A general framework for measuring inconsistency through minimal inconsistent sets

Hunter and Konieczny explored the relationships between measures of inconsistency for a belief base and the minimal inconsistent subsets of that belief base in several of their papers. In particular, an inconsistency value termed MIVC, defined from minimal inconsistent subsets, can be considered as a Shapley Inconsistency Value. Moreover, it can be axiomatized completely in terms of five simple axioms. MinInc, one of the five axioms, states that each minimal inconsistent set has the same amount of conflict. However, it conflicts with the intuition illustrated by the lottery paradox, which states that as the size of a minimal inconsistent belief base increases, the degree of inconsistency of that belief base becomes smaller. To address this, we present two kinds of revised inconsistency measures for a belief base from its minimal inconsistent subsets. Each of these measures considers the size of each minimal inconsistent subset as well as the number of minimal inconsistent subsets of a belief base. More specifically, we first present a vectorial measure to capture the inconsistency for a belief base, which is more discriminative than MIVC. Then we present a family of weighted inconsistency measures based on the vectorial inconsistency measure, which allow us to capture the inconsistency for a belief base in terms of a single numerical value as usual. We also show that each of the two kinds of revised inconsistency measures can be considered as a particular Shapley Inconsistency Value, and can be axiomatically characterized by the corresponding revised axioms presented in this paper.

Measuring conflict and agreement between two prioritized knowledge bases in possibilistic logic.

In this paper we investigate the relationship between two prioritized knowledge bases by measuring both the conflict and the agreement between them.First of all, a quantity of conflict and two quantities of agreement are defined. The former is shown to be a generalization of the well-known Dalal distance which is the hamming distance between two interpretations. The latter are, respectively, a quantity of strong agreement which measures the amount ofinformation on which two belief bases “totally” agree, and a quantity of weak agreement which measures the amount of information that is believed by onesource but is unknown to the other. All three quantity measures are based on the weighted prime implicant, which represents beliefs in a prioritized belief base. We then define a degree of conflict and two degrees of agreement based on our quantity of conflict and quantities of agreement. We also consider the impact of these measures on belief merging and information source ordering.

A Syntax-based approach to measuring the degree of inconsistency for belief bases

Measuring the degree of inconsistency of a belief base is an important issue in many real world applications. It has been increasingly recognized that deriving syntax sensitive inconsistency measures for a belief base from its minimal inconsistent subsets is a natural way forward. Most of the current proposals along this line do not take the impact of the size of each minimal inconsistent subset into account. However, as illustrated by the well-known Lottery Paradox, as the size of a minimal inconsistent subset increases, the degree of its inconsistency decreases. Another lack in current studies in this area is about the role of free formulas of a belief base in measuring the degree of inconsistency. This has not yet been characterized well. Adding free formulas to a belief base can enlarge the set of consistent subsets of that base. However, consistent subsets of a belief base also have an impact on the syntax sensitive normalized measures of the degree of inconsistency, the reason for this is that each consistent subset can be considered as a distinctive plausible perspective reflected by that belief base,whilst eachminimal inconsistent subset projects a distinctive viewof the inconsistency. To address these two issues,we propose a normalized framework formeasuring the degree of inconsistency of a belief base which unifies the impact of both consistent subsets and minimal inconsistent subsets. We also show that this normalized framework satisfies all the properties deemed necessary by common consent to characterize an intuitively satisfactory measure of the degree of inconsistency for belief bases. Finally, we use a simple but explanatory example in equirements engineering to illustrate the application of the normalized framework.

Measuring the blame of each formula for inconsistent prioritized knowledge bases.

It is increasingly recognized that identifying the degree of blame or responsibility of each formula for inconsistency of a knowledge base (i.e. a set of formulas) is useful for making rational decisions to resolve inconsistency in that knowledge base. Most current techniques for measuring the blame of each formula with regard to an inconsistent knowledge base focus on classical knowledge bases only. Proposals for measuring the blames of formulas with regard to an inconsistent prioritized knowledge base have not yet been given much consideration. However, the notion of priority is important in inconsistency-tolerant reasoning. This article investigates this issue and presents a family of measurements for the degree of blame of each formula in an inconsistent prioritized knowledge base by using the minimal inconsistent subsets of that knowledge base. First of all, we present a set of intuitive postulates as general criteria to characterize rational measurements for the blames of formulas of an inconsistent prioritized knowledge base. Then we present a family of measurements for the blame of each formula in an inconsistent prioritized knowledge base under the guidance of the principle of proportionality, one of the intuitive postulates. We also demonstrate that each of these measurements possesses the properties that it ought to have. Finally, we use a simple but explanatory example in requirements engineering to illustrate the application of these measurements. Compared to the related works, the postulates presented in this article consider the special characteristics of minimal inconsistent subsets as well as the priority levels of formulas. This makes them more appropriate to characterizing the inconsistency measures defined from minimal inconsistent subsets for prioritized knowledge bases as well as classical knowledge bases. Correspondingly, the measures guided by these postulates can intuitively capture the inconsistency for prioritized knowledge bases.

Fruits and vegetables:measuring intake and encouraging increased consumption

A high intake of fruit and vegetables (FV) is associated with reduced risk of chronic disease, although the evidence base is mostly observational. Blood biomarkers offer an objective indicator of FV intake, potentially improving estimates of intakes based on traditional methods. A valid biomarker of overall FV intake would be able to confirm population intakes, more precisely evaluate the association between intakes and health outcomes and confirm compliance in FV interventions. Several substances have been proposed as biomarkers of FV intake: vitamin C, the carotenoids and polyphenols. Certain biomarkers are strong predictors of single FV; however, the proposed single biomarkers of FV consumption are only modestly predictive of overall FV consumption. This is likely to be due to the complexity of the FV food group. While accurately measuring FV intake is important in nutrition research, another critical question is: how best can an increase in FV intake be achieved? Increased FV intake has been achieved in efficacy studies using intensive dietary advice. Alternative, less intensive methods for encouraging FV consumption need to be developed and tested for population level intervention. Systematic reviews suggest peer support to be an effective strategy to promote dietary change. This review will describe the evidence for a link between increased FV intake and good health, outline possible novel biomarkers of FV consumption, present the most recently available data on population intake of FV and examine the usefulness of different approaches to encourage increased consumption of FV.

Computational approaches to finding and measuring inconsistency in arbitrary knowledge bases

There is extensive theoretical work on measures of inconsistency for arbitrary formulae in knowledge bases. Many of these are defined in terms of the set of minimal inconsistent subsets (MISes) of the base. However, few have been implemented or experimentally evaluated to support their viability, since computing all MISes is intractable in the worst case. Fortunately, recent work on a related problem of minimal unsatisfiable sets of clauses (MUSes) offers a viable solution in many cases. In this paper, we begin by drawing connections between MISes and MUSes through algorithms based on a MUS generalization approach and a new optimized MUS transformation approach to finding MISes. We implement these algorithms, along with a selection of existing measures for flat and stratified knowledge bases, in a tool called mimus. We then carry out an extensive experimental evaluation of mimus using randomly generated arbitrary knowledge bases. We conclude that these measures are viable for many large and complex random instances. Moreover, they represent a practical and intuitive tool for inconsistency handling.

Measuring the health-related quality of life of people with ischaemic heart disease.

Objectives—To inform researchers and clinicians about the most appropriate generic and disease specific measures of health related quality of life for use among people with ischaemic heart disease. Methods—MEDLINE and BIDS were searched for research papers which contained a report of at least one of the three most common generic instruments or at least one of the five disease specific instruments used with ischaemic heart disease patients. Evidence for the validity, reliability, and sensitivity of these instruments was critically appraised. Results—Of the three generic measures—the Nottingham health profile, sickness impact profile, and short form 36 (SF-36)—the SF-36 appears to offer the most reliable, valid, and sensitive assessment of quality of life. However, a few of the SF-36 subscales lack a sufficient degree of sensitivity to detect change in a patient’s clinical condition. According to the best available evidence, the quality of life after myocardial infarction questionnaire should be preferred to the Seattle angina questionnaire, the quality of life index cardiac version, the angina pectoris quality of life questionnaire, and the summary index. Overall, research on disease specific measures is sparse compared to the number of studies which have investigated generic measures. Conclusions—An assessment of the quality of life of people with ischaemic heart disease should comprise a disease specific measure in addition to a generic measure. The SF-36 and the quality of life after myocardial infarction questionnaire (version 2) are the most appropriate currently available generic and disease specific measures of health related quality of life, respectively. Further research into the measurement of health related quality of life of people with ischaemic heart disease is required in order to address the problems (such as lack of sensitivity to detect change) identified by the review.

Generic, disease-specific and individualised approaches to measuring health-related quality of life among people with heart disease - a comparative analysis.

Increasing emphasis is being placed on the evaluation of health-related quality of life. However, there is no consensus on the definition of this concept and as a result there are a plethora of existing measurement instruments. Head-to-head comparisons of the psychometric properties of existing instruments are necessary to facilitate evidence-based decisions about which instrument should be chosen for routine use. Therefore, an individualised instrument (the modified Patient Generated Index), a generic instrument (the Short Form 36) and a disease-specific instrument (the Quality of Life after Myocardial Infarction questionnaire) were administered to patients with ischaemic heart disease (n=117) and the evidence for the validity, reliability and sensitivity of each instrument was examined and compared. The modified Patient Generated Index compared favourably with the other instruments but none of the instruments examined provided sound evidence for sensitivity to change. Therefore, any recommendation for the use of the individualised approach in the routine collection of health-related quality of life data in clinical practice must be conditional upon the submission of further evidence to support the sensitivity of such instruments.