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Learning LWF Chain Graphs: A Markov Blanket Discovery Approach

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LWF Chain graphs were introduced by Lauritzen, Wermuth, and Frydenberg as a generalization of graphical models based on undirected graphs and DAGs. From the causality point of view, in an LWF CG: Directed edges represent direct causal effects. Undirected edges represent causal effects due to interference, which occurs when an individual’s outcome is influenced by their social interaction with other population members, e.g., in situations that involve contagious agents, educational programs, or social networks. The construction of chain graph models is a challenging task that would be greatly facilitated by automation.

Markov blanket discovery has an important role in structure learning of Bayesian network. It is surprising, however, how little attention it has attracted in the context of learning LWF chain graphs. In this work, we provide a graphical characterization of Markov blankets in chain graphs. The characterization is different from the well-known one for Bayesian networks and generalizes it. We provide a novel scalable and sound algorithm for Markov blanket discovery in LWF chain graphs. We also provide a sound and scalable constraint-based framework for learning the structure of LWF CGs from faithful causally sufficient data. With the use of our algorithm, the problem of structure learning is reduced to finding an efficient algorithm for Markov blanket discovery in LWF chain graphs. This greatly simplifies the structure-learning task and makes a wide range of inference/learning problems computationally tractable because our approach exploits locality.

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Learning LWF Chain Graphs: A Markov Blanket Discovery Approach

  1. 1. Learning LWF Chain Graphs: A Markov Blanket Discovery Approach Mohammad Ali Javidian @ali javidian Marco Valtorta @MarcoGV2 Pooyan Jamshidi @PooyanJamshidi Department of Computer Science and Engineering University of South Carolina UAI 2020 1 / 9
  2. 2. LWF Chain Graphs (CGs) Chain graphs: admit both directed and undirected edges, 2 / 9
  3. 3. LWF Chain Graphs (CGs) Chain graphs: admit both directed and undirected edges, there are no partially directed cycles. 2 / 9
  4. 4. LWF Chain Graphs (CGs) Chain graphs: admit both directed and undirected edges, there are no partially directed cycles. A Partially directed cycle: is a sequence of n distinct vertices v1, v2, . . . , vn(n β‰₯ 3), and vn+1 ≑ v1, s.t. for all i (1 ≀ i ≀ n) either vi βˆ’ vi+1 or vi β†’ vi+1, and there exists a j (1 ≀ j ≀ n) such that vj ← vj+1. 2 / 9
  5. 5. LWF Chain Graphs (CGs) Chain graphs: admit both directed and undirected edges, there are no partially directed cycles. A Partially directed cycle: is a sequence of n distinct vertices v1, v2, . . . , vn(n β‰₯ 3), and vn+1 ≑ v1, s.t. for all i (1 ≀ i ≀ n) either vi βˆ’ vi+1 or vi β†’ vi+1, and there exists a j (1 ≀ j ≀ n) such that vj ← vj+1. Example: 2 / 9
  6. 6. LWF Chain Graphs (CGs) Chain graphs: admit both directed and undirected edges, there are no partially directed cycles. A Partially directed cycle: is a sequence of n distinct vertices v1, v2, . . . , vn(n β‰₯ 3), and vn+1 ≑ v1, s.t. for all i (1 ≀ i ≀ n) either vi βˆ’ vi+1 or vi β†’ vi+1, and there exists a j (1 ≀ j ≀ n) such that vj ← vj+1. Example: Chain graphs under different interpretations: LWF, AMP, MVR, ... Here, we focus on chain graphs (CGs) under the Lauritzen-Wermuth-Frydenberg (LWF) interpretation. 2 / 9
  7. 7. Markov Blankets Enable Locality in Causal Structure Recovery H B J ADE FC I OM N K L GT 𝑴𝒃(𝑇) H B J ADE FC I OM N K L GT 3 / 9
  8. 8. Markov Blankets Enable Locality in Causal Structure Recovery H B J ADE FC I OM N K L GT 𝑴𝒃(𝑇) H B J ADE FC I OM N K L GT 3 / 9
  9. 9. Markov Blankets Enable Locality in Causal Structure Recovery H B J ADE FC I OM N K L GT 𝑴𝒃(𝑇) H B J ADE FC I OM N K L GT 𝒄𝒉(𝑇) 3 / 9
  10. 10. Markov Blankets Enable Locality in Causal Structure Recovery H B J ADE FC I OM N K L GT 𝑴𝒃(𝑇) H B J ADE FC I OM N K L GT 3 / 9
  11. 11. Markov Blankets Enable Locality in Causal Structure Recovery H B J ADE FC I OM N K L GT 𝑴𝒃(𝑇) H B J ADE FC I OM N K L GT Markov blankets can be used as a powerful tool in: classification, local causal discovery 3 / 9
  12. 12. Markov Blankets: a Missing Concept in the Context of Chain Graph Models H B J ADE FC I OM N K L GT H B J ADE FC I OM N K L GT𝑴𝒃(𝑇) 4 / 9
  13. 13. Markov Blankets: a Missing Concept in the Context of Chain Graph Models H B J ADE FC I OM N K L GT H B J ADE FC I OM N K L GT𝑴𝒃(𝑇) 𝒑𝒂(𝑇) 4 / 9
  14. 14. Markov Blankets: a Missing Concept in the Context of Chain Graph Models H B J ADE FC I OM N K L GT H B J ADE FC I OM N K L GT𝑴𝒃(𝑇) 𝒄𝒉(𝑇) 4 / 9
  15. 15. Markov Blankets: a Missing Concept in the Context of Chain Graph Models H B J ADE FC I OM N K L GT H B J ADE FC I OM N K L GT𝑴𝒃(𝑇) 𝒏𝒆(𝑇) 4 / 9
  16. 16. Markov Blankets: a Missing Concept in the Context of Chain Graph Models H B J ADE FC I OM N K L GT H B J ADE FC I OM N K L GT𝑴𝒃(𝑇) 𝒄𝒔𝒑(𝑇) 4 / 9
  17. 17. Markov Blankets: a Missing Concept in the Context of Chain Graph Models H B J ADE FC I OM N K L GT H B J ADE FC I OM N K L GT𝑴𝒃(𝑇) 4 / 9
  18. 18. Markov Blankets in LWF Chain Graphs: Main Results Theorem (Characterization of Markov Blankets in LWF CGs) Let G = (V, E, P) be an LWF chain graph model. Then, G entails conditional independency TβŠ₯βŠ₯pV {T, Mb(T)}|Mb(T), i.e., the Markov blanket of the target variable T in an LWF CG probabilistically shields T from the rest of the variables. 5 / 9
  19. 19. Markov Blankets in LWF Chain Graphs: Main Results Theorem (Characterization of Markov Blankets in LWF CGs) Let G = (V, E, P) be an LWF chain graph model. Then, G entails conditional independency TβŠ₯βŠ₯pV {T, Mb(T)}|Mb(T), i.e., the Markov blanket of the target variable T in an LWF CG probabilistically shields T from the rest of the variables. Theorem (Standard algorithms for Markov blanket recovery in LWF CGs) Given the Markov assumption, the faithfulness assumption, a graphical model represented by an LWF CG, and i.i.d. sampling, in the large sample limit, the Markov blanket recovery algorithms Grow-Shrink, Incremental Association Markov blanket recovery, and its variants identify all Markov blankets for each variable. 5 / 9
  20. 20. Markov Blankets in LWF Chain Graphs: Main Results Theorem (Characterization of Markov Blankets in LWF CGs) Let G = (V, E, P) be an LWF chain graph model. Then, G entails conditional independency TβŠ₯βŠ₯pV {T, Mb(T)}|Mb(T), i.e., the Markov blanket of the target variable T in an LWF CG probabilistically shields T from the rest of the variables. Theorem (Standard algorithms for Markov blanket recovery in LWF CGs) Given the Markov assumption, the faithfulness assumption, a graphical model represented by an LWF CG, and i.i.d. sampling, in the large sample limit, the Markov blanket recovery algorithms Grow-Shrink, Incremental Association Markov blanket recovery, and its variants identify all Markov blankets for each variable. The characterization of Markov blankets in chain graphs enables us to develop new algorithms that are specifically designed for learning Markov blankets in chain graphs. 5 / 9
  21. 21. MBC-CSP Algorithm: Markov Blanket Discovery in LWF CGs H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-1Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-2Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 2Β  𝐜𝐬𝐩(𝑇) 6 / 9
  22. 22. MBC-CSP Algorithm: Markov Blanket Discovery in LWF CGs H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-1Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-2Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 2Β  𝐜𝐬𝐩(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-1Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-2Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 2Β  𝐜𝐬𝐩(𝑇) 6 / 9
  23. 23. MBC-CSP Algorithm: Markov Blanket Discovery in LWF CGs H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-1Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-2Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 2Β  𝐜𝐬𝐩(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-1Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-2Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 2Β  𝐜𝐬𝐩(𝑇) Shrink Phase: πŒπ›(𝑇) H B J A E FC I M K L GT D H B J A E FC I M K L GT D 6 / 9
  24. 24. MBC-CSP Algorithm: Markov Blanket Discovery in LWF CGs H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-1Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-2Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 2Β  𝐜𝐬𝐩(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-1Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-2Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 2Β  𝐜𝐬𝐩(𝑇) Shrink Phase: πŒπ›(𝑇) H B J A E FC I M K L GT D H B J A E FC I M K L GT D Such Markov blanket discovery algorithms help us to design new scalable algorithms for learning chain graphs based on local structure discovery. 6 / 9
  25. 25. MbLWF Algorithm: Learning LWF CGs via Markov Blanket Discovery ObservationalΒ Data Markov Blanket Discovery Algorithm A DC E B πŒπ› 𝐴 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐴, 𝐷 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐴, 𝐸 = {𝐢, 𝐡} πŒπ› 𝐡 = 𝐸, 𝐴 , π’πžπ©π¬πžπ­ 𝐡, 𝐢 = 𝐸, 𝐴 , π’πžπ©π¬πžπ­ 𝐡, 𝐷 = {𝐸, 𝐴} πŒπ› 𝐢 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐢, 𝐸 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐢, 𝐡 = {𝐴, 𝐷} πŒπ› 𝐷 = 𝐢, 𝐸 , π’πžπ©π¬πžπ­ 𝐷, 𝐴 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐷, 𝐡 = {𝐸, 𝐴} πŒπ› 𝐸 = 𝐷, 𝐡 , π’πžπ©π¬πžπ­ 𝐸, 𝐢 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐸, 𝐴 = {𝐢, 𝐡} 7 / 9
  26. 26. MbLWF Algorithm: Learning LWF CGs via Markov Blanket Discovery ObservationalΒ Data Markov Blanket Discovery Algorithm A DC E B πŒπ› 𝐴 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐴, 𝐷 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐴, 𝐸 = {𝐢, 𝐡} πŒπ› 𝐡 = 𝐸, 𝐴 , π’πžπ©π¬πžπ­ 𝐡, 𝐢 = 𝐸, 𝐴 , π’πžπ©π¬πžπ­ 𝐡, 𝐷 = {𝐸, 𝐴} πŒπ› 𝐢 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐢, 𝐸 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐢, 𝐡 = {𝐴, 𝐷} πŒπ› 𝐷 = 𝐢, 𝐸 , π’πžπ©π¬πžπ­ 𝐷, 𝐴 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐷, 𝐡 = {𝐸, 𝐴} πŒπ› 𝐸 = 𝐷, 𝐡 , π’πžπ©π¬πžπ­ 𝐸, 𝐢 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐸, 𝐴 = {𝐢, 𝐡} A DC E B Super Skeleton Recovery 7 / 9
  27. 27. MbLWF Algorithm: Learning LWF CGs via Markov Blanket Discovery ObservationalΒ Data Markov Blanket Discovery Algorithm A DC E B πŒπ› 𝐴 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐴, 𝐷 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐴, 𝐸 = {𝐢, 𝐡} πŒπ› 𝐡 = 𝐸, 𝐴 , π’πžπ©π¬πžπ­ 𝐡, 𝐢 = 𝐸, 𝐴 , π’πžπ©π¬πžπ­ 𝐡, 𝐷 = {𝐸, 𝐴} πŒπ› 𝐢 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐢, 𝐸 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐢, 𝐡 = {𝐴, 𝐷} πŒπ› 𝐷 = 𝐢, 𝐸 , π’πžπ©π¬πžπ­ 𝐷, 𝐴 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐷, 𝐡 = {𝐸, 𝐴} πŒπ› 𝐸 = 𝐷, 𝐡 , π’πžπ©π¬πžπ­ 𝐸, 𝐢 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐸, 𝐴 = {𝐢, 𝐡} A DC E B Super Skeleton Recovery π’πžπ©π¬πžπ­ 𝐴, 𝐷 = 𝐢, 𝐡 Skeleton A DC E B 7 / 9
  28. 28. MbLWF Algorithm: Learning LWF CGs via Markov Blanket Discovery ObservationalΒ Data Markov Blanket Discovery Algorithm A DC E B πŒπ› 𝐴 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐴, 𝐷 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐴, 𝐸 = {𝐢, 𝐡} πŒπ› 𝐡 = 𝐸, 𝐴 , π’πžπ©π¬πžπ­ 𝐡, 𝐢 = 𝐸, 𝐴 , π’πžπ©π¬πžπ­ 𝐡, 𝐷 = {𝐸, 𝐴} πŒπ› 𝐢 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐢, 𝐸 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐢, 𝐡 = {𝐴, 𝐷} πŒπ› 𝐷 = 𝐢, 𝐸 , π’πžπ©π¬πžπ­ 𝐷, 𝐴 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐷, 𝐡 = {𝐸, 𝐴} πŒπ› 𝐸 = 𝐷, 𝐡 , π’πžπ©π¬πžπ­ 𝐸, 𝐢 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐸, 𝐴 = {𝐢, 𝐡} A DC E B Super Skeleton Recovery π’πžπ©π¬πžπ­ 𝐴, 𝐷 = 𝐢, 𝐡 Skeleton A DC E B A DC E BComplex Recovery 7 / 9
  29. 29. Experimental Evaluation: Markov Blankets Make a Broad Range of Inference/Learning Problems Computationally Tractable and More Precise. ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●0.90 0.95 1.00 GSLWF fastIAMBLWF fdrIAMBLWF interIAMBLWF IAMBLWF MBCCSPLWF LCD Precision sample size size = 200 size = 2000 alpha = 0.05 8 / 9
  30. 30. Experimental Evaluation: Markov Blankets Make a Broad Range of Inference/Learning Problems Computationally Tractable and More Precise. ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●0.90 0.95 1.00 GSLWF fastIAMBLWF fdrIAMBLWF interIAMBLWF IAMBLWF MBCCSPLWF LCD Precision sample size size = 200 size = 2000 alpha = 0.05 ● ● ● ● ● ● 0.5 0.6 0.7 0.8 0.9 1.0 GSLWF fastIAMBLWF fdrIAMBLWF interIAMBLWF IAMBLWF MBCCSPLWF LCD Recall alpha = 0.05 8 / 9
  31. 31. Markov Blankets: a Missing Concept in the Context of Chain Graph Models H B J ADE FC I OM N K L GT H B J ADE FC I OM N K L GT𝑴𝒃(𝑇) 4 / 9
  32. 32. Markov Blankets: a Missing Concept in the Context of Chain Graph Models H B J ADE FC I OM N K L GT H B J ADE FC I OM N K L GT𝑴𝒃(𝑇) 4 / 9 Markov Blankets in LWF Chain Graphs: Main Results Theorem Let G = (V, E, P) be an LWF chain graph model. Then, G entails conditional independency TβŠ₯βŠ₯pV {T, Mb(T)}|Mb(T). Theorem Given the Markov assumption, the faithfulness assumption, a graphical model represented by an LWF CG, and i.i.d. sampling, in the large sample limit, the Markov blanket recovery algorithms Grow-Shrink, Incremental Association Markov blanket recovery, and its variants identify all Markov blankets for each variable. 5 / 9
  33. 33. Markov Blankets: a Missing Concept in the Context of Chain Graph Models H B J ADE FC I OM N K L GT H B J ADE FC I OM N K L GT𝑴𝒃(𝑇) 4 / 9 Markov Blankets in LWF Chain Graphs: Main Results Theorem Let G = (V, E, P) be an LWF chain graph model. Then, G entails conditional independency TβŠ₯βŠ₯pV {T, Mb(T)}|Mb(T). Theorem Given the Markov assumption, the faithfulness assumption, a graphical model represented by an LWF CG, and i.i.d. sampling, in the large sample limit, the Markov blanket recovery algorithms Grow-Shrink, Incremental Association Markov blanket recovery, and its variants identify all Markov blankets for each variable. 5 / 9 MBC-CSP Algorithm: Markov Blanket Discovery in LWF CGs H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-1Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-2Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 2Β  𝐜𝐬𝐩(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-1Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-2Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 2Β  𝐜𝐬𝐩(𝑇) Shrink Phase: πŒπ›(𝑇) H B J A E FC I M K L GT D H B J A E FC I M K L GT D 6 / 9
  34. 34. Markov Blankets: a Missing Concept in the Context of Chain Graph Models H B J ADE FC I OM N K L GT H B J ADE FC I OM N K L GT𝑴𝒃(𝑇) 4 / 9 Markov Blankets in LWF Chain Graphs: Main Results Theorem Let G = (V, E, P) be an LWF chain graph model. Then, G entails conditional independency TβŠ₯βŠ₯pV {T, Mb(T)}|Mb(T). Theorem Given the Markov assumption, the faithfulness assumption, a graphical model represented by an LWF CG, and i.i.d. sampling, in the large sample limit, the Markov blanket recovery algorithms Grow-Shrink, Incremental Association Markov blanket recovery, and its variants identify all Markov blankets for each variable. 5 / 9 MBC-CSP Algorithm: Markov Blanket Discovery in LWF CGs H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-1Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-2Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 2Β  𝐜𝐬𝐩(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-1Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-2Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 2Β  𝐜𝐬𝐩(𝑇) Shrink Phase: πŒπ›(𝑇) H B J A E FC I M K L GT D H B J A E FC I M K L GT D 6 / 9 MbLWF Algorithm: Learning LWF CGs via Markov Blanket Discovery ObservationalΒ Data Markov Blanket Discovery Algorithm A DC E B πŒπ› 𝐴 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐴, 𝐷 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐴, 𝐸 = {𝐢, 𝐡} πŒπ› 𝐡 = 𝐸, 𝐴 , π’πžπ©π¬πžπ­ 𝐡, 𝐢 = 𝐸, 𝐴 , π’πžπ©π¬πžπ­ 𝐡, 𝐷 = {𝐸, 𝐴} πŒπ› 𝐢 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐢, 𝐸 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐢, 𝐡 = {𝐴, 𝐷} πŒπ› 𝐷 = 𝐢, 𝐸 , π’πžπ©π¬πžπ­ 𝐷, 𝐴 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐷, 𝐡 = {𝐸, 𝐴} πŒπ› 𝐸 = 𝐷, 𝐡 , π’πžπ©π¬πžπ­ 𝐸, 𝐢 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐸, 𝐴 = {𝐢, 𝐡} A DC E B Super Skeleton Recovery π’πžπ©π¬πžπ­ 𝐴, 𝐷 = 𝐢, 𝐡 Skeleton A DC E B A DC E BComplex Recovery 7 / 9
  35. 35. Markov Blankets: a Missing Concept in the Context of Chain Graph Models H B J ADE FC I OM N K L GT H B J ADE FC I OM N K L GT𝑴𝒃(𝑇) 4 / 9 Markov Blankets in LWF Chain Graphs: Main Results Theorem Let G = (V, E, P) be an LWF chain graph model. Then, G entails conditional independency TβŠ₯βŠ₯pV {T, Mb(T)}|Mb(T). Theorem Given the Markov assumption, the faithfulness assumption, a graphical model represented by an LWF CG, and i.i.d. sampling, in the large sample limit, the Markov blanket recovery algorithms Grow-Shrink, Incremental Association Markov blanket recovery, and its variants identify all Markov blankets for each variable. 5 / 9 MBC-CSP Algorithm: Markov Blanket Discovery in LWF CGs H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-1Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-2Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 2Β  𝐜𝐬𝐩(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-1Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 1-2Β  𝒂𝒅𝒋(𝑇) H B J A E FC I M K L GT D Grow Phase:Β  StepΒ 2Β  𝐜𝐬𝐩(𝑇) Shrink Phase: πŒπ›(𝑇) H B J A E FC I M K L GT D H B J A E FC I M K L GT D 6 / 9 MbLWF Algorithm: Learning LWF CGs via Markov Blanket Discovery ObservationalΒ Data Markov Blanket Discovery Algorithm A DC E B πŒπ› 𝐴 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐴, 𝐷 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐴, 𝐸 = {𝐢, 𝐡} πŒπ› 𝐡 = 𝐸, 𝐴 , π’πžπ©π¬πžπ­ 𝐡, 𝐢 = 𝐸, 𝐴 , π’πžπ©π¬πžπ­ 𝐡, 𝐷 = {𝐸, 𝐴} πŒπ› 𝐢 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐢, 𝐸 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐢, 𝐡 = {𝐴, 𝐷} πŒπ› 𝐷 = 𝐢, 𝐸 , π’πžπ©π¬πžπ­ 𝐷, 𝐴 = 𝐢, 𝐡 , π’πžπ©π¬πžπ­ 𝐷, 𝐡 = {𝐸, 𝐴} πŒπ› 𝐸 = 𝐷, 𝐡 , π’πžπ©π¬πžπ­ 𝐸, 𝐢 = 𝐴, 𝐷 , π’πžπ©π¬πžπ­ 𝐸, 𝐴 = {𝐢, 𝐡} A DC E B Super Skeleton Recovery π’πžπ©π¬πžπ­ 𝐴, 𝐷 = 𝐢, 𝐡 Skeleton A DC E B A DC E BComplex Recovery 7 / 9 All code, data, and supplementary materials are available at: https://majavid.github.io/structurelearning/blog/2020/uai/

LWF Chain graphs were introduced by Lauritzen, Wermuth, and Frydenberg as a generalization of graphical models based on undirected graphs and DAGs. From the causality point of view, in an LWF CG: Directed edges represent direct causal effects. Undirected edges represent causal effects due to interference, which occurs when an individual’s outcome is influenced by their social interaction with other population members, e.g., in situations that involve contagious agents, educational programs, or social networks. The construction of chain graph models is a challenging task that would be greatly facilitated by automation. Markov blanket discovery has an important role in structure learning of Bayesian network. It is surprising, however, how little attention it has attracted in the context of learning LWF chain graphs. In this work, we provide a graphical characterization of Markov blankets in chain graphs. The characterization is different from the well-known one for Bayesian networks and generalizes it. We provide a novel scalable and sound algorithm for Markov blanket discovery in LWF chain graphs. We also provide a sound and scalable constraint-based framework for learning the structure of LWF CGs from faithful causally sufficient data. With the use of our algorithm, the problem of structure learning is reduced to finding an efficient algorithm for Markov blanket discovery in LWF chain graphs. This greatly simplifies the structure-learning task and makes a wide range of inference/learning problems computationally tractable because our approach exploits locality.

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