How Knowledge Is . . . Experts Are Usually . . . Asymmetric Information Measures: Problem Description of the . . . How to Extract Knowledge Asymptotically . . . Average Case . . . From an Expert so That Average Case: . . . the Expert’s Effort Is Minimal Title Page Hung T. Nguyen ◭◭ ◮◮ Department of Mathematical Sciences ◭ ◮ New Mexico State University Las Cruces, New Mexico 88003, USA Page 1 of 17 Email: hunguyen@nmsu.edu Go Back Vladik Kreinovich and Elizabeth Kamoroff Full Screen Department of Computer Science, University of Texas at El Paso, Close El Paso, TX 79968, USA vladik@utep.edu Quit
1. How Knowledge Is Extracted Now How Knowledge Is . . . Experts Are Usually . . . • Knowledge acquisition: we ask experts questions, and Problem put the answers into the computer system. Description of the . . . • Problem: it is a very time-consuming and therefore Asymptotically . . . expensive task. Average Case . . . Average Case: . . . • Objective: minimize the effort of an expert. Title Page • Related problem: how do we estimate this effort? ◭◭ ◮◮ • Reasonable idea: number of binary (“yes”-“no”) ques- tions. ◭ ◮ • Resulting strategy: binary search. Page 2 of 17 Go Back • Idea: we choose a question for which the answer is “yes” for exactly half of the remaining alternatives. Full Screen • Property: we need log 2 ( N ) questions to select one of Close N alternatives. Quit
2. Experts Are Usually More Comfortable with “Yes” How Knowledge Is . . . Answers Experts Are Usually . . . Problem • In practice: most people feel more comfortable answer- Description of the . . . ing “yes” than “no”. Asymptotically . . . • Fact: the expert’s time is valuable. Average Case . . . Average Case: . . . • Consequence: an expert is usually called after compe- tent people tried to solve the problem. Title Page • Expected situation: the expert mostly confirms their ◭◭ ◮◮ preliminary solutions. ◭ ◮ • Consequence: most expert’s answers are “yes”. Page 3 of 17 • Binary search case: half of the answers are “no”s. Go Back • Meaning: half of the previous decisions were wrong. Full Screen • Expert’s conclusion: no competent people tried this Close problem – so his/her valuable time was wasted. Quit
3. Experts Are Usually More Comfortable with “Yes” How Knowledge Is . . . Answers (cont-d) Experts Are Usually . . . Problem • Situation: a knowledge engineer interviews the expert. Description of the . . . • First alternative: most answers are “yes”; meaning: Asymptotically . . . Average Case . . . – the knowledge engineer already has some prelimi- Average Case: . . . nary knowledge of the area, and – he/she is appropriately asking these questions to Title Page improve this knowledge. ◭◭ ◮◮ • Binary search: half of the answers are “no” (same as ◭ ◮ for random questions); interpretation: Page 4 of 17 – the knowledge engineer did not bother to get pre- Go Back liminary knowledge; Full Screen – the highly skilled expert is inappropriately used to answer questions Close – which could be answered by consulting a textbook. Quit
4. Problem How Knowledge Is . . . Experts Are Usually . . . • Reminder: experts prefer “yes” answers. Problem • Additional phenomenon: Description of the . . . Asymptotically . . . – the larger the number of negative answers, Average Case . . . – the more discomfort the expert will experience, and Average Case: . . . – the larger effort he will have to make to continue this interview. Title Page • Previous objective: minimize the total number of ques- ◭◭ ◮◮ tions. ◭ ◮ • More appropriate objective: minimize the effort of an Page 5 of 17 expert. Go Back • How to describe the effort: assign more weight to “no” Full Screen answers than to “yes” ones. Close • What we do: find a search procedure which attains this objective. Quit
5. How to Describe Different Search Procedures How Knowledge Is . . . Experts Are Usually . . . • Let S be the set of N alternatives. Problem • We denote “yes” as 1, “no” as 0, so each sequence of Description of the . . . answers ω is a binary sequence. Asymptotically . . . Average Case . . . • To describe a search procedure, we must have: Average Case: . . . – the set Ω of possible answer sequences ω , and Title Page – a mapping A which maps each ω ∈ Ω to the set ◭◭ ◮◮ A ( ω ) of all alternatives which are consistent with ω . ◭ ◮ • Formally: A (Λ) = S , and for every ω ∈ Ω: Page 6 of 17 • if | A ( ω ) | = 1, then no extension of ω belongs to Ω; Go Back • otherwise, ω 0 ∈ Ω, ω 1 ∈ Ω, and we have Full Screen A ( ω ) = A ( ω 0) ∪ A ( ω 1) , A ( ω 0) ∩ A ( ω 1) = ∅ , Close A ( ω 0) � = ∅ , A ( ω 1) � = ∅ . Quit
6. How to Gauge Different Search Procedures How Knowledge Is . . . Experts Are Usually . . . • Let P = (Ω , A ) be a search procedure. Problem • Let W 0 be the cost of “no” answer, and W 1 < W 0 be Description of the . . . the cost of the “yes” answer. Asymptotically . . . Average Case . . . • For a ∈ Ω, let ω ( a, P ) = ω 1 ω 2 . . . ω k denote the se- quence of answers which leads to a . Average Case: . . . • The cost W ( ω ( a, P )) of finding a is defined as Title Page W ( ω ( a, P )) = W ( ω 1 ω 2 . . . ω k ) = W ω 1 + W ω 2 + . . . + W ω k . ◭◭ ◮◮ • The effort of a procedure is defined as the largest of its ◭ ◮ costs: Page 7 of 17 E ( P ) = max a ∈ S W ( ω ( a, P )) . Go Back • Objective: find a procedure P opt with the smallest pos- Full Screen sible effort: Close def E ( P opt ) = T ( N ) = min E ( P ) . P Quit
7. Example 1: Binary Search (Optimal for W 0 = W 1 ) How Knowledge Is . . . Experts Are Usually . . . • Situation: a doctor chooses between N = 4 possible Problem analgetics: Description of the . . . – aspirin ( as ), Asymptotically . . . Average Case . . . – acetaminophen ( ac ), Average Case: . . . – ibuprofen ( ib ), and – valium ( va ). Title Page ◭◭ ◮◮ • Binary search: A (Λ) ◭ ◮ ւ ց Page 8 of 17 A (0) A (1) Go Back ւ ց ւ ց Full Screen A (00) A (01) A (10) A (11) Close as ac ib va Quit
8. Example 2: A Search Procedure Which Is Better How Knowledge Is . . . Than Binary ( W 0 > W 1 ) Experts Are Usually . . . Problem • When W 1 = 1 and W 0 = 3, the effort of the binary Description of the . . . search is 6. Asymptotically . . . • We can decrease the effort to 5 by applying the follow- Average Case . . . ing alternative procedure: Average Case: . . . A (Λ) Title Page ւ ց ◭◭ ◮◮ A (0) A (1) ◭ ◮ ւ ց as Page 9 of 17 A (10) A (11) Go Back ւ ց ac Full Screen A (110) A (111) Close ib va Quit
9. Description of the Optimal Search Procedure How Knowledge Is . . . Experts Are Usually . . . • Auxiliary result: Problem Description of the . . . T ( N ) = 0 <N + <N { max { W 1 + T ( N + ) , W 0 + T ( N − N + ) }} . min Asymptotically . . . Average Case . . . • Conclusion: we can consequently compute T (1), T (2), . . . , T ( N ) in time N · O ( N ) = O ( N 2 ). Average Case: . . . • Notation: let N + ( N ) be the value where the minimum Title Page is attained. ◭◭ ◮◮ • Optimal procedure: for each sequence ω with ◭ ◮ def n = | A ( ω ) | > 1: Page 10 of 17 – we assign N + ( n ) values to the “yes” case A ( ω 1); Go Back – we assign the remaining n − N + ( n ) values to the Full Screen “no” case A ( ω 0). Close Quit
10. Example: N = 4 , W 0 = 3 , and W 1 = 1 How Knowledge Is . . . Experts Are Usually . . . • We take T (1) = 0. Then, Problem T (2) = 0 <N + < 2 { max { 1 + T ( N + ) , 3 + T (2 − N + ) }} = min Description of the . . . Asymptotically . . . max { 1+ T (1) , 3+ T (1) } = max { 1 , 3 } = 3 , with N + (2) = 1 . Average Case . . . • T (3) = 4, with min attained for N + (3) = 2. Average Case: . . . • T (4) = 5, with min attained for N + (4) = 3. Title Page • Optimal procedure: ◭◭ ◮◮ – since N + (4) = 3, we divide 4 elements A (Λ) into a ◭ ◮ 3-element set A (1) and a 1-element set A (0); Page 11 of 17 – since N + (3) = 2, we divide 3 elements A (1) into a Go Back 2-element set A (11) and a 1-element set A (10); Full Screen – since N + (2) = 1, we divide 2 elements A (10) into a 1-element set A (101) and a 1-element set A (100). Close • Observation: this is the procedure from Example 2. Quit
11. Asymptotically Optimal Search Procedure How Knowledge Is . . . Experts Are Usually . . . • We described: optimal search procedure: Problem def Description of the . . . E ( P opt ) = T ( N ) = min E ( P ) . P Asymptotically . . . Average Case . . . • Property: P opt takes time ≈ N 2 . Average Case: . . . • Problem: for large N , time N 2 is too large. Title Page • Alternative: asymptotically optimal procedure, with ◭◭ ◮◮ E ( P a ) ≤ T ( N ) + C for some constant C > 0. ◭ ◮ • Asymptotically optimal search procedure: Page 12 of 17 – find α such that α + α w = 1, where w def = W 0 /W 1 ; Go Back def – for each ω with n = | A ( ω ) | > 1, assign ⌊ α · n ⌋ Full Screen values to the “yes” case A ( ω 1); Close – assign the remaining values to the “no” case A ( ω 0). Quit
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