Modeling uncertainty using accept & reject statements

Uncertainty and preference is often modeled using linear previsions and linear orders. Some more expressive models use sets of probabilities, lower previsions, or partial orders (see, e.g., the work of Seidenfeld et al. and Walley). In the discussion of these more expressive models, or even to justify them, alternative representations in terms of sets of so-called acceptable, favorable, or desirable gambles appear (cf. the work of Williams, Seidenfeld et al., and Walley). Such ‘sets of gambles’-based models are attractive because of their geometric nature. We generalize these ‘sets of gambles’-based models by considering a pair of sets, one with accepted gambles and one with rejected gambles. We develop a framework based on a small number of axioms—No Confusion, Deductive Closure, No Limbo, and Indifference to Status Quo—and provide an interesting characterization of the resulting models. Furthermore, we define a pair of equivalent gamble relations that generalize the partial orders mentioned earlier; the corresponding characterization result is also given

Limit behaviour for imprecise Markov Chains

An imprecise Markov chain is defined by a closed convex set of transition matrices instead of a unique one for a classical precise Markov chain. These imprecise Markov chains allow us to model situations where we do not have enough information to specify a unique transition matrix, or to approximate the behaviour of non‐stationary Markov chains. We show that there are efficient, dynamic programming‐ like ways to work and reason with these imprecise Markov chains; e.g. to calculate the resulting distribution over the states at any time instant. We prove that this distribution converges in time, similarly to the precise case and under very mild conditions. We thus effectively prove a Perron‐Frobenius theorem for a special class of non‐linear systems

Belief propagation in imprecise Markov trees

We replace strong independence in credal networks with the weaker notion of epistemic irrelevance. Focusing on directed trees, we show how to combine the local credal sets in the networks into an overall joint model, and use this to construct and justify an exact message-passing algorithm that computes updated beliefs for a variable in the network. The algorithm, which is essentially linear in the number of nodes, is formulated entirely in terms of coherent lower previsions. We supply examples of the algorithm's operation, and report an application to on-line character recognition that illustrates the advantages of the model for prediction

Propagating imprecise probabilities through event trees

Event trees are a graphical model of a set of possible situations and the possible paths going through them, from the initial situation to the terminal situations. With each situation, there is associated a local uncertainty model that represents beliefs about the next situation. The uncertainty models can be classical, precise probabilities; they can also be of a more general, imprecise probabilistic type, in which case they can be seen as sets of classical probabilities (yielding probability intervals). To work with such event trees, we must combine these local uncertainty models. We show this can be done efficiently by back-propagation through the tree, both for precise and imprecise probabilistic models, and we illustrate this using an imprecise probabilistic counterpart of the classical Markov chain. This allows us to perform a robustness analysis for Markov chains very efficiently

Aristotle from York to Basra: an investigation into the simultaneous study of Aristotle's Categories in the Carolingian, the Byzantine and the Abbasid worlds

In the eighth and ninth centuries CE intellectuals in three different societies were studying the same classical text in three different languages. In Western Europe, Carolingian intellectuals were studying the Categories of Aristotle in Latin, while in Byzantium contemporary scholars were reading it in Greek and in the Middle East Abbasid scholars did so in Arabic. My dissertation addresses the question of why the Categories was studied at the same time in these three different culturo‐political worlds. The primary sources that I use include paraphrases and translations of the Categories that are found in the works of John of Damascus and Photius in the Byzantine world, Alcuin and John Scottus Eriugena in the Carolingian world and Ibn Al--‐Muqaffaʿ and Al--‐Kindī in the Islamic world. Rather than providing an analysis of the philosophical interpretations of the Categories by any of these intellectuals, I explore the possible explanations of the simultaneous study of the Categories, such as direct contact between these scholars, movement of manuscripts and coincidence. I conclude that the most likely explanation is that the late Roman educational curriculum which was established by the sixth century and which included Aristotle’s Categories, continued to exert its inTluence in all three cultural zones. As a result, I argue that early medieval scholars living as far apart as England and Iraq had a similar intellectual horizon in which exposure to Aristotelian logic in schools played an important role.Institute for the Study of the Ancient World, New York Universit

Imprecise Markov chains and their limit behaviour

When the initial and transition probabilities of a finite Markov chain in discrete time are not well known, we should perform a sensitivity analysis. This can be done by considering as basic uncertainty models the so-called credal sets that these probabilities are known or believed to belong to, and by allowing the probabilities to vary over such sets. This leads to the definition of an imprecise Markov chain. We show that the time evolution of such a system can be studied very efficiently using so-called lower and upper expectations, which are equivalent mathematical representations of credal sets. We also study how the inferred credal set about the state at time n evolves as n goes to infinity: under quite unrestrictive conditions, it converges to a uniquely invariant credal set, regardless of the credal set given for the initial state. This leads to a non-trivial generalisation of the classical Perron-Frobenius Theorem to imprecise Markov chains.Comment: v1: 28 pages, 8 figures; v2: 31 pages, 9 figures, major revision after review: added, modified, and removed material (no results dropped, results added), moved proofs to an appendi

Are 3D-printed Models of Tibial Plateau Fractures a Useful Addition to Understanding Fractures for Junior Surgeons?

Background Tibial plateau fractures are often complex, and they can be challenging to treat. Classifying fractures is often part of the treatment process, but intra- and interobserver reliability of fracture classification systems often is inadequate to the task, and classifications that lack reliability can mislead providers and result in harm to patients. Three-dimensionally (3D)-printed models might help in this regard, but whether that is the case for the classification of tibial plateau fractures, and whether the utility of such models might vary by the experience of the individual classifying the fractures, is unknown. Questions/purposes (1) Does the overall interobserver agreement improve when fractures are classified with 3D-printed models compared with conventional radiology? (2) Does interobserver agreement vary among attending and consultant trauma surgeons, senior surgical residents, and junior surgical residents? (3) Do surgeons' and surgical residents' confidence and accuracy improve when tibial plateau fractures are classified with an additional 3D model compared with conventional radiology? Methods Between 2012 and 2020, 113 patients with tibial plateau fractures were treated at a Level 1 trauma center. Forty-four patients were excluded based on the presence of bone diseases (such as osteoporosis) and the absence of a CT scan. To increase the chance to detect an improvement or deterioration and to prevent observers from losing focus during the classification, we decided to include 40 patients with tibial plateau fractures. Nine trauma surgeons, eight senior surgical residents, and eight junior surgical residents-none of whom underwent any study-specific pretraining-classified these fractures according to three often-used classification systems (Schatzker, OA/OTA, and the Luo three-column concept), with and without 3D-printed models, and they indicated their overall confidence on a 10-point Likert scale, with 0 meaning not confident at all and 10 absolutely certainty. To set the gold standard, a panel of three experienced trauma surgeons who had special expertise in knee surgery and 10 years to 25 years of experience in practice also classified the fractures until consensus was reached. The Fleiss kappa was used to determine interobserver agreement for fracture classification. Differences in confidence in assessing fractures with and without the 3D-printed model were compared using a paired t-test. Accuracy was calculated by comparing the participants' observations with the gold standard. Results The overall interobserver agreement improved minimally for fracture classification according to two of three classification systems (Schatzker: kappa(conv) = 0.514 versus kappa(3Dprint) = 0.539; p = 0.005; AO/OTA:kappa(conv) = 0.359 versus kappa(3Dprint) = 0.372; p = 0.03). However, none of the classification systems, even when used by our most experienced group of trauma surgeons, achieved more than moderate interobserver agreement, meaning that a large proportion of fractures were misclassified by at least one observer. Overall, there was no improvement in self-assessed confidence in classifying fractures or accuracy with 3D-printed models; confidence was high (about 7 points on a 10-point scale) as rated by all observers, despite moderate or worse accuracy and interobserver agreement Conclusion Although 3D-printed models minimally improved the overall interobserver agreement for two of three classification systems, none of the classification systems achieved more than moderate interobserver agreement. This suggests that even with 3D-printed models, many fractures would be misclassified, which could result in misleading communication, inaccurate prognostic assessments, unclear research, and incorrect treatment choices. Therefore, we cannot recommend the use of 3D-printed models in practice and research for classification of tibial plateau fractures

The prognostic value of the hypoxia markers CA IX and GLUT 1 and the cytokines VEGF and IL 6 in head and neck squamous cell carcinoma treated by radiotherapy ± chemotherapy

BACKGROUND: Several parameters of the tumor microenvironment, such as hypoxia, inflammation and angiogenesis, play a critical role in tumor aggressiveness and treatment response. A major question remains if these markers can be used to stratify patients to certain treatment protocols. The purpose of this study was to investigate the inter-relationship and the prognostic significance of several biological and clinicopathological parameters in patients with head and neck squamous cell carcinoma (HNSCC) treated by radiotherapy ± chemotherapy. METHODS: We used two subgroups of a retrospective series for which CT-determined tumoral perfusion correlated with local control. In the first subgroup (n = 67), immunohistochemistry for carbonic anhydrase IX (CA IX) and glucose transporter-1 (GLUT-1) was performed on the pretreatment tumor biopsy. In the second subgroup (n = 34), enzyme linked immunosorbent assay (ELISA) was used to determine pretreatment levels of the cytokines vascular endothelial growth factor (VEGF) and interleukin-6 (IL-6) in serum. Correlation was investigated between tumoral perfusion and each of these biological markers, as well as between the markers mutually. The prognostic value of these microenvironmental parameters was also evaluated. RESULTS: For CA IX and GLUT-1, the combined assessment of patients with both markers expressed above the median showed an independent correlation with local control (p = 0.02) and disease-free survival (p = 0.04) with a trend for regional control (p = 0.06). In the second subgroup, IL-6 pretreatment serum level above the median was the only independent predictor of local control (p = 0.009), disease-free survival (p = 0.02) and overall survival (p = 0.005). CONCLUSION: To our knowledge, we are the first to report a link in HNSCC between IL-6 pretreatment serum levels and radioresistance in vivo. This link is supported by the strong prognostic association of pretreatment IL-6 with local control, known to be the most important parameter to judge radiotherapy responses. Furthermore, the combined assessment of CA IX and GLUT-1 correlated independently with prognosis. This is a valuable indication that a combined approach is important in the investigation of prognostic markers