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Probabilistic Resolution

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Logical reasoning Absolute implications office meeting office talk office pick_book But what if my rules are not absolute?

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Migrating to Probabilities: Graphical Models noisy_office meeting talk pick_book Actually, the original model does not justify the last row

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Migrating to Probabilities: Graphical Models noisy_officemeeting talk pick_book

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Variable Elimination (VE) noisy_officemeeting talk pick_book

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Variable Elimination (VE) noisy_office talk pick_book meeting (noisy_office, pick_book, talk, meeting) (meeting) meeting

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Variable Elimination (VE) noisy_office talk pick_book

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Variable Elimination (VE) noisy_office pick_book

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Variable Elimination (VE) noisy_office

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Graphical Models generalize Logic officemeeting talk pick_book

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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.

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Story so far Logic uses absolute rules; Probabilistic models can deal with noise, and generalize logic; But...

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Logical reasoning ends early office meeting office talk office pick_book... Given evidence meeting, we are done after considering first rule alone.

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Ending early in deterministic graphical model Variable Elimination uses all nodes to calculate P(office | meeting) officemeeting talk pick_book

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

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

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

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

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Ending early in non-deterministic graphical models noisy_office talk pick_book But we already know P(noisy_office | meeting) [0.99, 0.9992] Can we take advantage of this?

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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.

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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

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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?

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

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

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But how to get bounds? QN2N2 N1N1 N4N4 N3N3

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QN

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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) )

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

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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) )

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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

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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

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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

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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

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

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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 / (0.7 + 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 / (0.3 + 0.6) = 0.67

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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)

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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)

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Example IV noisy_officemeeting talk pick_book

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Example IV noisy_office talk pick_book

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Example IV noisy_office talk pick_book 0.4

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Example IV noisy_office pick_book

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Example IV noisy_office pick_book 1

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Example IV noisy_office

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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.

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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.

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Preliminary Tests

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

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