Believing Probabilistic Contents: On the Expressive Power and Coherence of Sets of Sets of Probabilities

Moss (2018) argues that rational agents are best thought of not as having degrees of belief in various propositions but as having beliefs in probabilistic contents, or probabilistic beliefs. Probabilistic contents are sets of probability functions. Probabilistic belief states, in turn, are modeled by sets of probabilistic contents, or sets of sets of probability functions. We argue that this Mossean framework is of considerable interest quite independently of its role in Moss’ account of probabilistic knowledge or her semantics for epistemic modals and probability operators. It is an extremely general model of uncertainty. Indeed, it is at least as general and expressively powerful as every other current imprecise probability framework, including lower probabilities, lower previsions, sets of probabilities, sets of desirable gambles, and choice functions. In addition, we partially answer an important question that Moss leaves open, viz., why should rational agents have consistent probabilistic beliefs? We show that an important subclass of Mossean believers avoid Dutch bookability iff they have consistent probabilistic beliefs

Self-referential probability

This thesis focuses on expressively rich languages that can formalise talk about probability. These languages have sentences that say something about probabilities of probabilities, but also sentences that say something about the probability of themselves. For example: (π): “The probability of the sentence labelled π is not greater than 1/2.” Such sentences lead to philosophical and technical challenges. For example seemingly harmless principles, such as an introspection principle, lead to inconsistencies with the axioms of probability in this framework. This thesis aims to answer two questions relevant to such frameworks, which correspond to the two parts of the thesis: “How can one develop a formal semantics for this framework?” and “What rational constraints are there on an agent once such expressive frameworks are considered?”. In this second part we are considering probability as measuring an agent’s degrees of belief. In fact that concept of probability will be the motivating one throughout the thesis. The first chapter of the thesis provides an introduction to the framework. The following four chapters, which make up Part I, focus on the question of how to provide a semantics for this expressively rich framework. In Chapter 2, we discuss some preliminaries and why developing semantics for such a framework is challenging. We will generally base our semantics on certain possible world structures that we call probabilistic modal structures. These immediately allow for a definition of a natural semantics in restrictive languages but not in the expressively rich languages that this thesis focuses on. The chapter also presents an overview of the strategy that will be used throughout this part of the thesis: we will generalise theories and semantics developed for the liar paradox, which is the sentence: (λ): “The sentence labelled λ is not true”. In Chapter 3, we will present a semantics that generalises a very influential theory of truth: a Kripke-style theory (Kripke, 1975) using a strong Kleene evaluation scheme. A feature of this semantics is that we can understand it as assigning sentences intervals as probability values instead of single numbers. Certain axioms of probability have to be dropped, for example “P ‘λ ∨ ¬λ’ = 1” is not satisfied in the construction, but the semantics can be seen as assigning non-classical probabilities. This semantics allows one to further understand the languages, for example the conflict with introspection, where one can see that the appropriate way to express the principle of introspection in this case is in fact to use a truth predicate in its formulation. We also develop an axiomatic system and show that it is complete in the presence of the ω-rule which allows one to fix the standard model of arithmetic. In Chapter 4, we will consider another Kripke-style semantics but now based on a supervaluational evaluation scheme. This variation is particularly interesting because it bears a close relationship to imprecise probabilities where agents’ credal states are taken to be sets of probability functions. In this chapter, we will also consider how to use this language to describe imprecise agents reasoning about one another. These considerations provide us with an argument for using imprecise probabilities that is very different from traditional justifications: by allowing agents to have imprecise probabilities one can easily extend a semantics to languages with sentences that talk about their own probability, whereas the traditional precise probabilist cannot directly apply his semantics to such languages. In Chapter 5, a revision theory of probability will be developed. In this one retains classical logic and traditional probability theory but the price to pay is that one obtains a transfinite sequence of interpretations of the language and identifying any particular interpretation as “correct” is problematic. In developing this we are particularly interested in finding limit stage interpretations that can themselves be used as good models for probability and truth. We will require that the limit stages “sum up” the previous stages, understood in a strong way. In this chapter two strategies for defining the successor stages are discussed. We first discuss defining (successor) probabilities by considering relative frequencies in the revision sequence up to that stage, extending ideas from Leitgeb (2012). The second strategy is to base the construction on a probabilistic modal structure and use the accessibility measure from that to determine the interpretation of probability. That concludes Part I and the development of semantics. In Part II, we consider rationality requirements on agents who have beliefs about self-referential probability sentences like π. For such sentences, a choice of the agent’s credences will affect which worlds are possible. Caie (2013) has argued that the accuracy and Dutch book arguments should be modified because the agent should only care about her inaccuracy or payoffs in the world(s) that could be actual if she adopted the considered credences. We consider this suggestion for the accuracy argument in Chapter 7 and the Dutch book argument in Chapter 8. Chapter 6 acts as an introduction to these considerations. We will show that these modified accuracy and Dutch book criteria lead to an agent being rationally required to be probabilistically incoherent, have negative credences, fail to be introspective and fail to assign the same credence to logically equivalent sentences. We will also show that this accuracy criterion depends on how inaccuracy is measured and that the accuracy criterion differs from the Dutch book criterion. We will in fact suggest rejecting Caie’s suggested modifications. For the accuracy argument, we suggest in Section 7.3 that the agent should consider how accurate the considered credences are from the perspective of her current credences. We will also consider how to generalise this version of the accuracy criterion and present ideas suggesting that it connects to the vi semantics developed in Part I. For the Dutch book argument, in Section 8.6 we suggest that this is a case where an agent should not bet with his credences

How to Express Self-Referential Probability. A Kripkean Proposal

We present a semantics for a language that includes sentences that can talk about their own probabilities. This semantics applies a fixed point construction to possible world style structures. One feature of the construction is that some sentences only have their probability given as a range of values. We develop a corresponding axiomatic theory and show by a canonical model construction that it is complete in the presence of the ω-rule. By considering this semantics we argue that principles such as introspection, which lead to paradoxical contradictions if naively formulated, should be expressed by using a truth predicate to do the job of quotation and disquotation and observe that in the case of introspection the principle is then consistent

Probability for the Revision Theory of Truth

We investigate how to assign probabilities to sentences that contain a type-free truth predicate. These probability values track how often a sentence is satisfied in transfinite revision sequences, following Gupta and Belnap’s revision theory of truth. This answers an open problem by Leitgeb which asks how one might describe transfinite stages of the revision sequence using such probability functions. We offer a general construction, and explore additional constraints that lead to desirable properties of the resulting probability function. One such property is Leitgeb’s Probabilistic Convention T, which says that the probability of φ equals the probability that φ is true.publishe

Development and application of the DePtH framework for categorising the agentic demands of population health interventions [Pre-print]

The ‘agentic demand’ of population health interventions may influence intervention effectiveness and equity, yet the absence of an adequate framework to classify agentic demands limits the fields’ advancement. We systematically developed the DEmands for PopulaTion Health Interventions (DePtH) framework identifying three constructs influencing agentic demand - exposure (initial contact with intervention), mechanism of action (how the intervention enables or discourages behaviour), and engagement (recipient response), combined into twenty classifications. We conducted expert qualitative feedback and reliability testing, revised the framework and applied it in a proof-of-concept review, combining it with data on overall effectiveness and equity of dietary and physical activity interventions. Intervention components were concentrated in a small number of classifications; DePtH classification appeared to be related to intervention equity but not effectiveness. This framework holds potential for future research, policy and practice, facilitating the design, selection, evaluation and synthesis of evidence

Development and application of the Demands for Population Health Interventions (Depth) framework for categorising the agentic demands of population health interventions

Background: The ‘agentic demand’ of population health interventions (PHIs) refers to the capacity, resources and freedom to act that interventions demand of their recipients to benefit, which have a socio-economical pattern. Highly agentic interventions, e.g. information campaigns, rely on recipients noticing and responding to the intervention and thus might affect intervention effectiveness and equity. The absence of an adequate framework to classify agentic demands limits the fields’ ability to systematically explore these associations. Methods: We systematically developed the Demands for Population Health Interventions (Depth) framework using an iterative approach: (1) Developing the Depth framework by systematically identifying examples of PHIs aiming to promote healthier diets and physical activity, coding of intervention actors and actions and synthesising the data to develop the framework; (2) Testing the Depth framework in online workshops with academic and policy experts and a quantitative reliability assessment. We applied the final framework in a proof-of-concept review, extracting studies from three existing equity focused systematic reviews on framework category, overall effectiveness and differential socioeconomic effects and visualised the findings in Harvest Plots. Results: The Depth framework identifies three constructs influencing agentic demand: exposure - initial contact with intervention (2 levels), mechanism of action - how the intervention enables or discourages behaviour (5 levels), and engagement - recipient response (2 levels). When combined, these constructs form a matrix of twenty possible classifications. In the proof-of-concept review, we classified all components of 31 interventions according to the Depth framework. Intervention components were concentrated in a small number of Depth classifications; Depth classification appeared to be related to intervention equity but not effectiveness. Conclusions: This framework holds potential for future research, policy and practice, facilitating the design, selection and evaluation of interventions and evidence synthesis

Clinically and microbiologically derived azithromycin susceptibility breakpoints for Salmonella enterica serovars Typhi and Paratyphi A.

Azithromycin is an effective treatment for uncomplicated infections with Salmonella enterica serovar Typhi and serovar Paratyphi A (enteric fever), but there are no clinically validated MIC and disk zone size interpretative guidelines. We studied individual patient data from three randomized controlled trials (RCTs) of antimicrobial treatment in enteric fever in Vietnam, with azithromycin used in one treatment arm, to determine the relationship between azithromycin treatment response and the azithromycin MIC of the infecting isolate. We additionally compared the azithromycin MIC and the disk susceptibility zone sizes of 1,640 S. Typhi and S. Paratyphi A clinical isolates collected from seven Asian countries. In the RCTs, 214 patients who were treated with azithromycin at a dose of 10 to 20 mg/ml for 5 to 7 days were analyzed. Treatment was successful in 195 of 214 (91%) patients, with no significant difference in response (cure rate, fever clearance time) with MICs ranging from 4 to 16 μg/ml. The proportion of Asian enteric fever isolates with an MIC of ≤ 16 μg/ml was 1,452/1,460 (99.5%; 95% confidence interval [CI], 98.9 to 99.7) for S. Typhi and 207/240 (86.3%; 95% CI, 81.2 to 90.3) (P 16 μg/ml and to determine MIC and disk breakpoints for S. Paratyphi A

Ongoing strategies to improve antimicrobial utilization in hospitals across the Middle East and North Africa (MENA) : findings and implications

Antimicrobial resistance (AMR) is an increasing global concern, increasing costs, morbidity, and mortality. National action plans (NAPs) to minimize AMR are one of several global and national initiatives to slow down rising AMR rates. NAPs are also helping key stakeholders understand current antimicrobial utilization patterns and resistance rates. The Middle East is no exception with high AMR rates. Antibiotic point prevalence surveys (PPS) provide a better understanding of existing antimicrobial consumption trends in hospitals, and assist with the subsequent imple-mentation of antimicrobial stewardship programs (ASPs). These are important NAP activities. We examined current hospital consumption trends across the Middle East along with documented ASPs. A narrative assessment of 24 PPS studies in the Region found that, on average, more than 50% of in-patients received antibiotics, with Jordan having the highest rate at 98.1%. Published studies ranged in size from single to 18 hospitals. The most prescribed antibiotics were ceftriax-one, metronidazole, and penicillin. In addition, significant postoperative antibiotic prescribing lasting up to five days or longer was common to avoid surgical site infections. These findings have resulted in a variety of suggested short-, medium-, and long-term actions among key stakehold-ers, including governments and healthcare workers, to improve and sustain future antibiotic prescribing in order to decrease AMR throughout the Middle East

Ongoing efforts to improve antimicrobial utilisation in hospitals among African countries and implications for the future

There are serious concerns with rising antimicrobial resistance (AMR) across countries 67 increasing morbidity, mortality and costs. These concerns have resulted in a plethora of initiatives 68 globally and nationally including national action plans (NAPs) to reduce AMR. Africa is no excep-69 tion especially with the highest rates of AMR globally. Key activities in NAPs include gaining a 70 greater understanding of current antimicrobial utilisation patterns through point prevalence sur-71 veys (PPS) and subsequently instigating antimicrobial stewardship programmes (ASPs). Conse-72 quently, there is a need to comprehensively document current utilisation patterns among hospitals 73 across Africa coupled with ASP studies. 33 PPS studies ranging from single up to 18 hospitals were 74 documented from a narrative review with typically over 50% of in-patients prescribed antimicrobi-75 als, up to 97.6% in Nigeria. The penicillins, ceftriaxone and metronidazole were the most prescribed 76 antibiotics. Appreciable extended prescribing of antibiotics up to 6 days or more post-operatively 77 was seen across Africa to prevent surgical site infections. At least 19 ASPs have been instigated 78 across Africa in recent years to improve future prescribing utilising a range of prescribing indicators. 79 The various findings resulted in a range of suggested activities that key stakeholders, including 80 governments and healthcare professionals, should undertake in the short, medium and long term 81 to improve future antimicrobial prescribing and reduce AMR