Taylor-Type Techniques for Example: Intervals . . . New Approach - PowerPoint PPT Presentation

Traditional Approach Traditional Approach: . . . Main Idea Taylor Model-Type . . . Technical Details Example of Using LP Taylor-Type Techniques for Example: Intervals . . . New Approach Handling Uncertainty in Other Examples General Comment . . . Expert Systems, with Traditional Trust & Its . . . Probabilistic . . . Possibly Dependent . . . Potential Applications to Precise Formulation of . . . This Problem Is . . . Geoinformatics Possibly Dependent . . . Acknowledgments Algorithm: Justification Martine Ceberio, Vladik Kreinovich, Sanjeev Chopra Proof (cont-d) NASA Pan-American Center for Earth Title Page and Environmental Studies (PACES) University of Texas at El Paso ◭◭ ◮◮ mceberio@cs.utep.edu, vladik@utep.edu ◭ ◮ Bertram Ludaescher Page 1 of 23 Department of Computer Science, University of California, Davis Go Back Full Screen

Traditional Approach Traditional Approach: . . . 1. Formulation of the Problem Main Idea Taylor Model-Type . . . • Expert knowledge consists of statements S j : facts and rules. Technical Details Example of Using LP • Objective: given a query Q , check whether Q follows from the expert knowl- Example: Intervals . . . edge. New Approach • Example of a knowledge base: Other Examples General Comment . . . S 1 : a ← b. Traditional Trust & Its . . . Probabilistic . . . S 2 : b ← . Possibly Dependent . . . S 3 : a ← c. Precise Formulation of . . . This Problem Is . . . S 4 : c ← . Possibly Dependent . . . • In this example, S 1 and S 3 are rules, S 2 and S 4 are facts. Acknowledgments Algorithm: Justification • Example of a query Q : a ?. Proof (cont-d) • Answer: yes, e.g., Q follows from S 1 and S 2 . Title Page • Tools: Prolog-type inference engines. ◭◭ ◮◮ ◭ ◮ Page 2 of 23 Go Back Full Screen

Traditional Approach Traditional Approach: . . . 2. Enter Uncertainty Main Idea Taylor Model-Type . . . • Fact: experts are not 100% confident. Technical Details Example of Using LP • How: the expert’s degree of confidence in each statement S j can be described Example: Intervals . . . as a (subjective) probability p ( S j ). New Approach • Example: if we are interested in oil, we should look for certain geological Other Examples structures (confidence 80%). General Comment . . . Traditional Trust & Its . . . • Question: if a query Q is deducible from facts and rules, what is our confi- Probabilistic . . . dence p ( Q ) in Q ? Possibly Dependent . . . • Example: Precise Formulation of . . . This Problem Is . . . – to find oil, look for subterranean structures (80%); Possibly Dependent . . . – to find these structures, analyze gravity data (90%); Acknowledgments – what is our confidence that to find oil, we must look for gravity data? Algorithm: Justification Proof (cont-d) Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 23 Go Back Full Screen

Traditional Approach Traditional Approach: . . . 3. Representation Main Idea Taylor Model-Type . . . • Idea: we can usually describe Q as a propositional formula F in terms of S j . Technical Details Example of Using LP • Example: Example: Intervals . . . S 1 : a ← b. S 2 : b ← . New Approach S 3 : a ← c. S 4 : c ← . Other Examples General Comment . . . Here, F = ( S 1 & S 2 ) ∨ ( S 3 & S 4 ) . Traditional Trust & Its . . . • Resulting problem: Probabilistic . . . Possibly Dependent . . . – we have a propositional combination F of known statements S j ; Precise Formulation of . . . – we know the probabilities p ( S j ) of different statements; This Problem Is . . . – we must determine the probability p ( F ); Possibly Dependent . . . – to be more precise, we need the interval p ( F ) of possible values of p ( F ). Acknowledgments Algorithm: Justification Proof (cont-d) Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 23 Go Back Full Screen

Traditional Approach Traditional Approach: . . . 4. Traditional Approach Main Idea Taylor Model-Type . . . • Fact: the problem of finding the exact bounds for p ( F ) is NP-hard. Technical Details Example of Using LP • Traditionally: expert systems use technique similar to straightforward inter- Example: Intervals . . . val computations: New Approach – we parse F and Other Examples General Comment . . . – replace each computation step with corresponding probability operation. Traditional Trust & Its . . . • Operations: if we know the bounds [ a, a ] for p ( A ) and [ b, b ] for p ( B ), then: Probabilistic . . . Possibly Dependent . . . – p ( A & B ) is in the interval Precise Formulation of . . . [max( a + b − 1 , 0) , min( a, b )]; This Problem Is . . . Possibly Dependent . . . – p ( A ∨ B ) is in the interval Acknowledgments Algorithm: Justification [max( a, b ) , min( a + b, 1)] . Proof (cont-d) Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 23 Go Back Full Screen

Traditional Approach Traditional Approach: . . . 5. Traditional Approach: Too Wide Main Idea Taylor Model-Type . . . • Example: F = ( A & B ) ∨ ( A & ¬ B ), p ( A ) = p ( B ) = 0 . 6. Technical Details Example of Using LP • Parsing: Example: Intervals . . . New Approach • we first find the bounds for p ( ¬ B ), Other Examples • then for p ( A & B ) and p ( A & ¬ B ), and General Comment . . . • finally, the bounds for p ( F ). Traditional Trust & Its . . . Probabilistic . . . • Result: p ( ¬ B ) = 1 − 0 . 6 = 0 . 4; Possibly Dependent . . . • p ( A & B ) = [max(0 . 6 + 0 . 6 − 1 , 0) , min(0 . 6 , 0 . 6)] = [0 . 2 , 0 . 6]; Precise Formulation of . . . This Problem Is . . . • p ( A & ¬ B ) = [max(0 . 6 + 0 . 4 − 1 , 0) , min(0 . 6 , 0 . 4)] = [0 , 0 . 4]; Possibly Dependent . . . • p ( F ) = [max(0 , 0 . 2) , min(0 . 4 + 0 . 6 , 1)] = [0 . 2 , 1 . 0]. Acknowledgments Algorithm: Justification • Problem: F is equivalent to A , so p ( F ) = 0 . 6. Proof (cont-d) Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 23 Go Back Full Screen

Traditional Approach Traditional Approach: . . . 6. Main Idea Main Idea Taylor Model-Type . . . • Similar problem: excess width in straightforward interval computations. Technical Details Example of Using LP • Solution to the similar problem: Taylor methods narrow down the resulting Example: Intervals . . . intervals. New Approach • Idea behind this solution: if we use linear Taylor models, then, for each inter- Other Examples mediate result y j : General Comment . . . Traditional Trust & Its . . . – we not only keep the interval of its possible values, Probabilistic . . . – we also keep the relation between this value and the original inputs – Possibly Dependent . . . – in the form of a linear dependence Precise Formulation of . . . This Problem Is . . . y j = a 0 j + a 1 j · x 1 + . . . + a nj · x n . Possibly Dependent . . . Acknowledgments • For quadratic Taylor models, we also keep the relation between y j and pairs Algorithm: Justification of inputs (as terms a jkl · x k · x l ), Proof (cont-d) • etc. Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 23 Go Back Full Screen

Traditional Approach Traditional Approach: . . . 7. Taylor Model-Type Techniques Main Idea Taylor Model-Type . . . • Main idea: similarly to Taylor arithmetic, for each intermediate result F j : Technical Details Example of Using LP – besides an interval of possible values for p ( F j ), Example: Intervals . . . – we also compute intervals of possible values for pairs p ( F j & F i ) New Approach – (or even all Boolean functions of pairs); Other Examples General Comment . . . – on each step, use all such probabilities to get new estimates. Traditional Trust & Its . . . • If this is not enough: we use an analog of k -th order Taylor methods – Probabilistic . . . estimate intervals for Possibly Dependent . . . p ( F j 1 & . . . & F j k +1 ) . Precise Formulation of . . . This Problem Is . . . • The higher the order k : Possibly Dependent . . . – the more accurate the results, but Acknowledgments – the longer the computations. Algorithm: Justification Proof (cont-d) Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 23 Go Back Full Screen

Traditional Approach Traditional Approach: . . . 8. Technical Details Main Idea Taylor Model-Type . . . • Minor problem: even if we know the probability of triples, then, in general, Technical Details the problem is NP-hard. Example of Using LP Example: Intervals . . . • Proof: reduction to satisfiability of 3-CNF formulas. New Approach • Solution: when estimating interval for p ( F i & . . . ), we take into account only Other Examples ≤ l known probabilities. General Comment . . . Traditional Trust & Its . . . • How: Probabilistic . . . • we describe both known and estimated probabilities as sums of proba- Possibly Dependent . . . bilities of atomic statements S ε 1 i 1 & . . . & S ε m i m , where m ≤ k · l , and Precise Formulation of . . . This Problem Is . . . • use linear programming (LP) to get desired bounds on the unknown probability. Possibly Dependent . . . Acknowledgments + When k → ∞ and l → ∞ , we get exact results. Algorithm: Justification Proof (cont-d) − However, computation time grows exponentially with k and l . Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 23 Go Back Full Screen