How Can We Elicit . . . Problem Probabilistic . . . From Frequencies to a . . . Relation Between Polling Quantum Computing: . . . and Likert-Scale Approaches Superposition and Qubits Resulting Relation . . . to Eliciting Membership Superposition . . . Fuzzy Interpretation of . . . Degrees Clarified by Home Page Quantum Computing Title Page ◭◭ ◮◮ Renata Hax Sander Reiser 1 , Adriano Maron 1 ◭ ◮ Lidiana Visintin 1 , Ana Maria Abeijon 2 , Vladik Kreinovich 3 1 Universidade Federal de Pelotas, Brazil Page 1 of 17 { reiser, akmaron, lvisintin } @inf.ufpel.edu.br Go Back 2 Universidade Cat´ olica de Pelotas, Brazil anabeijon@terra.com.br Full Screen 3 University of Texas at El Paso, USA vladik@utep.edu Close Quit
How Can We Elicit . . . Problem 1. How Can We Elicit Membership Degrees? Probabilistic . . . • One of these methods is polling: we ask several experts From Frequencies to a . . . whether, e.g., a 1 cm blemish is small or not. Quantum Computing: . . . Superposition and Qubits • If 7 out of 10 experts say “small”, we assign a degree Resulting Relation . . . 7 / 10 = 0 . 7 to the statement “a 1 cm blemish is small”. Superposition . . . • In general, if m out of n experts agree with the state- Fuzzy Interpretation of . . . ment, we assign it a degree of certainty m/n . Home Page • When we only have one expert, we cannot use polling. Title Page • We can ask the expert to mark the degree of certainty ◭◭ ◮◮ in this statement on a scale, e.g., from 0 to 10. ◭ ◮ • Such scales are known as Likert scales . Page 2 of 17 • If the expert selects 7 on a scale from 0 to 10, we assign, Go Back to this statement, a degree 7 / 10 = 0 . 7. Full Screen • If the expert marks m on a scale from 0 to n , we assign a degree of certainty m/n . Close Quit
How Can We Elicit . . . Problem 2. Problem Probabilistic . . . • Both above elicitation methods are reasonable, both From Frequencies to a . . . lead to reasonable useful results. Quantum Computing: . . . Superposition and Qubits • However, usually, these two methods led to different Resulting Relation . . . membership degrees. Superposition . . . • It is therefore reasonable to find out how these different Fuzzy Interpretation of . . . degrees are connected. Home Page • Of course, degrees are subjective. Title Page • In general, different experts assign different Likert-scale ◭◭ ◮◮ degrees of certainty to the same statement. ◭ ◮ • We thus cannot expect an exact one-to-one correspon- Page 3 of 17 dence between the polling and Likert-scale degrees. Go Back • What we want to discover is an approximate relation Full Screen between the corresponding scales. Close Quit
How Can We Elicit . . . Problem 3. Probabilistic Description of Polling Uncertainty Probabilistic . . . • Our main objective in describing the expert’s knowl- From Frequencies to a . . . edge is to use it. Quantum Computing: . . . Superposition and Qubits • For example, we want to know whether a 1 cm blemish Resulting Relation . . . is small or not because: Superposition . . . – one cure is proposed for a small blemish, Fuzzy Interpretation of . . . – another for a large one. Home Page • A doctor on whose patient with a 1 cm blemish the Title Page small-blemish cure worked will vote “small”. ◭◭ ◮◮ • A doctor on whose patient it didn’t work will vote ◭ ◮ “No”. Page 4 of 17 • The polling ratio m/n is equal to the frequency with Go Back which the small-blemish cure cures a 1 cm blemish. Full Screen Close Quit
How Can We Elicit . . . Problem 4. From Frequencies to a Likert Scale: Main Idea Probabilistic . . . • If on average, P -method works on a half on x -objects, From Frequencies to a . . . it does not mean that we always get µ P ( x ) = 1 / 2. Quantum Computing: . . . Superposition and Qubits • We may get µ P ( x ) < 1 / 2 or µ P ( x ) > 1 / 2. Resulting Relation . . . • Usually, frequencies 0 /N and 1 /N may come from the Superposition . . . same probability p = p ′ . Fuzzy Interpretation of . . . • Similarly, 0 /N and 2 /N may come from the same prob. Home Page • Eventually, we reach m 1 for which f 0 = 0 and f 1 = Title Page m 1 /N cannot come from the same prob. ◭◭ ◮◮ • By repeating this procedure, we get a sequence of dis- ◭ ◮ tinguishable frequencies f 0 < f 1 < f 2 < . . . Page 5 of 17 • This is exactly what a Likert scale is about: Go Back – we have a finite number of possible estimates, and Full Screen – to each situation, we place into correspondence one of these estimates. Close Quit
How Can We Elicit . . . Problem 5. From Probabilities to a Likert Scale: Details Probabilistic . . . • The observed frequency is f = p + ∆ p , where From Frequencies to a . . . � p (1 − p ) Quantum Computing: . . . E [∆ p ] = 0 and σ [∆ p ] = . N Superposition and Qubits • If p = p ′ for two frequenices f � = f ′ , then Resulting Relation . . . � 2 p (1 − p ) Superposition . . . f − f ′ = ∆ p − ∆ p ′ , with σ [ f − f ′ ] = . N Fuzzy Interpretation of . . . Home Page • In statistics, such value is guaranteed to be different from 0 if | f − f ′ | ≥ k 0 · σ (for k 0 = 2 , 3 , or 6). Title Page � ◭◭ ◮◮ 2 f k (1 − f k ) • Thus, f k +1 − f k = k 0 · . N ◭ ◮ • For Likert-scale memb. f-n, µ ( f k ) = k Page 6 of 17 n , hence Go Back n = µ ( f k +1 ) − µ ( f k ) ≈ µ ′ ( f k ) · ( f k +1 − f k ) = µ ′ ( f k ) · k 0 · 2 f k (1 − f k ) 1 . Full Screen N Close Quit
How Can We Elicit . . . Problem 6. From Probabilities to a Likert Scale (cont-d) Probabilistic . . . n = µ ( f k +1 ) − µ ( f k ) ≈ µ ′ ( f k ) · ( f k +1 − f k ) = µ ′ ( f k ) · k 0 · 2 f k (1 − f k ) 1 From Frequencies to a . . . . N Quantum Computing: . . . c Superposition and Qubits • Thus, we get µ ′ ( f ) = . � f (1 − f ) Resulting Relation . . . • Solving this differential equation, we get f = sin 2 ( C · µ ). Superposition . . . Fuzzy Interpretation of . . . • The absolute confidence µ = 1 corresponds to f = 1, Home Page hence f ≈ sin 2 � π � Title Page 2 µ . ◭◭ ◮◮ • At first glance, this relation looks very mathematical ◭ ◮ and non-intuitive. Page 7 of 17 • We will show that it becomes much clearer if we use Go Back the techniques of quantum computing. Full Screen Close Quit
How Can We Elicit . . . Problem 7. Quantum Computing: Reminder Probabilistic . . . • In classical physics: From Frequencies to a . . . Quantum Computing: . . . – if we want to look for an element in an unsorted Superposition and Qubits array of n elements, Resulting Relation . . . – then we need at least n computational steps. Superposition . . . • If we use fewer steps, we will not look into all n cells Fuzzy Interpretation of . . . and thus, we may miss the desired element. Home Page • In quantum case, we can perform the search in √ n Title Page steps (and √ n ≪ n ). ◭◭ ◮◮ • This possibility comes from the fact that in quantum ◭ ◮ physics: Page 8 of 17 – in addition to the usual classical states, Go Back – we can also have superpositions of these states. Full Screen Close Quit
How Can We Elicit . . . Problem 8. Superposition and Qubits Probabilistic . . . • For a qubit (quantum bit), superposition is a state From Frequencies to a . . . a 0 � 0 | + a 1 � 1 | , where a i are complex numbers. Quantum Computing: . . . Superposition and Qubits • In quantum computing, only real values of a 0 and a 1 Resulting Relation . . . are used. Superposition . . . • Each such state can be described as a vector with co- Fuzzy Interpretation of . . . ordinates ( a 0 , a 1 ) in a 2-D vector space. Home Page • The probability p i of observing i is equal to a 2 i . Title Page • Since we always observe either 0 or 1, we must always ◭◭ ◮◮ have p 0 + p 1 = a 2 0 + a 2 1 = 1. ◭ ◮ • In geometric terms, this means that the vector ( a 0 , a 1 ) Page 9 of 17 must be on the unit circle with a center at 0. Go Back • Each such vector is uniquely described by its angle ϕ Full Screen with the � 0 | -axis: a 1 = sin( ϕ ), a 0 = cos( ϕ ). Close Quit
How Can We Elicit . . . Problem 9. Resulting Relation Between Polling and Likert- Probabilistic . . . Scale Degrees From Frequencies to a . . . • For each probability p , we can form a qubit state √ p � 1 | + Quantum Computing: . . . √ 1 − p � 0 | corresponding to this probability. Superposition and Qubits Resulting Relation . . . • For this state, p = a 2 1 = sin 2 ( ϕ ). Superposition . . . • Due to the above relation between frequencies and Likert- Fuzzy Interpretation of . . . scale values, we have p ≈ f ≈ sin 2 � π � 2 µ . Home Page • Thus, we have sin 2 ( ϕ ) ≈ sin 2 � π � Title Page 2 µ , hence ◭◭ ◮◮ ϕ ≈ π 2 µ. ◭ ◮ Page 10 of 17 • So, the Likert-scale degree µ can be geometrically inter- preted as (prop. to) the angle between the two states: Go Back µ ≈ 2 Full Screen πϕ. Close Quit
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