Probabilistic Resolution

Logical reasoning

• Absolute implications

• office meeting

• office talk

• office pick_book

• But what if my rules are not absolute?

Migrating to Probabilities:Graphical Models

noisy_office

meeting

talk

pick_bookmeeting talk pick_book P(noisy_office| m,p,t)

T T T 0.9992T T F 0.998T F T 0.996T F F 0.99F T T 0.92F T F 0.8F F T 0.6F F F 0

Actually, the original model does not justify the

last row

Migrating to Probabilities:Graphical Models

noisy_office meeting

talk

pick_book

meeting talk pick_book f(noisy_office = T)T T T 0.9992T T F 0.998T F T 0.996T F F 0.99F T T 0.92F T F 0.8F F T 0.6F F F 0

Variable Elimination (VE)

noisy_office meeting

talk

pick_book

Variable Elimination (VE)

noisy_office

talk

pick_book

meeting(noisy_office, pick_book, talk, meeting) (meeting)

meeting

Variable Elimination (VE)

noisy_office

talk

pick_book

Variable Elimination (VE)

noisy_office

pick_book

Variable Elimination (VE)

noisy_office

meeting talk pick_book f(office = T)T T T 1T T F 1T F T 1T F F 1F T T 1F T F 1F F T 1F F F 0

Graphical Models generalize Logic

office meeting

talk

pick_book

VE generalizes Resolution

Resolution

A or BB or C

A or C

A B C

A C

Variable Elimination

There is still an important difference, though.

Story so far

• Logic uses absolute rules;

• Probabilistic models can deal with noise, and generalize logic;

• But...

Logical reasoning “ends early”

• office meeting

• office talk

• office pick_book

• ...

• Given evidence ‘meeting’, we are done after considering first rule alone.

Ending early in deterministic graphical model

Variable Elimination uses all nodes to calculate P(office | meeting)

office meeting

talk

pick_book

meeting talk pick_book f(office = T)T T T 1T T F 1T F T 1T F F 1F T T 1F T F 1F F T 1F F F 0

Ending early in deterministic graphical model

But if ‘meeting’ is observed, we don’t need to look beyond it

office

talk

pick_book

talk pick_book f(office = T)T T 1T F 1F T 1F F 1

Ending early in deterministic graphical model

We can use “smarter” algorithms to end early here as well

office

talk

pick_book

talk pick_book f(office = T)T T 1T F 1F T 1F F 1

Ending early in non-deterministic graphical models

Calculating P(noisy_office | meeting)

noisy_office meeting

talk

pick_book

meeting talk pick_book f(noisy_office = T)T T T 0.9992T T F 0.998T F T 0.996T F F 0.99F T T 0.92F T F 0.8F F T 0.6F F F 0

Ending early in non-deterministic graphical models

P(noisy_office | meeting) depends on all nodes

noisy_office

talk

pick_book

talk pick_book f(noisy_office = T)T T 0.9992T F 0.998F T 0.996F F 0.99

Ending early in non-deterministic graphical models

noisy_office

talk

pick_book

talk pick_book f(noisy_office = T)T T 0.9992T F 0.998F T 0.996F F 0.99

But we already know P(noisy_office | meeting) [0.99, 0.9992]Can we take advantage of this?

Goal

• A graphical model inference algorithm that derives a bound on solution so far;

• Ends as soon as bound is “good enough”;

• An anytime algorithm.

Probabilistic Resolution

Resolution

A or BB or C

A or C

A B C

A C

Variable Elimination

•Variable Elimination generalizes Resolution, but neither provides intermediate results nor ends early.

•Probabilistic Resolution = VE + “ending early”

Story so far

• Logic uses absolute rules;

• Probabilistic models can deal with noise, and generalize logic;

• Logic ends as soon as possible, graphical models do not;

• They can if we are willing to use bounds;

• But how to calculate bounds?

But how to get bounds?

Q N2

N1

N4

N3

...

...

......

But how to get bounds?

Q N2

N1

N4

N3

...

...

......

But how to get bounds?

Q N2

N1

N4

N3

But how to get bounds?

Q N

But how to get bounds?

Q Nf1 f2

P(Q) N f1(Q,N) f2(N)

P(Q) N f1(Q,N) P2(N)

P(Q) f ( P2(N) )

But how to get bounds?

Q N

P(Q) f ( P2(N) )

0 1 0 1

f

P(Q) P2(N)

But how to get bounds?

Q N

P(Q) P2(N)

(0,0,1)

(1,0,0)

(0,1,0)

(0,0,1)

(1,0,0)

(0,1,0)

f

P(Q) f ( P2(N) )

But how to get bounds?

Q N

P(Q) P2(N)

(0,0,1)

(1,0,0)

(0,1,0)

(0,0,1)

(1,0,0)

(0,1,0) f

P(Q) f ( P2(N) )

bound

Infinite number of points!

Justify inner shape to be equal to outter one

But how to get bounds?

Q N

P(Q) P2(N)

(0,0,1)

(1,0,0)

(0,1,0)

(0,0,1)

(1,0,0)

(0,1,0)

f

P(Q) f ( P2(N) )Vertices are

enough

But how to get bounds?

Q N

P(Q)

(0,0,1)

(1,0,0)

(0,1,0)

(0,0,1)

(1,0,0)

(0,1,0)

f

P(Q) f ( P2(N) )

P2(N)

No necessary correspondence

But how to get bounds?

Q N

(0,0,1)

(1,0,0)

(0,1,0)

f

P(Q) f ( P2(N) )

P2(N)0 1P(Q)

Correspondence would be

impossible in this case

Make slide with opposite: segment to

triangle

But how to get bounds?

Q N

0 1 0 1

f

P(Q)

P(Q) f ( P2(N) )

P2(N)

Example I

Q N

Q N f(Q,N)1 1 0.61 0 0.40 1 0.30 0 0.7

[0,1][0.36, 0.67]

P(Q) f ( P2(N) )

P(Q) N f(Q,N) P2(N)

P(Q) f(Q,0)P2(N=0) + f(Q,1)P2(N=1)

For P2(N=0) = 1:

P(Q) f(Q,0) 1 + f(Q,1) 0

P(Q) f(Q,0)

P(Q=1) = f(1,0) / (f(0,0) + f(1,0))

P(Q=1) = 0.4 / (0.7 + 0.4) = 0.36

For P2(N=1) = 1:

P(Q) f(Q,0) 0 + f(Q,1) 1

P(Q) f(Q,1)

P(Q=1) = f(1,1) / (f(0,1) + f(1,1))

P(Q=1) = 0.6 / (0.3 + 0.6) = 0.67

P2(N)

Example II

Q N

[0,1][0.5]

Q N f(Q,N)1 1 11 0 10 1 10 0 1

P(Q)

(0,0,1)

(1,0,0)

(0,1,0)

(0,0,1)

(0,1,0)

f

(1,0,0)

0 1 0 1

f

P(Q) P2(N)

Example III

Q N

[0,1][0,1]

Q N f(Q,N)1 1 11 0 00 1 00 0 1

P2(N)P(Q)

(0,0,1)

(1,0,0)

(0,1,0)

(0,0,1)

(0,1,0)

f

(1,0,0)

0 1 0 1

f

P(Q) P2(N)

Example IV

noisy_office meeting

talk

pick_book

meeting talk pick_book f(noisy_office = T)T T T 0.9992T T F 0.998T F T 0.996T F F 0.99F T T 0.92F T F 0.8F F T 0.6F F F 0

Example IV

noisy_office

talk

pick_book

talk pick_book f(noisy_office = T)T T 0.9992T F 0.998F T 0.996F F 0.99

Example IV

noisy_office

talk

pick_book

talk pick_book f(noisy_office = T)T T 0.9992T F 0.998F T 0.996F F 0.99

0.4

Example IV

noisy_office

pick_book

pick_book f(noisy_office = T)T 0.9976F 0.994

Example IV

noisy_office

pick_book

pick_book f(noisy_office = T)T 0.9976F 0.994 1

Example IV

noisy_office

f(noisy_office = T)0.9976

Algorithm

• Same as Variable Elimination, but update bounds every time a neighbor is eliminated;

• Bounds always improve at each neighbor elimination;

• Trade-off between granularity of bound updates (explain granularity) and ordering efficiency.

Complexity Issues

• Calculating bound is exponential on the size of neighborhood component, so complexity is exponential on largest neighborhood component during execution;

• This can be larger than tree-width;

• But finding tree-width is hard anyway.

Preliminary Tests

0

0.2

0.4

0.6

0.8

1

1.2

0 0 1 2 2 4 8 9 92 102 105 105 108 125

% of exact computation time (not in scale)

Bo

un

d i

nte

rval

wid

th strokevolume=high

hrekg=low

pcwp=normal

Conclusions

• Making Probabilistic Inference more like Logic Inference;

• Getting an anytime algorithm in the process;

• Preparing ground for First-order Probabilistic Resolution.