first-order probabilistic inference rodrigo de salvo braz

First-OrderProbabilistic Inference

Rodrigo de Salvo Braz

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A remarkFeel free to ask clarification questions!

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Slides online Just search for

“Rodrigo de Salvo Braz” and check at the presentations page.(www.cs.berkeley.edu/~braz)

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Outline A goal: First-order Probabilistic Inference

An ultimate goal Propositionalization

Lifted Inference Inversion Elimination Counting Elimination

DBLOG BLOG background DBLOG Retro-instantiation Improving BLOG inference

Future Directions and Conclusion

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How to solvecommonsense

reasoning

How to solveNatural

LanguageProcessing

How to solvePlanning

How to solveVision

DomainKnowledge

Inferenceand

Learning

LanguageKnowledge

Inferenceand

Learning

Domain &Planning

Knowledge

Inferenceand

Learning

Objects &Optics

Knowledge

Inferenceand

Learning

Abstracting Inference and Learning

Commonsense reasoning

Natural LanguageProcessing

Planning Vision ...

Inferenceand

Learning

AI Problems

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Inference and LearningWhat capabilities should it have? Inference

andLearningIt should represent and use:

•predicates (relations) and quantification over objects: 8 X tank(X) Ç plane(X) 8 X tank(X) ) color(X)=greencolor(o121) = red

•varying degrees of uncertainty: 8 X { tank(X) ) color(X)=green }

since it may hold often but not absolutely.•essential for language, vision, pattern recognition,

commonsense reasoning etc•modal knowledge, utilities, and more.

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Domain knowledge:

tank(X) Ç plane(X){ tank(X) ) color(X)=green}color(o121) = red...

Commonsense reasoningLanguage knowledge:

{ Sentence(S) Æ Verb(S,”attacks”) Æ Subject(S,X) Æ Object(S,Y) ) attacking(X,Y) }...

Natural Language Processing

Inferenceand

Learning

Sharing inference and learning module

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Domain knowledge:

tank(X) Ç plane(X){ tank(X) ) color(X)=green}color(o121) = red...

Commonsense reasoning WITH Natural Language ProcessingLanguage knowledge:

{ Sentence(S) Æ Verb(S,”attacks”) Æ Subject(S,X) Æ Object(S,Y) ) attacking(X,Y) }...

Inferenceand

Learning

Joint solution made simpler

Verb(s13,”attacks”), Subject(s13,o121)...

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Standard solutions fall short Logic

Has objects and properties; but statements are absolute;

Graphical models and machine learning Have varying degrees of uncertainty; but propositional; treating objects and

quantification is awkward.

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Standard solutions fall shortGraphical models and objects

tank(X) Ç plane(X){ tank(X) )color(X)=green }{ next(X,Y) Æ tank(X) ) tank(Y) }

color(o121)=red color(o135)=green

color(121)

tank(o121) plane(o121)

next(o121, o135)

tank(o135) plane(o135)

color(o135)

•Transformation (propositionalization) not part of graphical model framework;

•Graphical model have only random variables, not objects;

•Different data creates distinct, formally unrelated model.

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First-Order Probabilistic Inference Many languages have been proposed

Knowledge-based model construction (Breese, 1992) Probabilistic Logic Programming (Ng & Subrahmanian, 1992) Probabilistic Logic Programming (Ngo and Haddawy, 1995) Probabilistic Relational Models (Friedman et al., 1999) Relational Dependency Networks (Neville & Jensen, 2004) Bayesian Logic (BLOG) (Milch & Russell, 2004) Bayesian Logic Programs (Kersting & DeRaedt, 2001) DAPER (Heckerman, Meek & Koller, 2007) Markov Logic Networks (Richardson & Domingos, 2004)

“Smart” propositionalization is the main method.

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Propositionalization Bayesian Logic Programs

Prolog-like statements such as

rescue(Battalion) | in(Soldier,Battalion), wounded(Soldier).

wounded(Soldier) | in(Soldier,Battalion),

attacked(Battalion).

associated with CPTs and combination rules.

|instead of

:-

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Propositionalizationrescue(Battalion) | in(Soldier,Battalion),

wounded(Soldier). (plus CPT, cr)BN grounded from and-or tree:

rescue(bat13)

in(john,bat13)

wounded(john)

in(peter,bat13)

wounded(peter)

...

...

...

...

...

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PropositionalizationMulti-Entity Bayesian Networks (Laskey, 2004)

in(Soldier,Battalion) wounded(Soldier)

rescue(Battalion)

in(Soldier,Battalion) attacked(Battalion)

wounded(Soldier)

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in(Soldier,Battalion) wounded(Soldier)

rescue(Battalion)

in(john,bat13) wounded(john)

rescue(bat13)

in(peter,bat13) wounded(peter)

rescue(bat13)

...

first-order fragments

instantiated

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rescue(bat13)


...


rescue(bat13)


rescue(bat13)

...

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Lifted Inference

rescue(bat13)

rescue(Battalion) ( in(Soldier,Battalion) Æ wounded(Soldier){ wounded(Soldier) ( in(Soldier,Battalion) Æ attacked(Battalion) }

wounded(Soldier)

attacked(bat13)

in(Soldier, bat13)

Unbound, general variable

•Faster•More compact•More intuitive•Higher level – more

information/structure available for optimization

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Factor Networks (undirected)

epidemic_france

P(epi_france, epi_belgium, epi_uk, epi_germany)/ (epi_france, epi_belgium) * (epi_france, epi_uk) * (epi_france, epi_germany) * (epi_belgium, epi_germany)

epidemic_belgium epidemic_germany

epidemic_uk

factor,or potential

function

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Bayesian Nets as Factor Networks

epidemic

P(sick_john, sick_mary, sick_bob, epidemic)/ P(sick_john | epidemic) * P(sick_mary | epidemic) * P(sick_bob | epidemic) * P(epidemic)



P(epidemic)



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Inference: Marginalization

epidemic

P(sick_john) / epidemic sick_mary sick_bob P(sick_john | epidemic) * P(sick_mary | epidemic) * P(sick_bob | epidemic) * P(epidemic)



P(epidemic)



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epidemic



P(epidemic)



P(sick_john) / epidemic P(sick_john | epidemic) * P(epidemic)

* sick_mary P(sick_mary | epidemic)

* sick_bob P(sick_bob | epidemic)

Inference: Variable Elimination (VE)

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epidemic

sick_john sick_mary

(epidemic)

P(epidemic)



P(sick_john) / epidemic P(sick_john | epidemic) * P(epidemic)

* sick_mary P(sick_mary | epidemic) * (epidemic)


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epidemic

sick_john sick_mary

(epidemic)

P(epidemic)



P(sick_john) / epidemic P(sick_john | epidemic) * P(epidemic) * (epidemic)

* sick_mary P(sick_mary | epidemic)


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epidemic

sick_john

(epidemic)

P(epidemic)


P(sick_john) / epidemic P(sick_john | epidemic) * P(epidemic) * (epidemic) * (epidemic)

(epidemic)


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sick_john

(sick_john)

P(sick_john) / (sick_john)


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A Factor Network

sick(mary,measles)

hospital(mary)

epidemic(measles) epidemic(flu)

sick(mary,flu)

…

… sick(bob,measles)

hospital(bob)

sick(bob,flu)……

…

… …

… …

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First-order Representation

sick(P,D)

hospital(P)

epidemic(D)

Atoms representa set of random

variables

Logical Variablesparameterize random

variables

Parfactors, for parameterized

factors

Contraints to logical

variables

P john

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Semantics

sick(mary,measles)

hospital(mary)

epidemic(measles) epidemic(flu)

sick(mary,flu)

…

… sick(bob,measles)

hospital(bob)

sick(bob,flu)……

…

… …

… …sick(mary,measles), epidemic(measles))

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Inversion Elimination (IE)

sick(P, D)

epidemic(D)

D measles

D measles

IE

…sick(john, flu)

epidemic(flu)

sick(mary, rubella)

epidemic(rubella)…

Grounding

…sick(john, flu)

epidemic(flu)

sick(mary, rubella)


VE

Abstraction

epidemic(D)

D measles

D measles

sick(P, D)

sick(P,D) (epidemic(D),sick(P,D))

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Inversion Elimination (IE) Proposed by Poole (2003); Lacked formalization; Did not identify cases in which it is

incorrect; Formalized and identified correctness

conditions (with Dan Roth and Eyal Amir).

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Requires eliminated RVs to occur in separate instances of parfactor

…sick(mary, flu)

epidemic(flu)

sick(mary, rubella)


sick(mary, D)

epidemic(D)

D measles

InversionElimination

correct

Inversion Elimination - Limitations

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IE not correct

Requires eliminated RVs to occur in separate instances of parfactor

Inversion Elimination - Limitations

epidemic(D2)

epidemic(D1)

D1 D2 month

epidemic(measles)

epidemic(flu)

epidemic(rubella)

…month

epidemic(D2)

epidemic(D1)

D1 D2 month

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Counting Elimination Need to consider joint assignments; Exponential number of those; But actually, potential depends on

histogram of values in assignment only: 00101 the same as 11000;

Polynomial number of assignments instead.

epidemic(D2)

epidemic(D1)

D1 D2 month

epidemic(measles)

epidemic(flu)

epidemic(rubella)

…month

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A Simple Experiment

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BLOG (Bayesian LOGic) Milch & Russell (2004); A probabilistic logic language; Current inference is propositional

sampling; Special characteristics:

Open Universe; Expressive language.

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type Battalion;#Battalion ~ Uniform(1,10);

random Boolean Large(Battalion bat) ~ Bernoulli(0.6);

random NaturalNum Region(Battalion bat) ~ TabularCPD[[0.3, 0.4, 0.2, 0.1]]();

type Soldier;#Soldier(BatallionOf = Battalion bat)

if Large(bat) then ~ Poisson(1500); else if Region(bat) = 2 then = 300; else ~ Poisson(500);

query Average( {#Soldier(b) for Battalion b} );

BLOG (Bayesian LOGic)Open Universe

Expressive language

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Inference in BLOGrandom Boolean Attacked(bat) ~ Bernoulli(0.7);

random Boolean Wounded(Soldier sol)if Attacked(BattalionOf(sol)) then if Experienced(sol) then ~ Bernoulli(0.1); else ~ Bernoulli(0.4); else = False;

random Boolean Experienced(Soldier sol) ~ Bernoulli(0.1);

random Boolean Rescue(Battalion bat)= exists Soldier sol BattalionOf(sol) = bat & Wounded(sol);

guaranteed Battalion bat13;query Rescue(bat13);

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Inference in BLOG - Sampling

Rescue(bat13)

#Soldier(bat13)

Wounded(soldier1)

Wounded(soldier73)

Attacked(bat13)

...

Sampled false

Open Universe

false

false

false

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Inference in BLOG - Sampling

Rescue(bat13)

#Soldier(bat13)

Wounded(soldier1)

Wounded(soldier49)

Attacked(bat13)

...

Sampled true

Open Universe

true Experienced(soldier1)

truetrue

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Temporal models in BLOG Expressive enough to directly write temporal

modelstype Aircraft;#Aircraft ~ Poisson[3];

random Real Position(Aircraft a, NaturalNum t) if t = 0 then ~ UniformReal[-10, 10]() else ~ Gaussian(Position(a, Pred(t)), 2);type Blip;// num of blips from aircraft a is 0 or 1#Blip(Source = Aircraft a, Time = NaturalNum t) ~ TabularCPD[[0.2, 0.8]]()

// num false alarms has Poisson distribution.#Blip(Time = NaturalNum t) ~ Poisson[2];

random Real ApparentPos(Blip b, NaturalNum t) ~ Gaussian(Position(Source(b), Pred(t)), 2));

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Temporal models in BLOG Inference algorithm does not use

temporal structure: Markov property: state depends on

previous state only; evidence and query come for successive

time steps; Dynamic BLOG (DBLOG); analogous to Dynamic Bayesian

networks (DBNs).

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DBLOG Syntax Only change in syntax is a special Timestep

type, with same semantics as NaturalNum:type Aircraft;#Aircraft ~ Poisson[3];

random Real Position(Aircraft a, Timestep t) if t = @0 then ~ UniformReal[-10, 10]() else ~ Gaussian(Position(a, Prev(t)), 2);type Blip;// num of blips from aircraft a is 0 or 1#Blip(Source = Aircraft a, Time = Timestep t) ~ TabularCPD[[0.2, 0.8]]()

// num false alarms has Poisson distribution.#Blip(Time = Timestep t) ~ Poisson[2];

random Real ApparentPos(Blip b, Timestep t) ~ Gaussian(Position(Source(b), Prev(t)), 2));

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Similar idea to DBN Particle Filtering

DBLOG Particle Filtering

evidence at t

State samples for t - 1

Samples weighed by evidence

resampling

State samples for t

Likelihoodweighting

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Weighting by evidence in DBNs

evidence

state

t-1 t

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Weighting evidence in DBLOG

evidence

state Position(a1,@1)

Position(a2,@1)

#Aircraft

t-1 t

obs {Blip b : Source(b) = a1} = {B1};

obs ApparentPos(B1, @2) = 3.2;ApparentPos(B1,@2)

#Blip(a1,@2)

Position(a1,@2)B1

Lazy instantiation

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Retro-instantiation in DBLOG

Position(a1,@1)

Position(a2,@1)

#Aircraft

ApparentPos(B1,@2)

#Blip(a1,@2)

Position(a1,@2)B1

ApparentPos(B8,@9)

#Blip(a2,@9)

Position(a2,@9)

B8

Position(a2,@2)

...

obs {Blip b : Source(b) = a2} = {B8};

obs ApparentPos(B8,@2) = 2.1;

...

...

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Coping with retro-instantiation Analyse possible queries and

evidence (provided by user) andpre-instantiate random variables that will be necessary later;

Use mixing time to decide when to stop sampling back.

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Improving BLOG inference Rejection sampling

evidence

evidence

= ?

We count samples that match the evidence.

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Improving BLOG inference Likelihood weighting

evidence

We count samples using evidence likelihood as weight.

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Improving BLOG inference

{Happy(p) : Person p}=

{true, true, false}

Happy(john)= true

Happy(mary)= false

Happy(peter)=false

Evidence is deterministic function, so likelihood is always 0 or 1.So likelihood weighting is no better than rejection sampling.

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{Salary(p) : Person p}=

{90,000, 75,000, 30,000}

Salary(john)= 100,000

Salary(mary)= 80,000

Salary(peter)=500,000

The more unlikely the evidence, the worse it gets.

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{ApparentPos(b) : Blip b}=

{1.2, 0.5, 5.1}

ApparentPos(b1) = 2.1



Doesn’t work at all for continuous evidence.

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Structure-dependent automatic importance sampling

{ApparentPos(b) : Blip b}=

{1.2, 0.5, 5.1}

ApparentPos(b1) ApparentPos(b2) ApparentPos(b3)

Position(Source(b1)) = 1.5



= 1.2 = 5.1 = 0.5 High-level language allows algorithm to automatically apply better sampling schemes

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For the future Lifted inference works on symmetric,

indistinguishable objects only; Adapt approximations for dealing with

similar objects; Parameterized queries:

P(sick(X, measles)) = ? Function symbols:

(diabetes(X), diabetes(father(X))) Equality (paramodulation); Learning.

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Conclusion Expressive, first-order probabilistic

representations in inference too! Lifted inference equivalent to

grounded inference; much faster, yet yielding the same exact answer;

DBLOG brings techniques for temporal processes reasoning to first-order probabilistic reasoning.

first-order probabilistic inference rodrigo de salvo braz

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