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_book Actually, the original model does not justify the last row

Migrating to Probabilities: Graphical Models noisy_officemeeting talk pick_book

Variable Elimination (VE) noisy_officemeeting 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

Graphical Models generalize Logic officemeeting talk pick_book

VE generalizes Resolution Resolution A or B B or C A or C A B C AC 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) officemeeting talk pick_book

Ending early in deterministic graphical model But if meeting is observed, we dont need to look beyond it office talk pick_book

Ending early in deterministic graphical model We can use smarter algorithms to end early here as well office talk pick_book

Ending early in non-deterministic graphical models Calculating P(noisy_office | meeting) noisy_officemeeting talk pick_book

Ending early in non-deterministic graphical models P(noisy_office | meeting) depends on all nodes noisy_office talk pick_book

Ending early in non-deterministic graphical models noisy_office talk pick_book But we already know P(noisy_office | meeting) [0.99, ] 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 B B or C A or C A B C AC 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? QN2N2 N1N1 N4N4 N3N3...

But how to get bounds? QN2N2 N1N1 N4N4 N3N3...

But how to get bounds? QN2N2 N1N1 N4N4 N3N3

QN

QN 1 2 P(Q) N 1 (Q,N) 2 (N) P(Q) N 1 (Q,N) P 2 (N) P(Q) f ( P 2 (N) )

But how to get bounds? QN P(Q) f ( P 2 (N) ) 0101 f P(Q)P 2 (N)

But how to get bounds? QN P(Q)P 2 (N) (0,0,1) (1,0,0) (0,1,0) (0,0,1) (1,0,0) (0,1,0) f P(Q) f ( P 2 (N) )

But how to get bounds? QN P(Q)P 2 (N) (0,0,1) (1,0,0) (0,1,0) (0,0,1) (1,0,0) (0,1,0) f P(Q) f ( P 2 (N) ) bound Infinite number of points! Justify inner shape to be equal to outter one

But how to get bounds? QN P(Q)P 2 (N) (0,0,1) (1,0,0) (0,1,0) (0,0,1) (1,0,0) (0,1,0) f P(Q) f ( P 2 (N) ) Vertices are enough

But how to get bounds? QN P(Q) (0,0,1) (1,0,0) (0,1,0) (0,0,1) (1,0,0) (0,1,0) f P(Q) f ( P 2 (N) ) P 2 (N) No necessary correspondence

But how to get bounds? QN (0,0,1) (1,0,0) (0,1,0) f P(Q) f ( P 2 (N) ) P 2 (N) 01 P(Q) Correspondence would be impossible in this case Make slide with opposite: segment to triangle

But how to get bounds? QN 0101 f P(Q) P(Q) f ( P 2 (N) ) P 2 (N)

Example I QN [0,1][0.36, 0.67] P(Q) f ( P 2 (N) ) P(Q) N (Q,N) P 2 (N) P(Q) (Q,0)P 2 (N=0) + (Q,1)P 2 (N=1) For P 2 (N=0) = 1: P(Q) (Q,0) 1 + (Q,1) 0 P(Q) (Q,0) P(Q=1) = (1,0) / ( (0,0) + (1,0)) P(Q=1) = 0.4 / ( ) = 0.36 For P 2 (N=1) = 1: P(Q) (Q,0) 0 + (Q,1) 1 P(Q) (Q,1) P(Q=1) = (1,1) / ( (0,1) + (1,1)) P(Q=1) = 0.6 / ( ) = 0.67

P 2 (N) Example II QN [0,1][0.5] P(Q) (0,0,1) (1,0,0) (0,1,0) (0,0,1) (0,1,0) f (1,0,0) 0101 f P(Q) P 2 (N)

Example III QN [0,1] P 2 (N)P(Q) (0,0,1) (1,0,0) (0,1,0) (0,0,1) (0,1,0) f (1,0,0) 0101 f P(Q) P 2 (N)

Example IV noisy_officemeeting talk pick_book

Example IV noisy_office talk pick_book

Example IV noisy_office talk pick_book 0.4

Example IV noisy_office pick_book

Example IV noisy_office pick_book 1

Example IV noisy_office

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

Conclusions Making Probabilistic Inference more like Logic Inference; Getting an anytime algorithm in the process; Preparing ground for First-order Probabilistic Resolution.