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

INSTITUTE OF INFORMATION SYSTEMS

Indirect Causes in Dynamic Bayesian Networks Revisited

24th International Joint Conference on Artificial Intelligence

Alexander Motzek❻ Ralf Möller❻

❻Universität zu Lübeck

Institute of Information Systems Ratzeburger Allee 160, 23562 Lübeck, Germany {motzek,moeller}@ifis.uni-luebeck.de

July, 27th 2015

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Introduction

▲ Dynamic Bayesian Networks. ▲ Indirect Causes. ▲ DAG constraints limit causal expressiveness. ▲ Solution in DBN semantics on cyclic graph.

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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: Claire, Don and Eearl.

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Problem as a DBN

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▲ Say, only Claire influences Don,

influences Earl.

▲ i.e. C influences E indirectly. ▲ Typical DBN. ✓ ▲ Problem correctly represented. ✓

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Problem as a DBN

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▲ Let’s add some more influences. ▲ Claire can also influence Earl directly. ▲ Typical DBN. ✓ ▲ Problem correctly represented. ✓

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Problem as a DBN

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▲ Say, everybody can influence everybody. ▲ ‘‘A BN is a DAG’’. ▲ Not a DBN.✗ ▲ Problem correctly represented. ✓?

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Problem as a DBN

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▲ Resolve cycles over time. ▲ ‘‘Diagonal’’ inter-state dependencies. ▲ Common DBN . ✓ ▲ Problem correctly represented.✗ ?

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DBN Restrictions

▲ ‘‘Diagonal’’ encodes ‘‘incubation time’’:

t: Receive Message. t ✔ 1: Read and become influenced. a) Enforces infinitesimal resolution of time (e.g., seconds) ✗ High computation cost.

Observations not available this fine (e.g., only daily)? Computation too costly? Transition only known hourly?

b) Indirect influences not considerable. ✗ Does not explain the world.

C0 D0 E0 C1 D1 E1 C2 D2 E2 C3 D3 E3 C4 D4 E4 C5 D5 E5 C0 D0 E0 C1 D1 E1 C2 D2 E2

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Classic DBNs spread indirect effects over time

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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

C0 D0 E0 C1 D1 E1 C2 D2 E2 C0 D0 E0 C1 D1 E1 C2 D2 E2

I.e., observations that require anticipations of indirect effects are not supported.

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Intuitive Design

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Activator Random Variables

▲ Random variables Mt XY representing exchanged messages are special ▲ We see Mt XY as Activator Random Variables

➛x, x➐ ❃ dom❼X➁, ➛y ❃ dom❼Y➁, ➛Ñ z ❃ dom❼Ñ Z➁ ✂ P❼y❙x, ✥aXY,Ñ z➁ P❼y❙x➐, ✥aXY,Ñ z➁ P❼y❙❻, ✥aXY,Ñ z➁

❻ wildcard, Ñ z further dependencies

➜x, x❻ ❃ dom❼X➁, ➜y ❃ dom❼Y➁, ➜Ñ z ❃ dom❼Ñ Z➁ P❼y❙x, aXY,Ñ z➁ ① P❼y❙x❻, aXY,Ñ z➁

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Activator Dynamic Bayesian Networks

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▲ Is an Activator Dynamic Bayesian Network ▲ We show: Semantically a (D)BN, despite

being based on a cylic graph!

▲ Straight forward semantic as joint probability

as usual.

▲

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Activator Dynamic Bayesian Networks

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▲ Is an Activator Dynamic Bayesian Network ▲ We show: Semantically a (D)BN, despite

being based on a cylic graph!

▲ Straight forward semantic as joint probability

as usual.

▲ Under some restrictions...

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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, y, z ❃ Ñ Xt ✂ A❼x, z➁t, A❼z, y➁t A❼x, y➁t ✥➜q ✂ A❼q, q➁t , A❼i, j➁t ➣ ➝ ➝ ➛ ➝ ➝ ↕ false if At

ij ✥at ij

true

• therwise

. ADBN’s semantics are well-defined as usual in a DBN, P❼Ñ X0✂t➋, Ñ ❆1✂t➋➁ P❼Ñ X0✂t✏1➋, Ñ ❆1✂t✏1➋➁ ▼

i

P❼Xt

i ❙Ñ

Xt➋❷Xt

i , Ñ

At➋

i , Xt✏1 i

➁ P❼ Ñ ❆t➋➁ .

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Restrictions Comparison

Cyclic ADBN

▲ No cyclic Mt XY observations allowed. ▲ Activator set must form DAG.

‘‘Diagonal’’ DBN

▲ No ‘‘interlocking’’ Mt XY obs. allowed. ▲ must form bipartite graph.

#DAG >> #Bipartite

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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