Exploiting Innocuousness in Bayesian Networks 28 th Australasian - PowerPoint PPT Presentation

INSTITUTE OF INFORMATION SYSTEMS Exploiting Innocuousness in Bayesian Networks 28 th Australasian 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 December, 4 th 2015 EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’15

INSTITUTE OF INFORMATION SYSTEMS Introduction ▲ (Dynamic) Bayesian Networks . ▲ New form of independence - Innocuousness . ▲ New form of DBNs - Activator DBNs . ▲ Formalize and Exploit. EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’15

INSTITUTE OF INFORMATION SYSTEMS Bayesian Networks ▲ Syntactically defined by a graph B . ▲ Local semantics as specifications of local CPD ▲ Global semantic as the joint probability P ❼Ñ X ➁ � ▼ P ❼ X ❙ parents ❼ X ➁➁ X ❃ Ñ X EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’15

INSTITUTE OF INFORMATION SYSTEMS Bayesian Networks ▲ Graph encodes guaranteed independencies . ▲ Not dependencies! ▲ Actual dependencies encoded/ specified in CPDs . EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’15

INSTITUTE OF INFORMATION SYSTEMS Independencies - A Gedankenexperiment ▲ Multiple causes can cause one effect . ▲ Our hand is exposed to various risks in a blackbox . H ▲ Exposures can cause H arm. EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’15

INSTITUTE OF INFORMATION SYSTEMS Independencies - A Gedankenexperiment ▲ E.g., exposures to S and , B unsen burner , O 2 ▲ P ❼ H ❙ S, B, O 2 ➁ ▲ ✔ s and is present or not ✥ s present, a ✔ b urner is turned on, ✔ o 2 is present S B O 2 ▲ Exposures might be part of a much deeper probabilistic process. H EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’15

INSTITUTE OF INFORMATION SYSTEMS Causal Independence ▲ Classic. S and is completely irrelevant, i.e., independent . ▲ Investigated by Zhang and Poole, et al. (i) change graph. (ii) P ❼ H ❙ S, B, O 2 ➁ � P ❼ H ❙ B, O 2 ➁ specifiable in a local CPD by ➛ h, b, o 2 ✂ P ❼ h ❙ ✔ s, b, o 2 ➁ � P ❼ h ❙ ✥ s, b, o 2 ➁ S B O 2 H EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’15

INSTITUTE OF INFORMATION SYSTEMS Context-Specific Independence ▲ A B unsen burner only works / can only causes harm if ✔ o 2 is present . ▲ Investigated by Boutelier et al. ▲ P ❼ H ❙ S, B, ✥ o 2 ➁ � P ❼ H ❙ S, ✥ o 2 ➁ specifiable in local CPD by ➛ s, h, b, o 2 ✂ P ❼ h ❙ s, ✔ b, ✥ o 2 ➁ � P ❼ h ❙ s, ✥ b, ✥ o 2 ➁ ➜ s, h, b, o 2 ✂ P ❼ h ❙ s, ✔ b, ✔ o 2 ➁ ① P ❼ h ❙ s, ✥ b, ✔ o 2 ➁ ✥ o 2 S B H EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’15

INSTITUTE OF INFORMATION SYSTEMS Innocuousness ▲ Allegedly, B urner is only relevant if it is turned on ✔ b . ▲ A turned off ✥ b urner is completely irrelevant , could have been left out. ▲ Very commonly found in Noisy-OR Assumptions. ‘‘A false dependence does not cause any harm ’’ . ▲ How to formalize? The relevant context ✥ b is the ‘‘ir relevant ’’ random variable to be removed... ▲ P ❼ H ❙ S, ✥ b, O 2 ➁ � P ❼ H ❙ S, O 2 ➁ not specifiable/expressible ? S ✥ b O 2 ??? H EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’15

INSTITUTE OF INFORMATION SYSTEMS Why formalize innocuousness? Expressing Innocuousness is interesting: ▲ More expressive and causal specifications ▲ Removing a link is always good (computation time...) ▲ Can actually be formalized with and are beneficial for ADBNs . IJCAI’15 EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’15

INSTITUTE OF INFORMATION SYSTEMS ADBNs - 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 . EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’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 EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’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 EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’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 EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’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 EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’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 EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’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 . EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’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 . EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’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 EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’15

INSTITUTE OF INFORMATION SYSTEMS Activator Random Variables ▲ Random variables M t XY representing exchanged messages are special ▲ M t XY have activator nature , i.e., are 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 ➜ x, x ❻ ❃ dom ❼ X ➁ , ➜ y ❃ dom ❼ Y ➁ , ➜ Ñ z ❃ dom ❼Ñ Z ➁ P ❼ y ❙ x, a XY , Ñ z ➁ ① P ❼ y ❙ x ❻ , a XY , Ñ z ➁ ▲ Hint: O 2 is an activator for B unsen EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’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 EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’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 EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’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 EXPLOITING INNOCUOUSNESS IN BAYESIAN NETWORKS, AI’15