7YVCNtiu4gN5av/EjsUC/Yoayq+pMjzZQYbycRjXdnLuEy7wjaAmIwhWezeantTF4SJSp9lgfc3kdFja46HwOHs68xSHqh0joOrer1WzstleaPo62WdWb5XZRV9vSxTzpGbru4HCbBe1siliANL+KUsPaqRCrEnKLqy+dUG9gXYa8c6L2Jt4UmNpI8eo1YV6qp8Umtfi4Ya5+wzK9v4cFz9L8p3oAu68dCNr2UNwu5i7fyaSE/uaRnC0n/kv4tJN/USErsDe6MxjLPa2uKuZ7Vcql8qtsTJO0m+fuLqrYmrhZ5kfdsR3cmQ+83nBVqdzxED2hu1J8X9x2u5E5tHY/RfS30oRianTW1c/zO0IoIjdYX4J8L86Xm9TXMrnN86aLOCzrjvmMegMp3kLy0NS2p9c4BcniwxvgRO9k+Xvg/vIABXDn6gXPqF3rCk0fUPvdx26Pq7sepui8jlrPtlGfeAkO/SfvUQ+9rlcSN9GtqvnJmqOh6Rhvw31afLgtvTZ4BVSJfRuxZ2zcBaZ16TeNofEpU5gFzWIR4vx1sf3+XI6p+SECeA4hz2LpDLZ8zN8IqcWf2PQN3/FKtPzxHd+3T10NETzfsQWehs0vqMeqEr96eugv/VXD+HlBVcO5mt27O6aviz59+Bzi9jomj4FGnmBTSilZGXHWOjiCT6rfFBGfO4mTdqpyUFd68bUkpr3yNd6j7kdS7xdbr6zud+X/fFgcvDlqd71rt5wc797X/8xdFV+KW2KP/HRX3KPKfEZ+tRt/Nv5u/NP4d3tv+n28faJYr3S0DJfiMpn+7f/AedmR4M=</latexit> Complete Graphs Are the Worst Case Theorem: Two graphs are Markov equivalent if and only if they have the same skeleton and same immoralities (Verma & Pearl, 1990; Frydenburg, 1990). In complete graphs, there are no immoralities, so we can only get the complete skeleton graph (e.g. from PC) as the essential graph in the worse case Immorality A B C Brady Neal Single-Node Interventions 7 / 28

  49. <latexit sha1_base64="wqYwkHdEApJm7kG8N6KiuvyY+8c=">AVpHicrVjbtGEF2lt8S9xSn81Bc2doEkBXZbeECgYEkSoKiSIoEtZO0thtQ5EoixFvJVSH0G/0M/oFfW3/oX/TmbNLkdSNchAJIpezM2dm57ZcdWPfS1W7/V/j0nvf/DhR5evbHz8yaefX5189rzNBoljx2Ij9KXnbtVPpeKI+Vp3z5Mk6kHXR9+aI7PD8i9cySb0oPFLnsTwL7H7o9TzHVkR6tdm4fdqVfS/MlDd8E3uOGiVysmGZzw3rp8iV6fT5NKTHk1TZSp5ZN+/dsmxl3Ww327esbOfezuTOEsb7t6yTrvSjseXLnrIOo5174xE7i8X6UxFEq8/KMl0WKZk4EO3XzYwtUAlkmis+Y7S+c683P3y3OYPJWhW/HNq6vb7VYbH2t+sGcG28J8nkabV/4Sp8IVkXDESARCilAoGvCFil9T8SeaIuYaGciI1pCIw/zUkzEBsmOiEsSh03UIV379HRiqCE9M2YKaYe0+PRLSNISNwyPS+MeqPrO+q0S7zIdGbDZxnO6dw1mQFQlBkStk8s515fr0jeoWbWiNXyP1Xq0khgU9oNTWXOP7j49K1ohX8+JU9LIJamERg7RfKJqCutI6K49z74ZIBI2+CSN2OpVdq+ymucj+g4Jy6ZxCkvZVks8MnEJoVnCVubxEdPliH/QCrV9q7B60zV4iLteBfMeEWUo3tCoirxKZzm7lnMpgzxZoIv5FThC4k6JEiHzPfKfRxzVXHUIk3XYiGQfa4mRQS2D/CPm82yOKZ67sCdF1Czkmwc9samswu7cTp/uXWAnJ/R3AB62BdNskcCO0H25l5sgjuhpzGeHNjozNBbqEyOZ5PWxNnVJNyIZrwplvaE9mdukbYuI5plvrviMfRLinsTXuG8bpJsBAt1FqfQFZrYH1JfYb8EdGUqezanNKEjgB0DZMuQaIU+jcT6btMqUtSPb1AzuvwWgyUJrSzf8YB1gTR3QE3WMau9D0hb1upY4MHZMKlrZNx607xWC/nHtas746xWlgtRqZbJykNoaotvzYo5Amx32WuO6Ya6J6dY3WTGqi7sL7oG52jVPpYq4lONlYtsHECqisteXLzORWvZR50EUv2fZ84DiHVM/QUtRkbTRul+uYKokQDwf3wOSErnfWNp6ZyYgeGE/5xKfrnPcknvmdMG3k1Cm8V5fhfjQyOe7AXf8DNTVcs9RWVXZhEbZdGa1vIeRhH/0iCOu8WL4OsZavpr+rNLMutjvBi+cQ+SMVQZpXQXubF45Udim3rBNmV1Z8caYkeDuJabwz5dwhXOYdQktIFK7wbDo/qY3BA6JMjMd6mMvrsLDBQ+dzwXlqMk9z6IpSMxc1au1cl4uyhtNXy3rTvO9Kqvpq2WZ8hq56Zl+kALrZY2cQhxYIihl6VGNVIQ9QdO1zb+sYV+IvXZk8iIxFr6okQzQY/SqIlOVT2rtU+iCifEJy/z6Fh58jf43wRvQRf1YyKoLebOQO38rnxby4wt6tpAMLujfQvJNjaTE3uBNaSzrLamOtpLZfOp7o9QdJukr+/6Gpr4tolL/Ke7ZrOZJn9hrNC746H6AHNtfrz/L7DlbxXW8cjdN8u+lACze6K2jl+Z2hFBEdrC9Fvhfny3Xqa5Hc+nTQRwWdcd8Rr+BFG8h+ekhLe1PHvCLk0WGt8Cx2cny98FdZICO4Q/UC5/QO9YEmr6h97s9uj6q7HrovI5a2T6ZRl3n5APaL96gH3t4rixOQ3tVs5MVR0PSUP+m5jTZcHtmzPAMqSL6F2MrLPOnTuLzGrSb5sD4tInsPMaxCKPF+Otju9iZP1PSYQTQHEOexfI5TPmenhFziz/R6Duf4rlp+eY7j26+ujoqeF9gKz0dkl9Rh9wtdvT525/yo4f/epKjh3s+vXZvRV8WdPv32cXkfEUfDoU4yCtKaVEZedo2PIKPO2mOLMWZysW5WTsabtS0t5Vg8A5NP5Jmt9h4dXV7b/bft/nB8/3W3net9rP97bv75p+5y+JLcV3cJD8diLtUmU/Jr07jz8bfjX8a/259vfV46+etY816qWFkvhCVz9Zv/wOb1Ebd</latexit> Three Variables A B C Brady Neal Single-Node Interventions 8 / 28

  50. <latexit sha1_base64="wqYwkHdEApJm7kG8N6KiuvyY+8c=">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</latexit> Three Variables Intervention A B C Brady Neal Single-Node Interventions 8 / 28

  51. <latexit 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<latexit 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Three Variables Interventional essential graphs Intervention <latexit 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I = { C } A A B C B C Brady Neal Single-Node Interventions 8 / 28

  52. <latexit 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<latexit 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Three Variables Interventional essential graphs Intervention <latexit 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I = { C } A What we’ve learned: A • No C à A edge • No C à B edge B C B C • A and B connected Brady Neal Single-Node Interventions 8 / 28

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Three Variables Interventional essential graphs Intervention <latexit 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I = { C } A What we’ve learned: A • No C à A edge • No C à B edge B C B C • A and B connected Brady Neal Single-Node Interventions 8 / 28

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Three Variables Interventional essential graphs Intervention <latexit 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I = { C } A What we’ve learned: A • No C à A edge • No C à B edge B C B C • A and B connected <latexit 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I = { B } A B C Brady Neal Single-Node Interventions 8 / 28

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Three Variables Interventional essential graphs Intervention <latexit 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I = { C } A What we’ve learned: A • No C à A edge • No C à B edge B C B C • A and B connected • No B à A edge <latexit 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I = { B } • Yes A à C edge A • Yes B à C edge B C Brady Neal Single-Node Interventions 8 / 28

  56. Single Variable Interventions: n – 1 Are Sufficient for n > 2 (Eberhardt et al., 2006) Brady Neal Single-Node Interventions 9 / 28

  57. Single Variable Interventions: n – 1 Are Sufficient for n > 2 (Eberhardt et al., 2006) Intuition: Brady Neal Single-Node Interventions 9 / 28

  58. Single Variable Interventions: n – 1 Are Sufficient for n > 2 (Eberhardt et al., 2006) Intuition: 1. First intervention identifies the adjacencies between the remaining n – 1 variables (skeleton of that subgraph). Brady Neal Single-Node Interventions 9 / 28

  59. Single Variable Interventions: n – 1 Are Sufficient for n > 2 (Eberhardt et al., 2006) Intuition: 1. First intervention identifies the adjacencies between the remaining n – 1 variables (skeleton of that subgraph). i th intervention orients the edges incident on X i 2. Brady Neal Single-Node Interventions 9 / 28

  60. <latexit sha1_base64="tGpbmpyp125YAcIyfEcSJXiVBC0=">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</latexit> <latexit sha1_base64="U0KLusF5sfuvC5VBUOiyiGiN/UE=">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</latexit> Single Variable Interventions: n – 1 Are Sufficient for n > 2 (Eberhardt et al., 2006) Intuition: 1. First intervention identifies the adjacencies between the remaining n – 1 variables (skeleton of that subgraph). i th intervention orients the edges incident on X i 2. or X j X j X i X i Brady Neal Single-Node Interventions 9 / 28

  61. <latexit sha1_base64="U0KLusF5sfuvC5VBUOiyiGiN/UE=">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</latexit> <latexit sha1_base64="tGpbmpyp125YAcIyfEcSJXiVBC0=">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</latexit> Single Variable Interventions: n – 1 Are Sufficient for n > 2 (Eberhardt et al., 2006) Intuition: 1. First intervention identifies the adjacencies between the remaining n – 1 variables (skeleton of that subgraph). i th intervention orients the edges incident on X i 2. If , then X i ß X j 1. <latexit sha1_base64="53o59j3Z1/sk+TzbO5smk/AWdFI=">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</latexit> X i ⊥ ⊥ X j or X j X j X i X i Brady Neal Single-Node Interventions 9 / 28

  62. <latexit sha1_base64="tGpbmpyp125YAcIyfEcSJXiVBC0=">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</latexit> <latexit sha1_base64="U0KLusF5sfuvC5VBUOiyiGiN/UE=">AVNnicrVhLb9tGEF6nr8R9OalvbCxA9iArMpuCxcoDCR1EhRFEziAX6ltGBS5kljxVXIV1yH0l3ruf+il96KXvsTOvPtUiT1ohxEgsjl7Mw3s/Nartqx76Wq1fpr6dY73/ge37yx/+NHn3y6cvfecRoNEkceOZEfJadtO5W+F8oj5SlfnsaJtIO2L0/a/X2eP3klk9SLwkN1HcuLwO6GXsdzbEWky5U/z9uy64WZ8vqvY89Rg0QOly3zeWA9j1yZjp7PQ3o8S5Wt5IW18WjTspW10Wq0Nq1s/fTSWx9+N4P1+03rLPG6PWXtR3r0QX4f2H+kq4nbresK7ZVD5KS6KzNMJ/L0K2Ye7my1mq28LEmB9tmsCbM5yC6e+d3cS5cEQlHDEQgpAiForEvbJHS90xsi5aIiXYhMqIlNPIwL8VQLJPsgLgkcdhE7dO1S09nhrSM2OmkHZIi0+/hCQt8cDwuDTugKrvrN8q8c7SkQGbyme9tgBkRVokfUOrmc3G5Nn2DmlUrWsO3WK1HK4lBYT84lTV36O7Ts6IV8vWaOCWNXJKaOQzSeqprCOhO7a8+ybHiJhg0/SiK2eZ/c8q3k+om+fsGwap7CUbXEUxOXEJolbGUeHzGdjfgbrVDbNw+rM1qDh7jrVTDvIVH64jWNqsjzdJazazaXMsjDKbqYX4EjJO6UKBEy3yP/ecRzVWHMFmHjUh2sZYGdQ0yD9iPs/mOK5BXtSRM1CvnQE5vKuzO7fTp3gZ2QvIZzfWgh3RIHsksBNkb+7FBrgTerCkwMbnTF6E5XJ8WzQmji7GoQb0Yw3wtKe0P7MLdLWZUSzHdL/AT9kuLegFc4rxskG8FCncUpdIUm9nvUV9gvAV2Zyp7NKQ3oCGBHD9nSJ1qhTyOxvi9pFSnqxzeoGd19eC0GSgPa2T9XGAdYE0d0AN1XNHah6Ut6nVNsWvsGFa0sm8dKdJrRbyj2tXd8ZxrSwXolItk5V70NQSX5sVcwTY7rLXHNMNdU9OsbrhmFVt2F90Dc7Rqn0sVcSnGisX2diDVBWXvTh9ndPWsoM6aSCW7PsucexBqmPoKWozNpqWS/W3b6okQjwc3AOTE7reWdvV2ExG9MB4yic+Xe8J/HMr4RpI6fO4b3y/DzEJ0Y+3w242egzpc7RmVZRMaZaOZ+fIeRhL+0SOuMaL4esYa/li9LNKM4tivx28cAKRM1YZpEXRXeTG9JUfijXqBWuUV/cqSZkuDtJKbx+ohznXCZtw8tIVG4wrPR/LA2Bo+JMjQe62Aur8PCBg+dzwXnuck8zaErSo1xcFXP18p5OS1vNH2+rDvK96qsps+XZcor5KZn+kEKrNMaOYU4sERQytLDGqkIe4Kma5tfLmBfiL12YPIiMRae1EgG6DF6VZGpyme19il0wcT4hGV+fgMPvkL/G+IN6KZ+LGTVjbxZyF2/kU8L+asberaQDG7o30LydY2kxN7gjWgs86K2pjroJZL51PdniBpN8nfX3S1NXBtkxd5z3ZNZ7LMfsNZoXfHPfSAxkL9eXLf4Urerq3jAbpvG30ogWZ3Tu0cvTW0IoKDBdaXIt+L8+Ui9TVNbvG82UcpnXHfEa/gRvIfnpIS3tTx7wi5NFhrfAK7OT5e+DW8gAHcMfqBc+o3esITR9Re9323R9Wtn1FkXlc9bA9Msy7g4h79J+9Rj72s1xY3Ma2qcmao6npCG/Dc0p8uC2zdngFlIN9E7HVlnTtxFhnXpN82e8SlT2DXNYhFHk/Hmx/f6cj6n5IJ4DiHPY2kMtnzMXwipyZ/Y9A3f8Us0/PMd07dPXR0VPD+xhZ56OzS+ox+oSv3572J/6r4Pzdoarg3M3u3xvTV8UfP/12cXodEfBo08xCtKaVkacdY6OIaPM2KM2dxsm5WTsoab9y2tJRXgcHbM/1Imt1i+XJlbXv837fJwfFOc/ubZuvFztrDHfP3G3xubgvNshPu+IhVeYB+dVZ2lh6vnSydLr6x+rfq/+s/qtZby0Zmc9E5bP63/9pWix3</latexit> Single Variable Interventions: n – 1 Are Sufficient for n > 2 (Eberhardt et al., 2006) Intuition: 1. First intervention identifies the adjacencies between the remaining n – 1 variables (skeleton of that subgraph). i th intervention orients the edges incident on X i 2. If , then X i ß X j 1. <latexit sha1_base64="53o59j3Z1/sk+TzbO5smk/AWdFI=">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</latexit> X i ⊥ ⊥ X j or X j X j X i X i If , then X i à X j 2. <latexit sha1_base64="GUlspICIlxaPetAvPQZk6OxcNYQ=">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</latexit> X i 6 ? ? X j Brady Neal Single-Node Interventions 9 / 28

  63. <latexit sha1_base64="tGpbmpyp125YAcIyfEcSJXiVBC0=">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</latexit> <latexit sha1_base64="U0KLusF5sfuvC5VBUOiyiGiN/UE=">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</latexit> Single Variable Interventions: n – 1 Are Sufficient for n > 2 (Eberhardt et al., 2006) Intuition: 1. First intervention identifies the adjacencies between the remaining n – 1 variables (skeleton of that subgraph). i th intervention orients the edges incident on X i 2. If , then X i ß X j 1. <latexit sha1_base64="53o59j3Z1/sk+TzbO5smk/AWdFI=">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</latexit> X i ⊥ ⊥ X j or X j X j X i X i If , then X i à X j 2. <latexit sha1_base64="GUlspICIlxaPetAvPQZk6OxcNYQ=">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</latexit> X i 6 ? ? X j 3. No intervention necessary on X n as all those edges would have been oriented in step 2. Brady Neal Single-Node Interventions 9 / 28

  64. <latexit sha1_base64="U0KLusF5sfuvC5VBUOiyiGiN/UE=">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</latexit> <latexit sha1_base64="tGpbmpyp125YAcIyfEcSJXiVBC0=">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</latexit> Single Variable Interventions: n – 1 Are Sufficient for n > 2 (Eberhardt et al., 2006) Intuition: 1. First intervention identifies the adjacencies between the remaining n – 1 variables (skeleton of that subgraph). i th intervention orients the edges incident on X i 2. If , then X i ß X j 1. <latexit sha1_base64="53o59j3Z1/sk+TzbO5smk/AWdFI=">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</latexit> X i ⊥ ⊥ X j or X j X j X i X i If , then X i à X j 2. <latexit sha1_base64="GUlspICIlxaPetAvPQZk6OxcNYQ=">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</latexit> X i 6 ? ? X j 3. No intervention necessary on X n as all those edges would have been oriented in step 2. Note: if you include the empty set (observational), then n are necessary in the worst case (Eberhardt et al., 2006). Brady Neal Single-Node Interventions 9 / 28

  65. Single Variable Interventions: n – 1 Are Necessary in the Worst Case (Complete Graph) Brady Neal Single-Node Interventions 10 / 28

  66. Single Variable Interventions: n – 1 Are Necessary in the Worst Case (Complete Graph) Intuition (Eberhardt et al., 2006): Brady Neal Single-Node Interventions 10 / 28

  67. Single Variable Interventions: n – 1 Are Necessary in the Worst Case (Complete Graph) Intuition (Eberhardt et al., 2006): 1. Choose arbitrary intervention ordering Brady Neal Single-Node Interventions 10 / 28

  68. Single Variable Interventions: n – 1 Are Necessary in the Worst Case (Complete Graph) Intuition (Eberhardt et al., 2006): 1. Choose arbitrary intervention ordering 2. n – 2 interventions leave the X n–1 – X n edge undirected Brady Neal Single-Node Interventions 10 / 28

  69. Single Variable Interventions: n – 1 Are Necessary in the Worst Case (Complete Graph) Intuition (Eberhardt et al., 2006): 1. Choose arbitrary intervention ordering 2. n – 2 interventions leave the X n–1 – X n edge undirected 3. Intervention n – 1 on X n–1 or X n necessary to direct this final edge Brady Neal Single-Node Interventions 10 / 28

  70. Single Variable Interventions: n – 1 Are Necessary in the Worst Case (Complete Graph) Intuition (Eberhardt et al., 2006): 1. Choose arbitrary intervention ordering 2. n – 2 interventions leave the X n–1 – X n edge undirected 3. Intervention n – 1 on X n–1 or X n necessary to direct this final edge Adaptivity doesn’t help in the worst case (Eberhardt et al., 2006). Brady Neal Single-Node Interventions 10 / 28

  71. Questions (where the goal is to identify the graph): 1. Show that n interventions are necessary in the worse case in the three- variable setting when you use the observational data (null intervention) as one of the interventions. 2. What is the minimum number of interventions necessary in the worst case when n = 3? 3. What is the minimum number of interventions necessary in the worst case when n = 2?

  72. Structural Interventions Single-Node Interventions Multi-Node Interventions Parametric Interventions Interventional Markov Equivalence Miscellaneous Other Settings Brady Neal Multi-Node Interventions 12 / 28

  73. Multiple-Node Interventions Brady Neal Multi-Node Interventions 13 / 28

  74. Multiple-Node Interventions Single-node interventions (Eberhardt et al., 2006): • n – 1 are sufficient • n – 1 are necessary in the worst case Brady Neal Multi-Node Interventions 13 / 28

  75. Multiple-Node Interventions Single-node interventions (Eberhardt et al., 2006): • n – 1 are sufficient • n – 1 are necessary in the worst case Multi-node interventions with no restrictions on the number of nodes per intervention: Brady Neal Multi-Node Interventions 13 / 28

  76. Multiple-Node Interventions Single-node interventions (Eberhardt et al., 2006): • n – 1 are sufficient • n – 1 are necessary in the worst case Multi-node interventions with no restrictions on the number of nodes per intervention: • interventions are sufficient (Eberhardt et al., 2005) <latexit sha1_base64="NcUwi+bH0tvWfJTwyI+DiPrwVy0=">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</latexit> b log 2 ( n ) c + 1 • Brady Neal Multi-Node Interventions 13 / 28

  77. Multiple-Node Interventions Single-node interventions (Eberhardt et al., 2006): • n – 1 are sufficient • n – 1 are necessary in the worst case Multi-node interventions with no restrictions on the number of nodes per intervention: • interventions are sufficient (Eberhardt et al., 2005) <latexit sha1_base64="NcUwi+bH0tvWfJTwyI+DiPrwVy0=">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</latexit> b log 2 ( n ) c + 1 • interventions are necessary in the worst case (Eberhardt et <latexit sha1_base64="NcUwi+bH0tvWfJTwyI+DiPrwVy0=">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</latexit> b log 2 ( n ) c + 1 al., 2005) Brady Neal Multi-Node Interventions 13 / 28

  78. Number of Interventions in the Worse Case Eberhardt et al. (2005) found that multi-node interventions <latexit sha1_base64="NcUwi+bH0tvWfJTwyI+DiPrwVy0=">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</latexit> b log 2 ( n ) c + 1 are necessary for worst case graphs (complete graph), but what about for non-complete graphs? Brady Neal Multi-Node Interventions 14 / 28

  79. Number of Interventions in the Worse Case Eberhardt et al. (2005) found that multi-node interventions <latexit sha1_base64="NcUwi+bH0tvWfJTwyI+DiPrwVy0=">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</latexit> b log 2 ( n ) c + 1 are necessary for worst case graphs (complete graph), but what about for non-complete graphs? Start with Markov equivalence class, then how many interventions are necessary, given that we can intervene on unlimited nodes per intervention? Brady Neal Multi-Node Interventions 14 / 28

  80. <latexit sha1_base64="4IaKNRi9BGPJI184ofiP0KVYTxU=">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</latexit> Number of Interventions in the Worse Case Eberhardt et al. (2005) found that multi-node interventions <latexit sha1_base64="NcUwi+bH0tvWfJTwyI+DiPrwVy0=">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</latexit> b log 2 ( n ) c + 1 are necessary for worst case graphs (complete graph), but what about for non-complete graphs? Start with Markov equivalence class, then how many interventions are necessary, given that we can intervene on unlimited nodes per intervention? Theorem: multi-node interventions are sufficient and necessary in d log 2 ( c ) e the worst case, where c is the size of the largest clique (conjectured by Eberhardt (2008) and proven by Hauser & Bühlmann (2014)). Brady Neal Multi-Node Interventions 14 / 28

  81. <latexit sha1_base64="7/vhHkBHeEqvfvrHRitA+4nNRXo=">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</latexit> Question: A In this graph, how many multi-node interventions are B C sufficient and necessary given D H that you know the essential graph? E F G

  82. Structural Interventions Single-Node Interventions Multi-Node Interventions Parametric Interventions Interventional Markov Equivalence Miscellaneous Other Settings Brady Neal Parametric Interventions 16 / 28

  83. Structural vs. Parametric Interventions Brady Neal Parametric Interventions 17 / 28

  84. <latexit sha1_base64="+3nlmifPMZTPoeix/DTUTw39aM=">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</latexit> Structural vs. Parametric Interventions Structural: A B C <latexit sha1_base64="snHKN1zLzjP27tiG+nJz1PVujA=">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</latexit> Y := f θ ( A, B, C, N Y ) N Y Y Brady Neal Parametric Interventions 17 / 28

  85. <latexit sha1_base64="NQvfRk5xl/MDYGANpuhe1z0emQY=">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</latexit> <latexit sha1_base64="SE6kTlm164tfVZG5VwJsJcMjzs=">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</latexit> Structural vs. Parametric Interventions Structural: N 0 Y Y := N 0 Y Y Brady Neal Parametric Interventions 17 / 28

  86. <latexit sha1_base64="NQvfRk5xl/MDYGANpuhe1z0emQY=">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</latexit> <latexit 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<latexit 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Structural vs. Parametric Interventions Structural: N 0 Y Y := N 0 Y Y Parametric: A B C N Y Y <latexit 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Y := f θ ( A, B, C, N Y ) Brady Neal Parametric Interventions 17 / 28

  87. <latexit 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<latexit 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<latexit 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Structural vs. Parametric Interventions Structural: N 0 Y Y := N 0 Y Y Parametric: A B C N Y θ 0 Y <latexit 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Y := f θ 0 ( A, B, C, N Y ) Brady Neal Parametric Interventions 17 / 28

  88. <latexit 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<latexit 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<latexit 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Structural vs. Parametric Interventions Structural: N 0 Y Y := N 0 Y Y Parametric: A B C N Y θ 0 Y <latexit 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Y := f θ 0 ( A, B, C, N Y ) Other names: hard vs. soft, perfect vs. imperfect, etc. Brady Neal Parametric Interventions 17 / 28

  89. <latexit 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<latexit 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Structural vs. Parametric Interventions Parametric Structural: N 0 Y Y := N 0 Y Y Parametric: A B C N Y θ 0 Structural Y <latexit 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Y := f θ 0 ( A, B, C, N Y ) Other names: hard vs. soft, perfect vs. imperfect, etc. Brady Neal Parametric Interventions 17 / 28

  90. <latexit 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Structural vs. Parametric Interventions Parametric Structural: N 0 Y Y := N 0 Y Y Parametric: A B C N Y θ 0 Structural Y <latexit 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Y := f θ 0 ( A, B, C, N Y ) Other names: hard vs. soft, perfect vs. imperfect, etc. Brady Neal Parametric Interventions 17 / 28

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Structural vs. Parametric Interventions Parametric Structural: N 0 Y Y := N 0 Y What people often mean by Y “soft/parametric” and what we’ll consider now Parametric: A B C N Y θ 0 Structural Y <latexit 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Y := f θ 0 ( A, B, C, N Y ) Other names: hard vs. soft, perfect vs. imperfect, etc. Brady Neal Parametric Interventions 17 / 28

  92. Number of Parametric Single-Node Interventions Brady Neal Parametric Interventions 18 / 28

  93. Number of Parametric Single-Node Interventions Number of interventions for identification (Eberhardt & Scheines, 2007): Brady Neal Parametric Interventions 18 / 28

  94. Number of Parametric Single-Node Interventions Number of interventions for identification (Eberhardt & Scheines, 2007): • n – 1 interventions are sufficient Brady Neal Parametric Interventions 18 / 28

  95. Number of Parametric Single-Node Interventions Number of interventions for identification (Eberhardt & Scheines, 2007): • n – 1 interventions are sufficient • n – 1 interventions are necessary in the worst case Brady Neal Parametric Interventions 18 / 28

  96. Number of Parametric Single-Node Interventions Number of interventions for identification (Eberhardt & Scheines, 2007): • n – 1 interventions are sufficient • n – 1 interventions are necessary in the worst case (same as with structural interventions) Brady Neal Parametric Interventions 18 / 28

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