Indirect Causes in Dynamic Bayesian Networks Revisited 24 th - PowerPoint PPT Presentation

INSTITUTE OF INFORMATION SYSTEMS Indirect Causes in Dynamic Bayesian Networks Revisited 24 th International Joint Conference on Artificial Intelligence Alexander Motzek ❻ Ralf Möller ❻ ❻ Universität zu Lübeck Institute of Information Systems Ratzeburger Allee 160, 23562 Lübeck, Germany {motzek,moeller}@ifis.uni-luebeck.de July, 27 th 2015 INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15

INSTITUTE OF INFORMATION SYSTEMS Introduction ▲ Dynamic Bayesian Networks . ▲ Indirect Causes. ▲ DAG constraints limit causal expressiveness . ▲ Solution in DBN semantics on cyclic graph . INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15

INSTITUTE OF INFORMATION SYSTEMS Running example ▲ Regulatory compliance of employees. ▲ A ‘‘creduluous’’ employee might manipulate documents . ▲ A credulous employee might (undeliberately) influence other employees . ▲ Might become credulous too, etc. ▲ Influences occur through exchanged messages . ▲ Track probabilistic credulousness-state over time . ▲ Employees: C laire, D on and E earl . INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15

INSTITUTE OF INFORMATION SYSTEMS Problem as a DBN M 1 DC C 0 C 1 C 2 ▲ Say, only C laire influences D on, influences E arl. M 1 M 2 CD CD ▲ i.e. C influences E indirectly. D 0 D 1 D 2 ▲ Typical DBN. ✓ E 0 E 1 E 2 ▲ Problem correctly represented. ✓ M 1 M 2 DE DE INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15

INSTITUTE OF INFORMATION SYSTEMS Problem as a DBN M 1 DC C 0 C 1 C 2 ▲ Let’s add some more influences . ▲ Claire can also influence Earl directly . M 1 M 2 CD CD D 0 D 1 D 2 ▲ Typical DBN. ✓ E 0 E 1 E 2 ▲ Problem correctly represented. ✓ M 1 M 1 M 2 M 2 CE DE CE DE INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15

INSTITUTE OF INFORMATION SYSTEMS Problem as a DBN M 1 M 1 M 1 M 2 M 2 DC DC EC DC EC C 0 C 1 C 2 ▲ Say, everybody can influence everybody . M 1 M 2 CD CD ▲ ‘‘A BN is a DAG ’’. D 0 D 1 D 2 ▲ Not a DBN. ✗ M 1 M 2 ED ED E 0 E 1 E 2 ▲ Problem correctly represented. ✓ ? M 1 M 1 M 2 M 2 CE DE CE DE INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15

INSTITUTE OF INFORMATION SYSTEMS Problem as a DBN M 1 M 1 M 1 M 2 M 2 DC DC EC DC EC C 0 C 1 C 2 ▲ Resolve cycles over time. ▲ ‘‘Diagonal’’ inter-state dependencies. M 1 M 2 CD CD D 0 D 1 D 2 ▲ Common DBN . ✓ M 1 M 2 ED ED ▲ Problem correctly represented. ✗ ? E 0 E 1 E 2 M 1 M 1 M 2 M 2 CE DE CE DE INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15

INSTITUTE OF INFORMATION SYSTEMS DBN Restrictions ▲ ‘‘Diagonal’’ encodes ‘‘ incubation time’’: t: Receive Message. t ✔ 1 : Read and become influenced. C 0 C 1 C 2 C 3 C 4 C 5 a) Enforces infinitesimal resolution of time (e.g., seconds) D 0 D 1 D 2 D 3 D 4 D 5 ✗ High computation cost. E 0 E 1 E 2 E 3 E 4 E 5 Observations not available this fine (e.g., only daily)? C 0 C 1 C 2 Computation too costly? Transition only known hourly? b) Indirect influences not considerable . D 0 D 1 D 2 ✗ Does not explain the world. E 0 E 1 E 2 INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15

INSTITUTE OF INFORMATION SYSTEMS Classic DBNs spread indirect effects over time C 0 C 1 C 2 C 0 C 1 C 2 D 0 D 1 D 2 D 0 D 1 D 2 E 0 E 1 E 2 E 0 E 1 E 2 I.e., observations that require anticipations of indirect effects are not supported . INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15

INSTITUTE OF INFORMATION SYSTEMS Classic DBNs spread indirect effects over time C 0 C 1 C 2 C 0 C 1 C 2 D 0 D 1 D 2 D 0 D 1 D 2 E 0 E 1 E 2 E 0 E 1 E 2 I.e., observations that require anticipations of indirect effects are not supported . INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15

INSTITUTE OF INFORMATION SYSTEMS Intuitive Design M 1 M 1 M 2 M 2 DC EC DC EC C 0 C 1 C 2 M 1 M 2 CD CD D 0 D 1 D 2 M 1 M 2 ED ED E 0 E 1 E 2 M 1 M 1 M 2 M 2 CE DE CE DE INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15

INSTITUTE OF INFORMATION SYSTEMS Activator Random Variables ▲ Random variables M t XY representing exchanged messages are special ▲ We see M t XY as Activator Random Variables ➛ x, x ➐ ❃ dom ❼ X ➁ , ➛ y ❃ dom ❼ Y ➁ , ➛ Ñ z ❃ dom ❼Ñ Z ➁ ✂ P ❼ y ❙ x, ✥ a XY , Ñ z ➁ � P ❼ y ❙ x ➐ , ✥ a XY , Ñ z ➁ � P ❼ y ❙ ❻ , ✥ a XY , Ñ z ➁ ❻ wildcard, Ñ z further dependencies z ❃ dom ❼Ñ ➜ x, x ❻ ❃ dom ❼ X ➁ , ➜ y ❃ dom ❼ Y ➁ , ➜ Ñ Z ➁ P ❼ y ❙ x, a XY , Ñ z ➁ ① P ❼ y ❙ x ❻ , a XY , Ñ z ➁ INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15

▲ INSTITUTE OF INFORMATION SYSTEMS Activator Dynamic Bayesian Networks M 1 M 1 M 2 M 2 DC EC DC EC ▲ Is an Activator Dynamic Bayesian Network C 0 C 1 C 2 M 1 M 2 ▲ We show: Semantically a (D)BN , despite CD CD being based on a cylic graph ! D 0 D 1 D 2 ▲ Straight forward semantic as joint probability M 1 M 2 ED ED as usual. E 0 E 1 E 2 M 1 M 1 M 2 M 2 CE DE CE DE INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15

INSTITUTE OF INFORMATION SYSTEMS Activator Dynamic Bayesian Networks M 1 M 1 M 2 M 2 DC EC DC EC ▲ Is an Activator Dynamic Bayesian Network C 0 C 1 C 2 M 1 M 2 ▲ We show: Semantically a (D)BN , despite CD CD being based on a cylic graph ! D 0 D 1 D 2 ▲ Straight forward semantic as joint probability M 1 M 2 ED ED as usual. E 0 E 1 E 2 ▲ Under some restrictions... M 1 M 1 M 2 M 2 CE DE CE DE INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15

INSTITUTE OF INFORMATION SYSTEMS ADBN Restrictions Formally Theorem (Bayesian Network Soundness) For every set of instantiations Ñ ❆ 1 ✂ t ❻ an ADBN corresponds to a Bayesian network (BN), if for all t, Ñ ❆ t ❻ satisfies a new acyclicity constraint: X t ✂ A ❼ x, z ➁ t , A ❼ z, y ➁ t � A ❼ x, y ➁ t ➣ ➛ x, y, z ❃ Ñ ➝ A t ij � ✥ a t ➝ false if A ❼ i, j ➁ t � ij ➛ . ✥➜ q ✂ A ❼ q, q ➁ t , ➝ ➝ true otherwise ↕ ADBN’s semantics are well-defined as usual in a DBN, X 0 ✂ t ➋ , Ñ ❆ 1 ✂ t ➋ ➁ � P ❼Ñ X 0 ✂ t ✏ 1 ➋ , Ñ ❆ 1 ✂ t ✏ 1 ➋ ➁ � ▼ X t ➋ ❷ X t A t ➋ ❆ t ➋ ➁ . P ❼Ñ i ❙Ñ i , Ñ ➁ � P ❼ Ñ P ❼ X t i , X t ✏ 1 i i INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15

INSTITUTE OF INFORMATION SYSTEMS Restrictions Comparison ‘‘Diagonal’’ DBN Cyclic ADBN ▲ No ‘‘interlocking’’ M t ▲ No cyclic M t XY obs. allowed. XY observations allowed. ▲ must form bipartite graph. ▲ Activator set must form DAG . #DAG >> #Bipartite ✥ m 1 ✥ m 1 M 2 M 2 ✥ m 1 M 1 ✥ m 1 M 2 M 2 DC EC DC EC DC DC EC DC EC C 0 C 1 C 2 C 0 C 1 C 2 m 1 M 2 m 1 M 2 CD CD CD CD D 0 D 1 D 2 D 0 D 1 D 2 ✥ m 1 M 2 ✥ m 1 M 2 ED ED ED ED E 0 E 1 E 2 E 0 E 1 E 2 M 1 m 1 M 2 M 2 M 1 m 1 M 2 M 2 CE DE CE DE CE DE CE DE INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15

INSTITUTE OF INFORMATION SYSTEMS ADBN Contributions ▲ Bayesian networks can syntactically be based on cyclic graphs. ▲ Cyclic graphs are causally required for some problems. ▲ Acyclic graphs run into causal problems. ADBNs provide ✓ Free choice of time granularity. ✓ Anticipation of indirect influences . ✓ Well-defined DBN semantics. ✓ Filtering, Smoothing same as usual. ✓ BN as world-representing first-class declaration. INDIRECT CAUSES IN DYNAMIC BAYESIAN NETWORKS REVISITED, IJCAI’15