Qualitative Probability Research Articles

Logic-based updating

We build on AGM belief revision (Alchourrón et al. (1985)) and propose a class of updating rules called pragmatic rules. Pragmatic updating applies to multiple priors and requires that the agent's posteriors be the subset of her priors under which the realized event occurs with probability 1, if such priors exist. We construct a propositional language based on qualitative probability and demonstrate the strong relation between belief updating rules and belief revision rules in this language. We show that an updating rule is consistent with AGM belief revision if and only if it is pragmatic. While maximum likelihood updating is pragmatic in general, full-Bayesian updating is not. We characterize maximum likelihood updating within the AGM framework, and show that full-Bayesian updating can be obtained by dropping one of AGM's postulates.

Pairwise summation as a method for the additive combination of probabilities in qualitative risk assessments.

Qualitative frameworks are widely employed to tackle urgent animal or public health issues when data are scarce and/or urgent decisions need to be made. In qualitative models, the degree of belief regarding the probabilities of the events occurring along the risk pathway(s) and the outcomes is described in nonnumerical terms, typically using words such as Low, Medium, or High. The main methodological challenge, intrinsic in qualitative models, relates to performing mathematical operations and adherence to the rule of probabilities when probabilities are nonnumerical. Although methods to obtain the qualitative probability from the conditional realization of n events are well-established and consistent with the multiplication rule of probabilities, there is a lack of accepted methods for addressing situations where the probability of an event occurring can increase, and the rule of probability P(AUB)=P(A)+P(B)-P(A∩B) should apply. In this work, we propose a method based on the pairwise summation to fill this methodological gap. Our method was tested on two qualitative models and compared by means of scenario analysis to other approaches found in literature. The qualitative nature of the models prevented formal validation; however, when using the pairwise summation, results consistently appeared more coherent with probability rules. Even if the final qualitative estimate can only represent an approximation of the actual probability of the event occurring, qualitative models have proven to be effective in providing scientific-based evidence to support decision-making. The method proposed in this study contributes to reducing the subjectivity that characterizes qualitative models, improving transparency and reproducibility.

Finite Additivity, Complete Additivity, and the Comparative Principle

AbstractIn the longstanding foundational debate whether to require that probability is countably additive, in addition to being finitely additive, those who resist the added condition raise two concerns that we take up in this paper. (1) Existence: Settings where no countably additive probability exists though finitely additive probabilities do. (2) Complete Additivity: Where reasons for countable additivity don’t stop there. Those reasons entail complete additivity—the (measurable) union of probability 0 sets has probability 0, regardless the cardinality of that union. Then probability distributions are discrete, not continuous. We use Easwaran’s (Easwaran, Thought 2:53–61, 2013) advocacy of the Comparative principle to illustrate these two concerns. Easwaran supports countable additivity, both for numerical probabilities and for finer, qualitative probabilities, by defending a condition he calls the Comparative principle [$${\mathcal{C}}$$ C ]. For numerical probabilities, principle $${\mathcal{C}}$$ C contrasts pairs, P1 and P2, defined over a common partition $$\prod$$ ∏ = {ai: i ∈ I} of measurable events. $${\mathcal{C}}$$ C requires that no P1 may be pointwise dominated, i.e., no (finitely additive) probability P2 exists such that for each i ∈ I, P2(ai) > P1(ai). By design, the cardinality of $$\prod$$ ∏ is not limited in $${\mathcal{C}}$$ C , which Easwaran asserts is important when arguing that the principle does not require more, or less, than that probability is countably additive. We agree that a numerical probability P satisfies principle $${\mathcal{C}}$$ C in all partitions just in case P is countably additive. However, we show that for numerical probabilities, by considering the size of the algebra of events to which probability is applied, principle $${\mathcal{C}}$$ C is subject to each of the above concerns, (1) and (2). Also, Easwaran considers principle $${\mathcal{C}}$$ C with non-numerical, qualitative probabilities, where a qualitative probability may be finer than an almost agreeing numerical probability P. A qualitative probability is regular if possible events are strictly more likely than impossible events. Easwaran motivates and illustrates regular qualitative probabilities using a continuous, almost agreeing quantitative probability that is uniform on the unit interval. We make explicit the conditions for applying principle $${\mathcal{C}}$$ C with qualitative probabilities and show that $${\mathcal{C}}$$ C restricts regular qualitative probabilities to those whose almost agreeing quantitative probabilities are completely additive. For instance, Easwaran’s motivating example of a regular qualitative probability is precluded by principle $${\mathcal{C}}$$ C .

Primary-school children’s knowledge of transitivity of probability and of bounds of probability

ABSTRACT Primary-school children aged between 7 and 10 years were questioned about their conceptual knowledge of the first two axioms of qualitative probability. Children were questioned with prespecified examples and it was subsequently evaluated whether their answers were consistent with axioms. In addition, children’s knowledge of qualitative probability was investigated with universal and existential statements about axioms. Children showed good knowledge of probability with prespecified examples. In contrast, majority showed poor knowledge when asked about some of the universal and existential statements. An age-related improvement was not detected. In particular, only 25% said that the transitivity was necessary, i.e. that it applied to all events. 60% said that an event may be both possible and impossible. Only 42% said that all impossible events were equally likely and 54% said that all certain events were equally likely. Only 56% said that probability has both upper and lower bounds.

Fair Infinite Lotteries, Qualitative Probability, and Regularity

AbstractA number of philosophers have thought that fair lotteries over countably infinite sets of outcomes are conceptually incoherent by virtue of violating countable additivity. In this article, I show that a qualitative analogue of this argument generalizes to an argument against the conceptual coherence of a much wider class of fair infinite lotteries—including continuous uniform distributions. I argue that this result suggests that fair lotteries over countably infinite sets of outcomes are no more conceptually problematic than continuous uniform distributions. Along the way, I provide a novel argument for a weak qualitative, epistemic version of regularity.

Foreseeing the Past: Probability and Ancient Greek Decision-Making


 
 
 The article explores the concept of probability in ancient Greece from a non-scientific perspective and shows how ancient decision-makers used historical data to make calculated decisions and speculate about the future. First, the paper considers how quantitative data was used by ancient Greek communities to make economic projections. It then shows how ancient Greek generals used the same conceptual tools to determine their odds of victory by tallying up and comparing the number and composition of armies and resources available to them and their enemy. In the third section, the paper examines how qualitative probability was articulated through the language of hope and likelihood to formulate chances of success in moments of crisis. Finally, the paper shows that ancient decision-makers implemented “power laws” to adapt to changing circumstances and the flow of new information, as they sought to improve their odds of success relative to their rivals.
 
 

Deciding Koopman's qualitative probability

In their recent paper in this journal, Delgrande, Renne and Sack study qualitative probability and discuss its role in Artificial Intelligence. Building on related work by de Finetti, Scott, Segerberg and others, the authors provide general axioms for qualitative probability. In this paper we investigate Koopman's conditional qualitative probability from the computational viewpoint. An important part of his work, published in the Annals of Mathematics in 1940-1941, deals with finite conjunctions K of statements of the form “the probability of a given h does not exceed the probability of b given k”, with a,b,h,k elements of a boolean algebra. Upon coding these elements by boolean formulas, we provide a decision procedure to check the consistency of any such K. As an immediate consequence, also inferences in Koopman's probability theory are shown to be computable. These problems of qualitative probability theory significantly generalize Boole's (typically quantitative) problem of estimating the possible probabilities of a new event given the probabilities of other events. Boole's classical problem today is known as the optimization version of the probabilistic satisfiability problem PSAT. In 1986 Nilsson published an influential paper on this subject in this journal. The scope of our results is much larger than that of PSAT, because Koopman's conjunctions K also formalize the key notion of independence. Some familiarity with boolean logic and finite boolean algebras is the only prerequisite for this paper.

Long-lived stable shelf deposition along Gondwana's southern margin during the Ordovician-Silurian: Inferences from U[sbnd]Pb detrital zircon ages of the Table Mountain Group (South Africa) and correlatives in Argentina and the Falklands/Malvinas Islands

The Ordovician-Silurian siliciclastic units of the Table Mountain Group (TMG; Cape Supergroup) of the Cape Fold Belt (CFB) are classically considered to signify a period of long-lived sedimentation along the southern margin of southern Africa. Despite its vast, prominent outcrop, meaningful interpretations regarding the provenance of the TMG, based on a representative number of UPb detrital zircon age fractions, remains lacking. Within southern Gondwana, the TMG succession is considered broadly time-correlative to the Sierra del Volcan and Balcarce formations of the Sierras Tandilia and lowermost successions of the Ventana Group of the Sierras Australes in Argentina and the Port Stephens Formation of the West Falkland Group (WFG) in the Falkland/Malvinas Islands and Natal Group (South Africa). UPb detrital zircon ages have previously been reported for the Balcarce Formation and Ventana and West Falkland groups. However, their age fractions have not necessarily been compared to the broadly time –equivalent units of the TMG. In addition to the overall scarcity of detrital zircon age data on the TMG, the validity of using conventional, qualitative probability density diagrams to display and interpret such data sets have come into question, with some authors arguing for a quantitative approach for comparing detrital zircon age data sets. A total of 1521 concordant UPb detrital zircon ages and some 330 LuHf isotope analysis for the Peninsula- and Skurweberg formations (TMG) are therefore contributed, from samples obtained throughout the CFB, along with detrital zircon age fractions for the Balcarce, Providencia and Port Stephens formations. A qualitative as well as quantitative approach to compare the UPb detrital zircon age fractions measured for these units were adapted. Noticeable similarities among detrital zircon age fractions of the Peninsula- and Skurweberg formations throughout the western to south-eastern reaches of the CFB could be confirmed by quantitative pairwise comparisons. These point towards very little change in sediment source during the deposition of the lower to upper formations of the TMG during the Ordovician-Silurian. Pairwise comparisons of the TMG zircon age fractions to those of the Balcarce-, Providencia and Port Stephens formations revealed that they all have major late Neoproterozoic to Cambrian and a late Mesoproterozoic to early Neoproterozoic age fractions in common, with some minor contributions from Ordovician-Silurian sources. Due to the overlap in prevalent detrital zircon age fractions as well as Hf-isotope signature of these units, it is difficult to assign a specific provenance area for them with a fair amount of confidence, even if dominant palaeocurrent directions are considered. A quantitative pairwise comparison among data sets of the same or time-equivalent formation did not necessarily allow for age fractions from different source areas to be sufficiently distinguished from each other. In contrast to the classically inferred Namaqua-Natal Metamorphic Complex (NNMC) as source major proto-source for the TMG, it is argued that the sediment sources were located much further to the north such as the Damara and Mozambique belts, given that parts of the NNMC were likely denudated and covered by the sediments such as those of the Nama and Vanrhynsdorp Groups during the Ordovician-Silurian. Older sedimentary successions such as the Nama Group, were likely recycled during the deposition of the TMG: a scenario which also provides a feasible explanation for the noticeable lack of Archean ages among the detrital zircon age fractions. The ages of the youngest detrital zircon fraction conforms to the maximum age of deposition as proposed in literature, although the newly reported ages does not allow for more stringent cross-basin correlations. Regions within Patagonia, considered to have been in relatively close proximity to the relevant depositories, and possibly the Famatinian Belt (Argentina), are considered the source of the youngest Ordovician-Silurian detrital zircons.

On atom-swarming and Luce\u2019s theorem for probabilistic beliefs

For qualitative probability spaces, monotone continuity and third-order atom-swarming are together sufficient for a unique countably additive probability measure representation that may have atoms (Mackenzie in Theor Econ 14:709–778, 2019). We provide a new proof by appealing to a theorem of Luce (Ann Math Stat 38:780–786, 1967), highlighting the usefulness of extensive measurement theory (Krantz et al. in Foundations of Measurement Volume I: Additive and Polynomial Representations. Academic Press, New York, 1971) for economists.

Intersection of REE and Probability Measure Theory: Getting Down to Brass Tacks

This study reviews the literature on modeling of Rational Expectations Equilibriums (REE) with respect to pragmatism of inferences. Regardless of evidence for feasibility or existence of REE, none of studies which assume either of Bayesian updating of information, and/or Borel (Quantitative) Probability Measures provide any pragmatic rubric for modeling of REE. While studies, which proffer econometric (`bounded rationality') approaches to modeling of REE have some pragmatism, all such approaches have robustness only if they are adopted prior to entry into a market. In presence of highlighted restriction, and normative importance of metrics that enable a regaining of REE subsequent to arrival of some perturbation, econometric approaches generate REE that inherently are unstable, as such, are lacking in pragmatism. The evidence shows all studies that pragmatically model REE specify qualitative probabilities over preferences, and explicitly characterize qualitative probabilities as having character of probability measures. Modeling of qualitative probability measures is shown to generate a Borel Space. While quantitative probability measures that facilitate Bayesian updating within Borel Spaces are required, however, to have concave distributions, qualitative probability measures can have either of concave or convex distributions, as such, are robust to any specification of preferences. Importantly, all studies that adopt qualitative probability measures for modeling of REE generate procedures, equivalently, mechanisms that either enable reestablishment of a preceding REE, or tattonement towards a new REE.

Prognostication in post-stroke aphasia: How do speech pathologists formulate and deliver information about recovery?

Prognostication is a complex clinical task that involves forming a prediction about recovery and communicating prognostic information to patients and families. In aphasia, recovery is difficult to predict and evidence-based guidance on prognosis delivery is lacking. Questions about aphasia prognosis commonly arise, but it is unknown how speech pathologists formulate and deliver information about expected recovery. An understanding of current practice in prognostication is needed to develop evidence-based guidelines for this process, and is necessary in order to ensure successful future implementation of recommended practice regarding prognosis delivery. To identify the factors speech pathologists consider important when responding to questions about aphasia prognosis; to examine how they respond in different scenarios; and to evaluate their current attitudes towards aphasia prognostication. A total of 54 speech pathologists participated in an online survey featuring hypothetical aphasia prognosis delivery scenarios, short-response questions and ratings. Open responses were analysed thematically. Multiple-choice responses were analysed using descriptive statistics and non-parametric tests. Speech pathologists regarded factors relating to the nature and severity of post-stroke deficits, an individual's level of motivation and the availability of social support as most important for forming an aphasia prognosis. When delivering prognostic information, considerations of the recipient's emotional well-being, hope and expectations, and comprehension of information were regarded as most important. Speech pathologists' prognosis responses varied in content and manner of communication. The content of the responses included predictions of recovery and information about various attributes and activities contributing to recovery. Prognostic information was most frequently communicated through qualitative probability expressions, general statements and uncertainty-based expressions. A need for more professional support in aphasia prognostication was indicated. There is variation in the way speech pathologists respond to questions about aphasia prognosis, and it is unknown how these conversations affect people with aphasia and their significant others. Further research to understand speech pathologists' clinical reasoning and professional support needs, and the perspectives of people living with aphasia, may help to develop an evidence-based approach to prognostication in aphasia. What this paper adds What is already known on this subject Questions about aphasia prognosis are difficult to answer. Current methods for predicting aphasia recovery are yet to demonstrate the reliability and individual specificity required for clinical application. At present, there is no evidence-based guidance or support for prognosis delivery in aphasia. What this study adds to existing knowledge There is variation in the way speech pathologists predict aphasia recovery, the prognostic information they deliver and the manner in which they communicate prognoses. Current approaches to aphasia prognostication are reliant on experience and professional judgement, and speech pathologists desire more support for undertaking this task. What are the potential or actual clinical implications of this work? Speech pathologists lack a consistent approach to aphasia prognostication and desire more support in this task. The present findings offer insight into clinician attitudes and practice, and provide a direction for future research to establish best-practice guidelines for this complex and demanding area of aphasia management.

Cross-Validation of Generic Risk Assessment Tools for Animal Disease Incursion Based on a Case Study for African Swine Fever.

In recent years, several generic risk assessment (RA) tools have been developed that can be applied to assess the incursion risk of multiple infectious animal diseases allowing for a rapid response to a variety of newly emerging or re-emerging diseases. Although these tools were originally developed for different purposes, they can be used to answer similar or even identical risk questions. To explore the opportunities for cross-validation, seven generic RA tools were used to assess the incursion risk of African swine fever (ASF) to the Netherlands and Finland for the 2017 situation and for two hypothetical scenarios in which ASF cases were reported in wild boar and/or domestic pigs in Germany. The generic tools ranged from qualitative risk assessment tools to stochastic spatial risk models but were all parameterized using the same global databases for disease occurrence and trade in live animals and animal products. A comparison of absolute results was not possible, because output parameters represented different endpoints, varied from qualitative probability levels to quantitative numbers, and were expressed in different units. Therefore, relative risks across countries and scenarios were calculated for each tool, for the three pathways most in common (trade in live animals, trade in animal products, and wild boar movements) and compared. For the 2017 situation, all tools evaluated the risk to the Netherlands to be higher than Finland for the live animal trade pathway, the risk to Finland the same or higher as the Netherlands for the wild boar pathway, while the tools were inconclusive on the animal products pathway. All tools agreed that the hypothetical presence of ASF in Germany increased the risk to the Netherlands, but not to Finland. The ultimate aim of generic RA tools is to provide risk-based evidence to support risk managers in making informed decisions to mitigate the incursion risk of infectious animal diseases. The case study illustrated that conclusions on the ASF risk were similar across the generic RA tools, despite differences observed in calculated risks. Hence, it was concluded that the cross-validation contributed to the credibility of their results.

Şiddetin ve Suçun Kamusallaşması: Reality Showlar ve Toplumsal Etkileri

Şiddet ve suç temalarının izlenme oranlarına olumlu katkısı olduğu varsayımı, söz konusu temaların haber bültenleri, suç dizileri ve reality showlar aracılığıyla izleyicilere sunulmasına yol açmaktadır. Yapılan araştırmalar şiddet ve suç konularının işlendiği bu programların içerik, izlenme oranları, kurgu biçimi, söylem dili ve ifade tarzına bağlı olarak izleyici tutumlarını farklı düzeyde etkilediğini tespit etmiştir. Özellikle kurgu olmayan reality show izleyiciliği söz konusu olduğunda, televizyon, bireyin şiddet eğilimini açığa çıkarabilmekte ancak bu eğilimi pekiştirici rol oynamamaktadır. Öte yandan bu programları takip eden kişide şiddete maruz kalma ve suçun mağduru pozisyonuna düşme korkusu oluşmaktadır. Bu makalede şiddet eğilimi ve şiddet içerikli ve suç teşkil eden davranışlar ile özellikle kurgu olmayan reality showları seyretmek arasındaki ilişki analiz edilmeye çalışılmıştır. Bu ilişkiyi çözümlemek amacıyla öncelikle şiddet içerikli davranışların kökeni değerlendirilmiş, ardından program türleri arasında içerik, yöntem ve etki bağlamında karşılaştırma yapılmış, nihayetinde suç vakalarının konu edildiği, tartışıldığı ve kimi zaman da emniyet güçleri ile işbirliği içinde sonuçlandığı bir mecra olan reality showların risk ve korku bağlamında toplumsal etkileri tespit edilmeye çalışılmıştır. Bu çalışma göstermiştir ki, söz konusu mağduriyet korkusunun gerçekçiliği, şiddet ve suç vakalarına ilişkin sayısal verilerden ziyade mağduriyetin niteliksel olasılığı dikkate alınarak değerlendirilmelidir.

Keynes’s Logical Theory of Probability Is Concerned with Rational Degrees of Belief Based on Cognition, Which Are Not Mysterious or Mystical, Rank Ordered or Qualitative: F P Ramsey Failed to Grasp Keynes’s Use of Non–Additivity in His Reviews of the a Treatise on Probability

Recent exchanges between two academics on the Internet ,both experts in conventional and unconventional approaches to decision theory, reveal an astounding lack of knowledge and understanding regarding the makeup of Keynes’s logical theory of probability as put forth by Keynes in 1921, approximately 100 years ago. Given this lacuna concerning Keynes, even after nearly 100 years has passed by, there is thus a need to restate some basic fundamental facts about Keynes's approach that can only be attained by reading beyond Part I of the A Treatise on Probability or Chapter III of the A Treatise on Probability. Unfortunately, what academics mean when they say that they have read the A Treatise on Probability is that they have read Part I and/or Chapter III of A Treatise on Probability or Chapter III of the A Treatise on Probability. Parts II, III, IV and V are simply skipped or skimmed over, as was done by F P Ramsey in his reviews of 1922 and 1926. Keynes’s logical theory of probability is based on the upper–lower probability approach of George Boole. Boole is the explicit founder of the logical theory of probability that Keynes built on when writing the TP. Keynes called his approach inexact measurement and approximation. Keynes’s interval valued theory was designed, like Boole’s, to explicitly deal with non measurability, non comparability and incommensurability. Qualitative probability, comparative probability, ordinal probability, or rank ordered probability, just like numerical probability, can’t deal with non measurability, non comparability and incommensurability. The failure to absorb this basic point lies at the heart of all Post Keynesian and Fundamentalist Keynesian interpretations that all ignore George Boole’s fundamental contributions to Keynes’s logical theory of probability.

The logic of qualitative probability

In this paper we present a theory of qualitative probability. The usual approach of earlier work was to specify a binary operator ⪯ on formulas with ϕ⪯ψ having the intended interpretation that the event expressed by ϕ is no more probable than that expressed by ψ. We generalise these approaches by extending the domain of the operator ⪯ from the set of events to the set of finite sequences of events. If Φ and Ψ are finite sequences of events, Φ⪯Ψ has the intended interpretation that the combined probabilities of the elements of Φ are no greater than those of Ψ. A sound and complete axiomatisation for this operator over finite outcome sets is given. We argue that our approach is more perspicuous and intuitive than previous accounts. As well, we show that the approach is sufficiently expressive to capture the results of axiomatic probability theory and to encode rational linear inequalities. We also prove that our approach generalises the two major accounts for finite outcome sets.

A foundation for probabilistic beliefs with or without atoms

We propose two novel axioms for qualitative probability spaces: (i) unlikely atoms, which requires that there is an event containing no atoms that is at least as likely as its complement; and (ii) third‐order atom‐swarming, which requires that for each atom, there is a countable pairwise‐disjoint collection of less‐likely events that can be partitioned into three groups, each with union at least as likely as the given atom. We prove that under monotone continuity, each of these axioms is sufficient to guarantee a unique countably‐additive probability measure representation, generalizing work by Villegas to allow atoms. Unlike previous contributions that allow atoms, we impose no cancellation or solvability axiom.

Communication About the Probability of Cancer in Indeterminate Pulmonary Nodules

Clinical guidelines recommend that clinicians estimate the probability of malignancy for patients with indeterminate pulmonary nodules (IPNs) larger than 8 mm. Adherence to these guidelines is unknown. To determine whether clinicians document the probability of malignancy in high-risk IPNs and to compare these quantitative or qualitative predictions with the validated Mayo Clinic Model. Single-institution, retrospective cohort study of patients from a tertiary care Department of Veterans Affairs hospital from January 1, 2003, through December 31, 2015. Cohort 1 included 291 veterans undergoing surgical resection of known or suspected lung cancer from January 1, 2003, through December 31, 2015. Cohort 2 included a random sample of 239 veterans undergoing inpatient or outpatient pulmonary evaluation of IPNs at the hospital from January 1, 2003, through December 31, 2012. Clinician documentation of the quantitative or qualitative probability of malignancy. Documentation from pulmonary and/or thoracic surgery clinicians as well as information from multidisciplinary tumor board presentations was reviewed. Any documented quantitative or qualitative predictions of malignancy were extracted and summarized using descriptive statistics. Clinicians' predictions were compared with risk estimates from the Mayo Clinic Model. Of 291 patients in cohort 1, 282 (96.9%) were men; mean (SD) age was 64.6 (9.0) years. Of 239 patients in cohort 2, 233 (97.5%) were men; mean (SD) age was 65.5 (10.8) years. Cancer prevalence was 258 of 291 cases (88.7%) in cohort 1 and 110 of 225 patients with a definitive diagnosis (48.9%) in cohort 2. Only 13 patients (4.5%) in cohort 1 and 3 (1.3%) in cohort 2 had a documented quantitative prediction of malignancy prior to tissue diagnosis. Of the remaining patients, 217 of 278 (78.1%) in cohort 1 and 149 of 236 (63.1%) in cohort 2 had qualitative statements of cancer risk. In cohort 2, 23 of 79 patients (29.1%) without any documented malignancy risk statements had a final diagnosis of cancer. Qualitative risk statements were distributed among 32 broad categories. The most frequently used statements aligned well with Mayo Clinic Model predictions for cohort 1 compared with cohort 2. The median Mayo Clinic Model-predicted probability of cancer was 68.7% (range, 2.4%-100.0%). Qualitative risk statements roughly aligned with Mayo predictions. Clinicians rarely provide quantitative documentation of cancer probability for high-risk IPNs, even among patients drawn from a broad range of cancer probabilities. Qualitative statements of cancer risk in current practice are imprecise and highly variable. A standard scale that correlates with predicted cancer risk for IPNs should be used to communicate with patients and other clinicians.

The wisdom of collective grading and the effects of epistemic and semantic diversity

A computer simulation is used to study collective judgements that an expert panel reaches on the basis of qualitative probability judgements contributed by individual members. The simulated panel displays a strong and robust crowd wisdom effect. The panel’s performance is better when members contribute precise probability estimates instead of qualitative judgements, but not by much. Surprisingly, it does not always hurt for panel members to interpret the probability expressions differently. Indeed, coordinating their understandings can be much worse.

On J M Keynes's Clear Cut Discussion of Interval Valued Probability in Chapter III of the a Treatise on Probability

In chapter III of the A Treatise on Probability, Keynes made it very clear that ordinal, qualitative probability did not apply in many cases. Further, ordinal probability was very weak as far as providing useful information upon which a decision maker could act. However, this did not mean that probabilities could not be analyzed technically if numerical and ordinal probability were inapplicable. In fact, Keynes’s argument was that, while numerical and ordinal probability could not be applied in many cases, interval valued probability could be used to make judgments about the choice of an option, given alternatives, that would be valuable to the decision maker. Of course, a problem would occur if the intervals were to overlap. Interval valued probabilities that overlapped would be ” non numerical”, “non comparable“, “non measurable” or “ incommensurable”. Keynes’s discussions on pp. 32-36 of chapter III of the A Treatise on Probability have been severely misinterpreted to mean that no probabilities could be provided because there was a category of “unknown” probabilities. T. Seidenfeld’s 2004 academic journal summary puts an end to the claims of the Keynesian Fundamentalists and their assertions about Keynes’s supposed unknown probabilities.

Pragmatic Considerations on Comparative Probability

While pragmatic arguments for numerical probability axioms have received much attention, justifications for axioms of qualitative probability have been less discussed. We offer an argument for the requirement that an agent’s qualitative (comparative) judgments be probabilistically representable, inspired by, but importantly different from, the Money Pump argument for transitivity of preference and Dutch book arguments for quantitative coherence. The argument is supported by a theorem, to the effect that a subject is systematically susceptible to dominance given her preferred acts, if and only if the subject’s comparative judgments preclude representation by a standard probability measure (or set of such measures).