probabilistic inference modulo theories - sri …braz/presentations/hr 2015.pdfprobabilistic...

Rodrigo de Salvo Braz Artificial Intelligence Center - SRI International

joint work with Ciaran O’Reilly

Artificial Intelligence Center - SRI International

Vibhav Gogate University of Texas at Dallas

Rina Dechter

University of California, Irvine

Probabilistic Inference Modulo Theories

Hybrid Reasoning Workshop – IJCAI 2015 - July 26, 2015

Probabilistic Programming for Advanced Machine Learning

Outline

• Two extremes: logic and probabilistic inference

• Probabilistic inference in Graphical Models

– Factorized joint probability distribution

– Marginalization

– Variable elimination

– Mainstream probabilistic inference representation

• Logic Inference

– Theories

– Inference: satisfiability

– DPLL

– DPLL(T)

• Symbolic Generalized DPLL Modulo Theories (SGDPLL(T))

• Experiments

• Conclusion

Two extremes: logic and probabilistic inference • Logic – Initial mainstream approach to AI: represent knowledge declaratively,

have a system using it

– Can use rich theories: data structures, numeric constraints, functions

– Basic logic does not offer treatment for uncertain knowledge

• Probabilistic Inference – Follows the idea of representing knowledge declaratively and having a

system use it

– Very good treatment of uncertainty

– Poor representation, equivalent to discrete variables with equality, lacking interpreted functions (data structures, arithmetic) and even uninterpreted non-nullary functions (friends(X,Y))

– Way to use such constructs is to ground them into tables or formulas

Probabilistic Inference in Graphical Models

epidemic

P(sick_john, sick_mary, sick_bob, epidemic) (chain rule) = P(sick_john | epidemic) * P(sick_mary | epidemic) * P(sick_bob | epidemic) * P(epidemic)

sick_john sick_mary sick_bob

P(sick_bob | epidemic)

P(epidemic)

P(sick_john | epidemic)

P(sick_mary | epidemic)

Graphical models 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)



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)



Inference: Marginalization via Variable Elimination

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)


epidemic

sick_john sick_mary

1(epidemic)

P(epidemic)





* 1 (epidemic)


epidemic

sick_john sick_mary

1(epidemic)

P(epidemic)




* 1 (epidemic)



epidemic

sick_john

1(epidemic)

P(epidemic)



* 1 (epidemic)

* 2 (epidemic)

2(epidemic)


sick_john

3(sick_john)

P(sick_john) 3 (sick_john)

A Graphical Model with a Theory

• Consider an undirected graphical model defined on strings:

and query P(speaker| announcement = “Efficient PP Solving

Prof. Paul Smith We did this and that”)= ?

• Can we do probabilistic inference using theories?

if announcement = title + speaker + venue + abstract then 0.7 else

if announcement = title + venue + speaker + abstract then 0.3 else 0

if contains(title, "complexity") then 0.1 else 0.9

if contains(title, "efficient") then 0.3 else 0.7

if length(title) > 5 then 0.9 else 0.1

if length(title) < 15 then 0.9 else 0.1

if speaker = "Prof." + name then 0.1 else

if speaker = name then 0.9 else 0

... // more statements, for example defining knowledge about names

Basic Step: summing a variable out

• A basic operation is summing a variable out

epidemic P(sick_john | epidemic) * P(epidemic) * 1 (epidemic) * 2 (epidemic)

=

3 (sick_john)

• Typically, factors are represented as tables

• This is fine for variables with a few values such as booleans, but what about, for example

x in {1,...,1000} (if x > 20 then 0.3 else 0.7) * (if x < z then 0.4 else 0.6)

or

s in Strings (if length(s) < 40 then 0.8 else if length(s) < length(s2) then 0.2 else 0 )

* (if s starts with “Announcement” then 0.7 else 0.3)

• It is wasteful to go over all values of x and impossible for s.

• Results are symbolic in z and s2

A parallel track: Satisfiability

• The Davis-Putnam-Logemann-Loveland (DPLL) algorithm solves the problem of satisfiability:

∃𝑝 ∃𝑞 ∃𝑟 (𝑝 ∨ 𝑞) ∧ (𝑞 ∨ ¬𝑟) ∧ (¬𝑝 ∨ 𝑟)

• This is similar to what we need, but for

– Existential quantification instead of summation

– Propositional variables (no theories)

– Total quantification (no free variables)

Solving Satisfiability with DPLL

xyz (x y) ( x y z)

yz y yz y z

z true z false z z z true

x = false x = true

y = false y = true y = false y = true

false true

z = false z = true

A parallel track: Satisfiability Modulo Theories (SMT)

• Satisfiability modulo theories generalizes satisfiability to non-propositional logic (includes arithmetic, inequalities, lists, uninterpreted functions, and others)

∃𝑥 ∃𝑦 ∃𝑙 𝑥 ≤ 𝑦 ∨ 𝑙 = 𝑥, 3 ∧ 𝑓𝑖𝑟𝑠𝑡 𝑙 = 𝑦

• This is closer to what we need (since it works on theories), but for

– Existential quantification instead of summation

– Total quantification (no free variables)

Solving Satisfiability Modulo Theories with DPLL(T)

∃𝑥 ∃𝑦 ∃𝑙 𝑥 ≤ 𝑦 ∨ 𝑙 = 𝑥, 3 ∧ 𝑓𝑖𝑟𝑠𝑡 𝑙 = 𝑦

𝑓𝑖𝑟𝑠𝑡 𝑙 = 𝑦 𝑙 = 𝑥, 3 ∧ 𝑓𝑖𝑟𝑠𝑡 𝑙 = 𝑦

true false

𝑥 ≤ 𝑦 𝑥 > 𝑦

𝑓𝑖𝑟𝑠𝑡 𝑙 = 𝑦 𝑓𝑖𝑟𝑠𝑡 𝑙 ≠ 𝑦 𝑙 = 𝑥, 3 𝑙 ≠ 𝑥, 3

𝑓𝑖𝑟𝑠𝑡 𝑙 = 𝑦 false

𝑓𝑖𝑟𝑠𝑡 𝑙 = 𝑦 𝑓𝑖𝑟𝑠𝑡 𝑙 ≠ 𝑦

false true

Our solution: Probabilistic Inference Modulo Theories

• Similar to SMT, but based on – Summation (or other quantifiers) instead of

– Partial quantification (free variables)

x{1,…,100}z{1,…,100} (if x > y y ≠ 5 then 0.1 else 0.9)

(if z < y y < 3 then 0.4 else 0.6)

• Note that y is a free variable

• Summed expression is not Boolean

• Language is not propositional (≠, <, …)

Probabilistic Inference Modulo Theories (SGDPLL(T))

xyz (x y) ( x y z)

yz y yz y z

z true z false z z z true

x = false x = true

y = false y = true y = false y = true

false true

z = false z = true

…

x {1,…,100} z {1,…,100} [ (if x > y y ≠ 5 then 0.1 else 0.9)

(if z < y y < 3 then 0.4 else 0.6) ]

x z (if y ≠ 5 then 0.1 else 0.9)

: x > y (if z < y y < 3 then 0.4 else 0.6)

x ≤ y +

x z 0.1 if z < y y < 3 then 0.4 else 0.6

: x > y

+ … …

x z 0.9

: x ≤ y (if z < y y < 3 then 0.4 else 0.6)

x > y

else then

if y ≠ 5

x z 0.04 : x > y : z < y

= x: y < x ≤ 100 z: 1 ≤ z < y 0.04

= x: y < x ≤ 100 (y – 1) 0.04

= (100 – y) (y – 1) 0.04

= –0.04y2 + 4.04y – 4

…

Condition on literals until base case with no literals in main expression:

Probabilistic Inference Modulo Theories (SGDPLL(T))

= x: y < x ≤ 100 z: 1 ≤ z < y 0.04

= x: y < x ≤ 100 (y – 1) 0.04

= (100 – y) (y – 1) 0.04

= –0.04y2 + 4.04y – 4

Condition on literals until base case with no literals in main expression

…

x {1,…,100} z {1,…,100} [ (if x > y y ≠ 5 then 0.1 else 0.9)

(if z < y y < 3 then 0.4 else 0.6) ]

x z (if y ≠ 5 then 0.1 else 0.9)

: x > y (if z < y y < 3 then 0.4 else 0.6)

x ≤ y +

x z 0.1 if z < y y < 3 then 0.4 else 0.6

: x > y

+ … …

x z 0.9

: x ≤ y (if z < y y < 3 then 0.4 else 0.6)

x > y

else then

if y ≠ 5

x z 0.04 : x > y : z < y

…

Unifying Logic and Probabilistic Inference

Variable Elimination DPLL (satisfiability)

SMT (satisfiability modulo theories)

SGVE(T), with SGDPLL(T)

for inner summations

Propositional

Modulo Theories

Symbolic Modulo Theories

Satisfiability () Sum (), max and others

Evaluation

• Generated random graphical models defined on equalities on bounded integers

• Evaluated against VEC (Gogate & Dechter 2011), a state-of-the-art graphical model solver, after grounding into tables with increasing random variable domain sizes

• For domain of size 16, our solver was already 20 times faster than VEC.

Final Remarks on SGDPLL(T)

• It is symbolic (S)

• It is generic (not only Boolean expressions) (G)

• Can use theories (T)

• Can re-use SMT techniques – Satisfiability solvers

– Modern SAT solver techniques: – Unit propagation and Watched literals

– Clause learning

• Requires new solvers on theories for base cases (at least as powerful as model counting)

Thanks!

probabilistic inference modulo theories - sri …braz/presentations/hr 2015.pdfprobabilistic...

Documents