Hykel Hosni

A logical characterization of coherence for imprecise probabilities

Whilst supported by compelling arguments, the representation of uncertainty by means of (subjective) probability does not enjoy a unanimous consensus. A substantial part of the relevant criticisms point to its alleged inadequacy for representing ignorance as opposed to uncertainty. The purpose of this paper is to show how a strong justification for taking belief as probability, namely the Dutch Book argument, can be extended naturally so as to provide a logical characterization of coherence for imprecise probability, a framework which is widely believed to accommodate some fundamental features of reasoning under ignorance. The appropriate logic for our purposes is an algebraizable logic whose equivalent algebraic semantics is a variety of MV-algebras with an additional internal unary operation representing upper probability (these algebras will be called UMV-algebras).

Forecasting in Light of Big Data

Predicting the future state of a system has always been a natural motivation for science and practical applications. Such a topic, beyond its obvious technical and societal relevance, is also interesting from a conceptual point of view. This owes to the fact that forecasting lends itself to two equally radical, yet opposite methodologies. A reductionist one, based on first principles, and the naïve-inductivist one, based only on data. This latter view has recently gained some attention in response to the availability of unprecedented amounts of data and increasingly sophisticated algorithmic analytic techniques. The purpose of this note is to assess critically the role of big data in reshaping the key aspects of forecasting and in particular the claim that bigger data leads to better predictions. Drawing on the representative example of weather forecasts we argue that this is not generally the case. We conclude by suggesting that a clever and context-dependent compromise between modelling and quantitative analysis stands out as the best forecasting strategy, as anticipated nearly a century ago by Richardson and von Neumann.

Coherence in the aggregate

We investigate a generalised betting method and isolate its general form of coherence.We provide an operational semantics for belief and plausibility functions.We investigate generalised uncertainty measures on MV-algebras. Betting methods, of which de Finettis Dutch Book is by far the most well-known, are uncertainty modelling devices which accomplish a twofold aim. Whilst providing an (operational) interpretation of the relevant measure of uncertainty, they also provide a formal definition of coherence. The main purpose of this paper is to put forward a betting method for belief functions on MV-algebras of many-valued events which allows us to isolate the corresponding coherence criterion, which we term coherence in the aggregate. Our framework generalises the classical Dutch Book method.

Rationality as conformity

We argue in favour of identifying one aspect of rational choice with the tendency to conform to the choice you expect another like-minded, but non-communicating, agent to make and study this idea in the very basic case where the choice is from a non-empty subset K of 2^A and no further structure or knowledge of A is assumed.

Benefits and Risks of Machine Learning Decision Support Systems

In Reply Dr Zuckerman and colleagues are concerned that the definition of CTE used in our study was overly sensitive. The definitive diagnosis of Alzheimer disease, Lewy body disease, or CTE is based on strictly defined neuropathologic criteria.1-3 In CTE and Lewy body disease, there is a pathognomonic lesion or a unique pathologic feature that is specific to the disorder and not found in healthy control patients. In Lewy body disease, the pathognomonic feature is the Lewy body, and a diagnosis is made even if only a single Lewy body is found.2,4 In CTE, the pathognomonic lesion is a cluster of tau-immunostained neurofibrillary tangles and neurites around a small vessel,3 and again, 1 lesion is sufficient for a pathologic diagnosis. The situation is analogous to finding a single microscopic focus of tumor—even a single small focus of tumor is sufficient for a diagnosis. Of the 177 CTE cases reported in our study, only 11 (6%) had 1 or 2 isolated CTE lesions and were considered stage I CTE; most cases (133 [75%]) showed advanced pathology and were stage III and IV. Unlike CTE and Lewy body disease, the diagnosis of Alzheimer disease requires the presence of 2 pathologic features, amyloid-β plaques and neurofibrillary tangles, distributed throughout the brain. Small amounts of amyloid-β plaques or neurofibrillary tangles are not sufficient for the diagnosis of Alzheimer disease. Criteria exist for the clinical diagnoses of probable and possible Alzheimer disease and Lewy body disease, and criteria for traumatic encephalopathy syndrome have been proposed,5 but for all 3 disorders, the clinical diagnosis is not definitive and does not contribute to the pathologic diagnosis. Zuckerman and colleagues and Dr Yari suggest that other diagnoses may have been responsible for the participants’ symptoms. All the clinical evaluations for CTE in our study were conducted retrospectively after death with an informant. Of the CTE cases, 91% had behavior or mood symptoms in the year before death and 43% initially presented with mood or behavior symptoms. There is no claim in the article that the reported symptoms were due to CTE pathology alone. Moreover, participants who initially presented with mood or behavior symptoms commonly had a family history of psychiatric illness, suggesting that “CTE ptau pathology may lower the threshold for psychiatric manifestations in susceptible individuals.” Although football data were collected via an online questionnaire beginning in 2014, all data gathered about TBI and concussions were only obtained via postmortem telephone interviews with informants by a clinician. These data were collected in an unstructured format prior to 2014 and via a structured telephone questionnaire after 2014. In the structured telephone questionnaire, the clinician asked about concussion number after reading a lay definition of concussion to the informant.6 The median number of concussions reported was substantially higher after 2014, suggesting that reading the definition led to inclusion of milder concussions. We acknowledged the possibility of recall bias due to the retrospective nature of data collection in the article.

On the Algebraic Structure of Conditional Events

This paper initiates an investigation of conditional measures as simple measures on conditional events. As a first step towards this end we investigate the construction of conditional algebras which allow us to distinguish between the logical properties of conditional events and those of the conditional measures which we can be attached to them. This distinction, we argue, helps us clarifying both concepts.

Stable Non-standard Imprecise Probabilities

Stability arises as the consistency criterion in a betting interpretation for hyperreal imprecise previsions, that is imprecise previsions (and probabilities) which may take infinitesimal values. The purpose of this work is to extend the notion of stable coherence introduced in [8] to conditional hyperreal imprecise probabilities. Our investigation extends the de Finetti-Walley operational characterisation of (imprecise) prevision to conditioning on events which are considered “practically impossible” but not “logically impossible”.

Towards a Bayesian Theory of Second-Order Uncertainty: Lessons from Non-Standard Logics

Second-order uncertainty, also known as model uncertainty and Knightian uncertainty, arises when decision-makers can (partly) model the parameters of their decision problems. It is widely believed that subjective probability, and more generally Bayesian theory, are ill-suited to represent a number of interesting second-order uncertainty features, especially “ignorance” and “ambiguity”. This failure is sometimes taken as an argument for the rejection of the whole Bayesian approach, triggering a Bayes versus anti-Bayes debate which is in many ways analogous to what the classical versus non-classical debate used to be in logic. This paper attempts to unfold this analogy and suggests that the development of non-standard logics offers very useful lessons on the contextualisation of justified norms of rationality. By putting those lessons to work I will flesh out an epistemological framework suitable for extending the expressive power of standard Bayesian norms of rationality to second-order uncertainty in a way which is both formally and foundationally conservative.

Convex MV-Algebras: Many-Valued Logics Meet Decision Theory

This paper introduces a logical analysis of convex combinations within the framework of Łukasiewicz real-valued logic. This provides a natural link between the fields of many-valued logics and decision theory under uncertainty, where the notion of convexity plays a central role. We set out to explore such a link by defining convex operators on MV-algebras, which are the equivalent algebraic semantics of Łukasiewicz logic. This gives us a formal language to reason about the expected value of bounded random variables. As an illustration of the applicability of our framework we present a logical version of the Anscombe–Aumann representation result.

STRICT COHERENCE ON MANY-VALUED EVENTS

We investigate the property of strict coherence in the setting of many-valued logics. Our main results read as follows: (i) a map from an MV-algebra to [0,1] is strictly coherent if and only if it satisfies Carnap’s regularity condition, and (ii) a [0,1]-valued book on a finite set of many-valued events is strictly coherent if and only if it extends to a faithful state of an MV-algebra that contains them. Remarkably this latter result allows us to relax the rather demanding conditions for the Shimony-Kemeny characterisation of strict coherence put forward in the mid 1950s in this Journal .