Jean-Claude Falmagne

Spaces for the assessment of knowledge

The information regarding a particular field of knowledge is conceptualized as a large, specified set of questions (or problems). The knowledge state of an individual with respect to that domain is formalized as the subset of all the questions that this individual is capable of solving. A particularly appealing postulate on the family of all possible knowledge states is that it is closed under arbitrary unions. A family of sets satisfying this condition is called a knowledge space. Generalizing a theorem of Birkhoff on partial orders, we show that knowledge spaces are in a one-to-one correspondence with AND/OR graphs of a particular kind. Two types of economical representations of knowledge spaces are analysed: bases, and Hasse systems, a concept generalizing that of a Hasse diagram of a partial order. The structures analysed here provide the foundation for later work on algorithmic procedures for the assessment of knowledge.

A representation theorem for finite random scale systems

Abstract This paper investigates necessary and sufficient conditions on choice probabilities Pa,B (of picking an element a in an offered set B), for the existence of random variables Ua, satisfying the equation Pa,B = P {Ua = max {Ub | b ∈ B}} for all nonempty finite subsets B in a fixed set A, and all a ∈ B. A complete solution to this representation problem is obtained in the case where A is finite. The proof of the representation theorem provides an algorithm to construct the random variables Ua, up to some uniqueness properties. Investigation of these uniqueness properties show that an important part of the ordinal structure of the underlying random variables can be recovered.

On realizable biorders and the biorder dimension of a relation

Abstract The paper discusses the mathematical foundations of a technique of multidimensional scaling, generalizing Guttman scaling, in which the structure of the embedding space relies only on ordinal concepts. An empirical relation is represented as an intersection of a minimal number (called bidimension) of Guttman relations. Fairly complete results are given for the cases of bidimensions 1 and 2. In the general case, the main results are based on the equivalence between the bidimension and the dimension of a certain partial order. A characterization of the bidimension as the chromatic number of some hypergraph is also provided.

Statistical issues in measurement

Abstract Measurement theories are traditionally couched in algebraic terms, which makes them unsuitable for statistical testing. A probabilistic recasting of these theories is proposed here. It is observed then that an axiom of probabilistic measurement has typically the form of a logical polynomial, the structure of which induces a particular partition of the parameter space, giving rise to a calss of statistical problems for which the null hypothesis is a union of convex polyhedrons. This is a consequence of the fact that a logical polynomial can always be rewritten in normal form, that is, as a disjunction of conjunctions. A likelihood ratio method is worked out in a couple of exemplary cases. One of these examples provides a test of transitivity, a property which lies at the heart of ordinal measurement.

Knowledge assessment

Abstract The QUERY procedure is designed to systematically question an expert, and construct the unique knowledge space consistent with the experts responses. Such a knowledge space can then serve as the core of a knowledge assessment system. The essentials of the theory of knowledge spaces are given here, together with the theoretical underpinnings of the QUERY procedure. A full scale application of the procedure is then described, which consists in constructing the knowledge spaces of five expert-teachers, pertaining to 50 mathematics items of the standard high school curriculum. The results show that the technique is applicable in a realistic setting. However, the analysis of the data indicates that, despite a good agreement across experts concerning item difficulty and other coarse measures, the constructed knowledge spaces obtained for the different experts are not as consistent as one might expect or hope. Some experts appear to be considerably more skillful than others at generating a usable knowledge space, at least by this technique.

Well-graded families of relations

Any semiorder on a finite set can be reached from any other semiorder on the same set by elementary steps consisting either in the addition or in the removal of a single ordered pair, in such a way that only semiorders are generated at every step, and also that the number of steps equals the distance between the two semiorders. Similar results are also established for other families of relations (partial orders, biorders, interval orders). These combinatorial results are used in another paper to develop a stochastic theory describing the emergence and the evolution of preference relations (Falmagne and Doignon, [7]).

A polynomial time algorithm for unidimensional unfolding representations

Abstract Two conditions on a collection of simple orders - unimodality and straightness - are necessary but not jointly sufficient for unidimensional unfolding representations. From the analysis of these conditions, a polynomial time algorithm is derived for the testing of unidimensionality and for the construction of a representation when one exists.

Random utility representation of binary choice probabilities: a new class of necessary conditions

Abstract Marshalk conjectured that binary, symmetric choice probabilities Pij had a representation P ij = P{U i ⩾ U j } (i≠j), where Ui, Uj are random variables, if and only if P ij + P jk + P ki ⩽ 2. Starting from a counterexample of McFadden and Richter ((1970), unpublished manuscript), this note establishes the necessity of an additional condition.

On a class of probabilistic conjoint measurement models: Some diagnostic properties

Abstract Let P ax , by be the probability of picking the two-dimensional object ax in the set { ax, by }. This paper investigates necessary and/or sufficient conditions for a number of representations for these probabilities: ( f , g , F , K , k , G , u , and v are functions) 1. 1. P ax,by ≤ 1 2 iff f ( a ) + g ( x ) ≤ f ( b ) + g ( y ), 2. 2. P ax , by = F [ f ( a ) + g ( x ), f ( b ) + g ( y )], 3. 3. P ax , by = K { k [ f ( a ) + g ( x )] − k [ f ( b ) + g ( y )]}, 4. 4. P ax , by = K [ u ( a ) v ( x ) − u ( b ) v ( y )], 5. 5. P ax,by = G[(f(a) + g(x)) (f(b) + g(y))] . The results concerning 1, 2, and 3 are essentially trivial consequences of known facts. The results concerning 4 and 5 are new, and amount to representation theorems for these equations. The cases in which k is a convex or concave function in 3 (of which 4 and 5 are special cases) are also analyzed.

Languages for the assessment of knowledge

Abstract Any element S in a family ψ of subsets of a finite set X can be specified by a sequence of statements such as: x ∈ S, y ∉ S, t ∉ S,…, z ϵ S. This sequence can be coded as a “word” x y t …z and a complete set of such words forms a “descriptive language” for the family ψ. This class of languages is defined precisely, and some connections between such languages and families of sets are investigated. It is shown in particular that when ψ is closed under intersections and unions, and satisfies the topological condition known as T0, then ψ can be recovered exactly from any of its descriptive languages. These results have an application in the assessment of knowledge. In this framework, the set X is a set of questions, and any set S ∈ ψ represents a possible knowledge state, containing all the questions that some individual is capable of solving. A subclass of the descriptive languages are then the “assessment languages.” Any such language defines a nonredundant algorithm for determining the knowledge state of any individual.