How to Assign Numerical 2- and 3-Element Sets Values to Partially - PowerPoint PPT Presentation

Need to Assign . . . How to Assign? Robustness as a . . . What Is Known: Case . . . How to Assign Numerical 2- and 3-Element Sets Values to Partially Ordered Case of 2 Minimal . . . Case of a Single . . . Levels of Confidence: 4 Elements, 3 or 4 . . . 4 Elements, 2 Minimal . . . Robustness Approach Home Page Title Page Kimberly Kato ◭◭ ◮◮ Department of Computer Science ◭ ◮ University of Texas at El Paso 500 W. University Page 1 of 18 El Paso, Texas 79968, USA kekato@miners.utep.edu Go Back Full Screen Close Quit

Need to Assign . . . How to Assign? 1. Outline Robustness as a . . . • In many practical situations, expert’s levels of confi- What Is Known: Case . . . dence are described by words from natural language. 2- and 3-Element Sets Case of 2 Minimal . . . • These words are often only partially ordered. Case of a Single . . . • Computers are much more efficient in processing num- 4 Elements, 3 or 4 . . . bers than words. 4 Elements, 2 Minimal . . . • So, it is desirable to assign numerical values to these Home Page degrees. Title Page • Of course, there are many possible assignments that ◭◭ ◮◮ preserve order between words. ◭ ◮ • It is reasonable to select an assignment which is the Page 2 of 18 most robust, i.e., for which Go Back – the largest possible deviation – still preserves the order. Full Screen • In this talk, we analyze cases of up to 4 different words. Close Quit

Need to Assign . . . How to Assign? 2. Need to Assign Numerical Values to Levels of Robustness as a . . . Confidence What Is Known: Case . . . • In many cases, it is desirable to describe experts’ 2- and 3-Element Sets knowledge in a computer-understandable form. Case of 2 Minimal . . . Case of a Single . . . • Experts are often not 100% confident in their state- 4 Elements, 3 or 4 . . . ments. 4 Elements, 2 Minimal . . . • The corresponding degrees of confidence are an impor- Home Page tant part of their knowledge. Title Page • It is therefore desirable to describe these levels of con- ◭◭ ◮◮ fidence in a computer-understandable form. ◭ ◮ • Experts often describe their levels of confidence by us- Page 3 of 18 ing words such as “most probably”, “usually”, etc. Go Back • Computers are much more efficient when they process numbers than when they process words. Full Screen Close Quit

Need to Assign . . . How to Assign? 3. Need to Assign Numerical Values (cont-d) Robustness as a . . . • So, it is desirable to describe these levels of confidence What Is Known: Case . . . by numbers. 2- and 3-Element Sets Case of 2 Minimal . . . • In other words, it is desirable to assign numerical values Case of a Single . . . to different levels of certainty. 4 Elements, 3 or 4 . . . • These numerical values are usually selected from the 4 Elements, 2 Minimal . . . interval [0 , 1], so that: Home Page • 1 corresponds to complete certainty, and Title Page • 0 to full certainty that the statement is true. ◭◭ ◮◮ • There is an order ≺ between levels, with a ≺ b meaning ◭ ◮ that level b corresponds to higher confidence than a . Page 4 of 18 • This order is often partial: Go Back • There exist levels a and b for which it is not clear which Full Screen of them corresponds to higher confidence. Close Quit

Need to Assign . . . How to Assign? 4. Need to Assign Numerical Values (cont-d) Robustness as a . . . • It is reasonable to assign degree in such a way that: What Is Known: Case . . . 2- and 3-Element Sets – if a ≺ b , Case of 2 Minimal . . . – then the degree assigned to level b is larger than Case of a Single . . . the degree assigned to level a . 4 Elements, 3 or 4 . . . • Also, all assigned degrees should be strictly between 0 4 Elements, 2 Minimal . . . and 1 – since Home Page – they describe different levels of certainty, Title Page – not absolute certainty. ◭◭ ◮◮ ◭ ◮ Page 5 of 18 Go Back Full Screen Close Quit

Need to Assign . . . How to Assign? 5. Notations Robustness as a . . . • For simplicity, let us number all levels by 1, 2, . . . , n . What Is Known: Case . . . 2- and 3-Element Sets • To these levels, we add ideal levels 0 (absolutely false) Case of 2 Minimal . . . and n + 1 (absolutely true), for which Case of a Single . . . 0 ≺ i ≺ n + 1 for all i from 1 to n. 4 Elements, 3 or 4 . . . 4 Elements, 2 Minimal . . . • Let us denote the numerical value assigned to the i -th Home Page level by n i ∈ [0 , 1]. Title Page • In these terms, our requirement means that i ≺ j im- ◭◭ ◮◮ plies n i < n j . ◭ ◮ Page 6 of 18 Go Back Full Screen Close Quit

Need to Assign . . . How to Assign? 6. How to Assign? Robustness as a . . . • There are many way to assign numbers to levels. What Is Known: Case . . . 2- and 3-Element Sets • For example: Case of 2 Minimal . . . – if we have n = 2 levels with 1 ≺ 2, Case of a Single . . . – then possible assignments are possible tuples 4 Elements, 3 or 4 . . . ( n 0 , n 1 , n 2 , n 3 ) for which 4 Elements, 2 Minimal . . . Home Page n 0 < n 1 < n 2 < n 3 . Title Page • Of course, there are many such tuples. ◭◭ ◮◮ • Which of the possible assignments should we select? ◭ ◮ Page 7 of 18 Go Back Full Screen Close Quit

Need to Assign . . . How to Assign? 7. Robustness as a Possible Criterion Robustness as a . . . • Computers are approximate machines. What Is Known: Case . . . 2- and 3-Element Sets • The higher accuracy we need: Case of 2 Minimal . . . – the more digits we should keep in our computa- Case of a Single . . . tions, and thus, 4 Elements, 3 or 4 . . . – the slower are these computations. 4 Elements, 2 Minimal . . . • Therefore, to speed up computations, we would like to Home Page store as few digits as possible. Title Page • So, we approximate the original values. ◭◭ ◮◮ • We want to make sure that this approximation pre- ◭ ◮ serves the order, i.e., that: Page 8 of 18 – if we replace the original values n i with approxi- Go Back mate values n ′ i for which | n ′ i − n i | ≤ ε , Full Screen – we will still have the same order between the new values n ′ i as between the old values. Close Quit

Need to Assign . . . How to Assign? 8. Robustness as a Possible Criterion (cont-d) Robustness as a . . . • So, we want the numerical assignment which is, in this What Is Known: Case . . . sense, robust . 2- and 3-Element Sets Case of 2 Minimal . . . • The larger ε , the fewer digits we can keep and thus, Case of a Single . . . the faster the computations. 4 Elements, 3 or 4 . . . • Thus, it is desirable to select the assignment for which 4 Elements, 2 Minimal . . . the robustness ε is the largest possible. Home Page • So, we want to select numbers n i for which: Title Page – i ≺ j implies n ′ i < n ′ j whenever | n ′ i − n i | ≤ ε and ◭◭ ◮◮ | n ′ j − n j | ≤ ε ◭ ◮ – for the largest possible value ε . Page 9 of 18 • If we have two arrangements with the same ε : Go Back – if one of them allows for larger deviations of at least Full Screen one of the values n i than the other one, – then we should select this one. Close Quit

Need to Assign . . . How to Assign? 9. What Is Known: Case of a Linear Order Robustness as a . . . • In our previous work, we have shown that: What Is Known: Case . . . 2- and 3-Element Sets – for the case of linear order, when 1 ≺ 2 ≺ . . . ≺ n , Case of 2 Minimal . . . i – the most robust assignment is n i = n + 1, with the Case of a Single . . . 1 4 Elements, 3 or 4 . . . robustness ε = 2( n + 1) . 4 Elements, 2 Minimal . . . Home Page • In this paper, we extend this result to partially ordered sets with up to 4 elements. Title Page • 1-element set: This case is the easiest, since a 1- ◭◭ ◮◮ element set is, by definition, linearly ordered. ◭ ◮ • So, in this case, we assign n 1 = 1 / 2. Page 10 of 18 Go Back Full Screen Close Quit

Need to Assign . . . How to Assign? 10. 2- and 3-Element Sets Robustness as a . . . • If the two elements are ordered (1 ≺ 2), then we assign What Is Known: Case . . . n 1 = 1 / 3 and n 2 = 2 / 3. 2- and 3-Element Sets Case of 2 Minimal . . . • If the elements are not related, then the most robust Case of a Single . . . assignment is when n 1 = n 2 = 1 / 2. 4 Elements, 3 or 4 . . . • 3-element set. Let us analyze cases based on the num- 4 Elements, 2 Minimal . . . ber of minimal elements – not preceded by others. Home Page • 3 minimal elements: Title Page – the three elements 1, 2, and 3 are unrelated: ◭◭ ◮◮ ◭ ◮ 1 2 3 Page 11 of 18 – so, the most robust assignment is n 1 = n 2 = n 3 = Go Back 1 / 2, with degree of robustness 1/4: Full Screen Close Quit