Nebel argues for the Repugnant Conclusion via the “Intrapersonal Repugnant Conclusion,” on which certainty of a mediocre life is better for individuals than a sufficiently small chance of an excellent life. In this article, I deny that acceptance of the Intrapersonal Repugnant Conclusion leads us to the Repugnant Conclusion. I point out that on many views which avoid the Repugnant Conclusion we should discount very small probabilities down to zero. If we do, then Nebel’s crucial premise of (...) Ex Ante Pareto fails, because discounting at the individual level can fail to match up with discounting at the population level. (shrink)

A small probability space representation of quantum mechanical probabilities is defined as a collection of Kolmogorovian probability spaces, each of which is associated with a context of a maximal set of compatible measurements, that portrays quantum probabilities as Kolmogorovian probabilities of classical events. Bell's theorem is stated and analyzed in terms of the small probability space formalism.

Shoot an arrow at a wall, and then paint a target around it so that the arrow sticks squarely in the bull's eye. Alternatively, paint a fixed target on a wall, and then shoot an arrow so that it sticks squarely in the bull's eye. How do these situations differ? In both instances the precise place where the arrow lands is highly improbable. Yet in the one, one can do no better than attribute the arrow's landing to chance, whereas in (...) the other one rightly attributes the arrow's landing to the archer's skill. ;Highly improbable events occur all the time, and by themselves may legitimately be attributed to chance. Yet when an event also conforms to a pattern given independently of that event , one rightly refuses to attribute the event to chance. Patterns given independently of events are called specifications. This dissertation explicates the relation of independence between specification and event, and sets precise bounds on what may count as a probability small enough to eliminate chance. Although this dissertation connects with how statistics eliminates chance, it considerably extends and clarifies the range of application for chance-elimination arguments based on small probabilities. ;The dual notions of specification and small probability together help justify a regulative principle of probability called the Law of Small Probability, which asserts that specified events of small probability do not occur by chance. This principle engenders a form of inference called the design inference, and a mode of explanation called design. To infer design in explaining an event is to eliminate decisively explanations of that event which appeal to law-like regularities or chance. (shrink)

Many scientists, businessmen, and government regulators believe that the criteria for acceptable societal risk are too stringent. Those who subscribe to this belief often accept the view which I call the probability-threshold position. Proponents of this stance maintain that society ought to ignore very small risks, i.e., those causing an average annual probability of fatality of less than 10–6.After examining the three major views in the risk-evaluation debate, viz., the probability-threshold position, the zero-risk position, and the weighted-risk position, I (...) focus on the arguments for the first of these views, since it is the position which currently undergirds most public policy (especially in the U.S.) regarding acceptable risk. After analyzing Arrow's argument from decision theory, Comar's and Gibson's argument from ontology, and Starr's and Whipple's argument from epistemology, I conclude that these defenses of the probability-threshold position err in a variety of ways. Most commonly, they fail because they tacitly accept the assumption that magnitude of probability, alone, provides a sufficient condition for judging the acceptability of a given risk. In the light of these errors, I suggest that it might be more desirable for risk assessors, decision theorists, and policymakers to weight various risk-cost-benefit parameters according to alternative ethical criteria, rather than to evaluate risks solely in terms of mathematical considerations. (shrink)

The design inference uncovers intelligent causes by isolating their key trademark: specified events of small probability. Just about anything that happens is highly improbable, but when a highly improbable event is also specified undirected natural causes lose their explanatory power. Design inferences can be found in a range of scientific pursuits from forensic science to research into the origins of life to the search for extraterrestrial intelligence. This challenging and provocative 1998 book shows how incomplete undirected causes are for (...) science and breathes new life into classical design arguments. It will be read with particular interest by philosophers of science and religion, other philosophers concerned with epistemology and logic, probability and complexity theorists, and statisticians. (shrink)

Analogies are offered to guide our explanatory responses to the fine-tuning of the universe. Situations that prompt us to reject an explanation involving a single chance event are presented as analogous to the fine-tuning. Thus, by analogy, we are prompted to reject an explanation of the fine-tuning involving a single universe fine-tuned by chance. But if the alleged analogues are not analogous they misguide us. I argue that the alleged analogues are not analogous and hence they do misguide our explanatory (...) responses to the fine-tuning. I use William Dembski’s work on eliminating chance explanations for “specified” events of small probability to illustrate the misguiding nature of the analogies. (shrink)

The design inference uncovers intelligent causes by isolating the key trademark of intelligent causes: specified events of small probability. Just about anything that happens is highly improbable, but when a highly improbable event is also specified (i.e., conforms to an independently given pattern) undirected natural causes lose their explanatory power. Design inferences can be found in a range of scientific pursuits from forensic science to research into the origins of life to the search for extraterrestrial intelligence.

Organic evolution, or change, among the diatoms has proceeded with considerable regularity. The origins, the extinctions, and the increases in numbers of species and genera, on the whole, have submitted to law and order, as rules-within-limits . The changes of evolution have followed from two sorts of phenomena — 1) the origin and extinction of shortlived or unstable species, in a proportion which has been more or less constant, but different in the two groups of diatoms, Centricae and Pennatae; these (...) short-lived species died out in the subperiod of their origin even under favourable enough environment or surrounding conditions, because of the innate instability of these particular species: and 2) the origin of new species which demonstrated their innate capacity for permanence by remaining in the living condition from their origins mostly up to the present time. The continued or continual additions of new and long-living species, in proportion to the number available as possible and parental species, has given the geometrical progressions.Consequently evolution in the diatoms has been monopodial, or one-stemmed, and not sympodial or not with many substitutions of one species after another in successions. The two kinds of species-durations are well-known in other classes of plants and animals. Exactly the same processes of additions, subtractions, and multiplications, particularly as found in the centric diatoms, are very probable in these classes of plants and animals which have given the same answer or arithmetical result for the whole of their previous histories, in the form of the frequency distribution of the present sizes of their genera.The rules-within-limits of this basic evolution are described, defined, and discussed in their relations to various theories of evolution which perhaps only concern the working of factors of superimposed modifications that reduce the results of the pattern or plan of basic evolution.Die phylogenetische Entwicklung wurde im Bereich der Kieselalgen von bestimmten Gesetzmässigkeiten beherrscht. Die Entstehung neuer Arten und Gattungen, ihr Wiederverschwinden und die Zunahme des Formenreichtums, resultierend aus der Zahl der neuentstandenen, vermindert um die der wieder verschwindenen Arten und Gattungen und vollzog sich nach Regeln , die durch folgende Phänomene gekennzeichnet sind: Zwischen der Zahl der neuentstehenden und der Zahl der in der gleichen geologischen Subperiode aus inneren Gründen wieder aussterbenden Arten besteht eine mehr oder wenig feste Beziehung. Diese ist für die Centricae und Pennatae verschieden. Ein bestimmter Anteil der neuentstandenen Arten und Gattungen erwies sich aus inneren Gründen als „ausdauernd” und blieb bis auf den heutigen Tag erhalten. Die Zunahme der Arten- Zahl durch das Hinzukommen langlebiger Formen war stetig und vollzog sich nach Massgabe geometrischer Progressionen.Diese Tatsachen führen zu dem Schluss, dass die phylogenetische Entwicklung der Kieselalgen nach monopodialem, und nicht nach sympodialem Muster vorsichging. Auch in andern Pflanzengruppen und im Tierreich durften die gleichen Gesetzmässigkeiten herrschend sein. Am klarsten gelangen sie bei den Centricae zum, Ausdruck und werden vom Autor als das Fazit von Additions-, Subtraktions-, und Multiplikationsprozessen aufgefasst. Die heutige Situation bezüglich Häufigkeit der Grössen der Gattungen, in Zahl der Arten, ist das Endergebnis der summativen Wirkung dieser Einzelprozesse.Diese als ‚rules-within-limits’ bezeichneten Gesetzmässigkeiten werden im einzelnen beschrieben, erklärt und zusammenfassend dargestellt. Zu andern Evolutionstheorien in Beziehung gebracht, werden sie schliesslich für Erörterung des Problems benutzt, inwieweit noch andere evolutionsgeschichtliche Prozesse als übergeordnete Faktoren in den vom Verfasser dargestellten Grundplan des Evolutionsgeschehens eingreifen können.L'évolution organique ou le changement parmi les Diatomées s'est effectué avec une régularité considérable. Les origines et les extinctions, et les accroissements des nombres des espèces et des genres, se sont soumis en tout aux lois et aux ordres, commes des régularités à l'intérieur de légères déviations. Le changement de l'évolution fut la conséquence de deux sortes de phénomène: l'origine et l'extinction des espèces éphémères dans une proportion plus ou moins déterminée mais différente pour les deux classes des Diatomées, Centricae et Pennatae; ces espèces éphémères moururent dans la subpériode de leur origines malgré le caractère assez favorable des circonstances environnantes, à cause de l'évanescence inhérente aux espèces particulières, et l'origine des espèces nouvelles qui démontrèrent leur qualité innée de permanence tout en continuant de vivre depuis leurs origines, pour la plupart, jusqu'à présent. Les additions continuelles d'espèces nouvelles et permanentes, en nombres proportionnés aux nombres disponibles comme espèces parentes et possibles, donnèrent des accroissements par progressions géométriques.Par conséquent, l'évolution parmi les Diatomées fut monopodiale et non pas sympodiale. Les deux sortes d'espèces-durées sont bien connues parmi beaucoup d'autres classes de plantes et d'animaux. Les mêmes procédés d'additions, de soustractions et de multiplications, particulièrement tels qu'on les trouve dans les Centricae, sont très probables pour ces classes de plantes et d'animaux qui ont donné un résultat le plus semblable pour la somme totale de leur histoire précédente, sous la forme de la répartition des fréquences de leurs grandeurs générique actuelles.Les régularités à l'intérieur de légères déviations de cette évolution fondamentale sont détaillées, définies et discutées dans leurs relations aux diverses théories touchant les autres méthodes de l'évolution qui jouent le rôle de facteurs modifiants superposés au dessin général ou au dessein de l'évolution fondamentale. (shrink)

In a controlled laboratory experiment we use one sample of college students and one of mature executives to investigate how positive skew influences risky choices. In reduced-form regressions we find that both students and executives make riskier choices when lotteries display positive skew. We estimate decision models to explore three explanations for skew seeking choices: risk-loving, optimism and likelihood insensitivity. We find no role for love for risk as neither students nor executives have convex utility. Both optimism and likelihood insensitivity (...) play a part in skew seeking choices. Likelihood insensitivity is larger for students than for executives. Executives have more concave utility and are more optimistic than students, but this is found to be largely due to them being older. (shrink)

For a finite universe of discourse, if Φ → and ~(Ψ → Φ) , then P(Ψ) > P(Φ), i.e., there is always a loss of information, there is an increase in probability, in a non reversible implication. But consider the two propositions, "All ravens are black", (i.e., "(x)(Rx ⊃ Bx)"), and "Some ravens are black" (i.e., "(∃x)(Rx & Bx)"). In a world of one individual, called "a", these two propositions are equivalent to "~Ra ∨ Ba" and "Ra & Ba" respectively. (...) However, (Ra & Ba) → (~Ra ∨ Ba) and ~[(~Ra ∨ Ba) → (Ra & Ba)]. Consequently, in a world of one individual it is more probable that all ravens are black than that some ravens are black! (shrink)

In response to cases that involve tiny probabilities of huge payoffs, some argue that we ought to discount small probabilities down to zero. However, this paper shows that doing so violates Independence and Continuity, and as a result of these violations, those who discount small probabilities can be exploited by money pumps. Various possible ways of avoiding exploitation will be discussed. This paper concludes that the money pump for Independence undermines the plausibility of discounting (...) class='Hi'>small probabilities. Much of the discussion on Independence generalizes to other views that also violate Independence. (shrink)

The topic of this thesis is how we should treat tiny probabilities of vast value. This thesis consists of six independent papers. Chapter 1 discusses the idea that utilities are bounded. It shows that bounded decision theories prescribe prospects that are better for no one and worse for some if combined with an additive axiology. Chapter 2, in turn, points out that standard axiomatizations of Expected Utility Theory violate dominance in cases that involve possible states of zero probability. Chapters (...) 3–6 discuss the idea that we should ignore tiny probabilities in practical decision-making. Chapter 3 argues that discounting small probabilities solves the ‘Intrapersonal Addition Paradox’ and thus helps avoid the Repugnant Conclusion. Chapter 4 explores what the most plausible version of this view might look like and what problems the different versions have. Chapter 5 focuses on one type of problem, namely, money pumps. The Independence Money Pump, in particular, presents a difficult challenge for those who wish to discount small probabilities. Finally, Chapter 6 discusses the implications of discounting small probabilities for the value of the far future. (shrink)

We show that infinitesimal probabilities are much too small for modeling the individual outcome of a countably infinite fair lottery.

We describe a dual-process theory of how individuals estimate the probabilities of unique events, such as Hillary Clinton becoming U.S. President. It postulates that uncertainty is a guide to improbability. In its computer implementation, an intuitive system 1 simulates evidence in mental models and forms analog non-numerical representations of the magnitude of degrees of belief. This system has minimal computational power and combines evidence using a small repertoire of primitive operations. It resolves the uncertainty of divergent evidence for (...) single events, for conjunctions of events, and for inclusive disjunctions of events, by taking a primitive average of non-numerical probabilities. It computes conditional probabilities in a tractable way, treating the given event as evidence that may be relevant to the probability of the dependent event. A deliberative system 2 maps the resulting representations into numerical probabilities. With access to working memory, it carries out arithmetical operations in combining numerical estimates. Experiments corroborated the theory's predictions. Participants concurred in estimates of real possibilities. They violated the complete joint probability distribution in the predicted ways, when they made estimates about conjunctions: P, P, P, disjunctions: P, P, P, and conditional probabilities P, P, P. They were faster to estimate the probabilities of compound propositions when they had already estimated the probabilities of each of their components. We discuss the implications of these results for theories of probabilistic reasoning. (shrink)

A proposal for an objective interpretation of probability is introduced and discussed: probabilities as deriving from ranges in suitably structured initial-state spaces. Roughly, the probability of an event on a chance trial is the proportion of initial states that lead to the event in question within the space of all possible initial states associated with this type of experiment, provided that the proportion is approximately the same in any not too small subregion of the space. This I would (...) like to call the “natural-range conception” of probability. Providing a substantial alternative to frequency or propensity accounts of probability in a deterministic setting, it is closely related to the so-called “method of arbitrary functions”. It is explicated, confronted with certain problems, and some ideas how these might be overcome are sketched and discussed. (shrink)

When probability discounting (or probability weighting), one multiplies the value of an outcome by one's subjective probability that the outcome will obtain in decision-making. The broader import of defending probability discounting is to help justify cost-benefit analyses in contexts such as climate change. This chapter defends probability discounting under risk both negatively, from arguments by Simon Caney (2008, 2009), and with a new positive argument. First, in responding to Caney, I argue that small costs and benefits need to be (...) evaluated, and that viewing practices at the social level is too coarse-grained. Second, I argue for probability discounting using a distinction between causal responsibility and moral responsibility. Moral responsibility can be cashed out in terms of blameworthiness and praiseworthiness, while causal responsibility obtains in full for any effect which is part of a causal chain linked to one's act. With this distinction in hand, unlike causal responsibility, moral responsibility can be seen as coming in degrees. My argument is, given that we can limit our deliberation and consideration to that which we are morally responsible for and that our moral responsibility for outcomes is limited by our subjective probabilities, our subjective probabilities can ground probability discounting. (shrink)

“Negative probability” in practice. Quantum Communication: Very small phase space regions turn out to be thermodynamically analogical to those of superconductors. Macro-bodies or signals might exist in coherent or entangled state. Such physical objects having unusual properties could be the basis of quantum communication channels or even normal physical ones … Questions and a few answers about negative probability: Why does it appear in quantum mechanics? It appears in phase-space formulated quantum mechanics; next, in quantum correlations … and for (...) wave-particle dualism. Its meaning:- mathematically: a ratio of two measures (of sets), which are not collinear; physically: the ratio of the measurements of two physical quantities, which are not simultaneously measurable. The main innovation is in the mapping between phase and Hilbert space, since both are sums. Phase space is a sum of cells, and Hilbert space is a sum of qubits. The mapping is reduced to the mapping of a cell into a qubit and vice versa. Negative probability helps quantum mechanics to be represented quasi-statistically by quasi-probabilistic distributions. Pure states of negative probability cannot exist, but they, where the conditions for their expression exists, decrease the sum probability of the integrally positive regions of the distributions. They reflect the immediate interaction (interference) of probabilities common in quantum mechanics. (shrink)

Analyses of singular causation often make use of the idea that a cause increases the probability of its effect. Of particular salience in such accounts are the values of the probability function of the effect, conditional on the presence and absence of the putative cause, analysed around the times of the events in question: causes are characterized by the effect’s probability function being greater when conditionalized upon them. Put this way, it becomes clearer that the ‘behaviour’ of probability functions in (...) small intervals about the times in question ought to be of concern. In this article, I make an extended case that causal theorists employing the ‘probability raising’ idea should pay attention to the continuity question. Specifically, if the probability functions are ‘jumping about’ in ways typical of discontinuous functions, then the stability of the relevant probability increase is called into question. The rub, however, is that sweeping requirements for either continuity or discontinuity are problematic and, as I argue, this constitutes a ‘continuity bind’. Hence more subtle considerations and constraints are needed, two of which I consider: utilizing discontinuous first derivatives of continuous probability functions, and abandoning point probability for imprecise probability. _1_ Introduction _2_ Probability Trajectories and Continuity _2.1_ Probability trajectories _2.2_ Causation as discontinuous jumps _2.3_ Against systematic discontinuity _3_ Broader Discontinuity Concerns _4_ The Continuity Bind _4.1_ Retaining continuity with discontinuous first derivatives _4.2_ Imprecise probability trajectories _5_ Concluding Remarks Appendix. (shrink)

We present a model for studying communities of epistemically interacting agents who update their belief states by averaging the belief states of other agents in the community. The agents in our model have a rich belief state, involving multiple independent issues which are interrelated in such a way that they form a theory of the world. Our main goal is to calculate the probability for an agent to end up in an inconsistent belief state due to updating. To that end, (...) an analytical expression is given and evaluated numerically, both exactly and using statistical sampling. It is shown that, under the assumptions of our model, an agent always has a probability of less than 2% of ending up in an inconsistent belief state. Moreover, this probability can be made arbitrarily small by increasing the number of independent issues the agents have to judge or by increasing the group size. A real-world situation to which this model applies is a group of experts participating in a Delphi-study. (shrink)

The origin of eukaryotes is one of the major challenges of evolutionary cell biology. Other than the endosymbiotic origin of mitochondria and chloroplasts, the steps leading to eukaryotic endomembranes and endoskeleton are poorly understood. Ras‐family small GTPases are key regulators of cytoskeleton dynamics, vesicular trafficking and nuclear function. They are specific for eukaryotes and their expansion probably traces the evolution of core eukaryote features. The phylogeny of small GTPases suggests that the first endomembranes to evolve during eukaryote evolution (...) had secretory, and not phagocytic, function. Based on the reconstruction of putative roles for ancestral small GTPases, a hypothetical scenario on the origins of the first endomembranes, the nucleus, and phagocytosis is presented. BioEssays 25:1129–1138, 2003. © 2003 Wiley Periodicals, Inc. (shrink)

What is the epistemological structure of situations where many small risks amount to a large one? Lottery and preface paradoxes and puzzles about quantum-mechanical blips threaten the idea that competent deduction is a way of extending our knowledge. Seemingly, everyday knowledge involves small risks, and competently deducing the conjunction of many such truths from them yields a conclusion too risky to constitute knowledge. But the dilemma between scepticism and abandoning MPC is false. In extreme cases, objectively improbable truths (...) are known. Safety is modal, not probabilistic, in structure, with closure and factiveness conditions. It is modelled using closeness of worlds. Safety is analogous to knowledge. It suggests an interpretation of possible worlds semantics for epistemic logic. To avoid logical omniscience, a relation of epistemic counterparthood between formulas is introduced. This supports a safety conception of knowledge and formalizes how extending knowledge by deduction depends on logical competence. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Christopher SmallAlbi Odendaal and Heidi WesterlundNeville Charles Christopher Small, musician, composer, teacher, lecturer, and author, died in Sitges, Spain, on 6 September 2011 at the age of 84. A funeral was held close to Sitges, the community he had made home for the last 25 years of his life and where he had lived with his long-time partner Neville Braithwaithe (1927-2006).Christopher studied Zoology and completed a Bachelor of (...) Science degree in 1949 at Otago University in his country of birth, New Zealand, before deciding to study music. He completed a bachelor's degree in music at the newly-formed Victoria University in Wellington in 1955. A scholarship from the New Zealand government allowed him to pursue composition studies in England, which became his home for the next 25 years. He taught music and lectured at Alperton High School, Anstey College of Education, and Ealing College of Higher Education during this time. He also wrote his first book Music, Society, Education after the prompting of John Calder, who published the book in 1977. Two other books followed after an early retirement to Spain in 1986: Music of the Common Tongue (1987) and Musicking (1988). He also published several journal articles and recently contributed an afterword to Ruth Wright, ed., Sociology and Music Education (2010).Christopher will be remembered for his lucid writing and willingness to ask big questions. His three books (all of which are still in print) address significant [End Page 93] issues, notably the role of music in society and that of society in music. Perhaps his most enduring contribution to the thinking on music is the coining of the term, "musicking," which has been widely adopted by authors and researchers who wish to share his conception that music is not a noun but a verb, not a thing that has separate existence but an action that humans perform.A common theme in his writings is to challenge conventional wisdom, especially when he perceives it to be entrenched in an elitist position. In this vein he questions the scientific-industrial world view and the effects it has on music and education in his first book. Then he goes on to show the importance of community and communality for the musics of the African diaspora in his second book, a direct challenge to the individualist and elitist attitude that he found in Western classical music.His third book, Musicking, investigates a typical Western classical symphony concert in an attempt to answer the question, "What is really going on here?" Understanding music as human action allows him to interrogate aspects of this event that are usually unexamined and leads him to emphasize that all music has social origins and implications. He argues that the enjoyment that comes from musical events is directly related to the significance of the relationships that are either formed or pre-existing, real or imagined, between performers, audience, musical sounds, and physical space. He argues that the significance of these relationships, and thus of the musical experience, is related to how well they fit our image of who we are or want to be.In his writings, Christopher offers a positive and buoyant view of human society, one in which everyone musicks and where the musicking is communal and interactive. Writing with such a broad and positive perspective may sometimes lead him to oversimplify an argument or to gloss over inconsistencies that would show themselves with more careful analysis. However, many of the conclusions he draws from these glosses carry their own weight and do not weaken the argument significantly when a more complex understanding is substituted. This is partly because he is ultimately not writing as an ethnographer but, rather, as a visionary.It was our honor to meet him, for probably the last formal interview he gave, in March 2011. He was kind, generous, and friendly to a fault and, despite being in some discomfort, welcomed a group of eleven researchers from the Sibelius Academy to his house for some intense discussions. His thoughts still enliven ours and challenge us to think more deeply and widely about the world that we live in. [End Page 94]Albi Odendaal and Heidi... (shrink)

One result of this note is about the nonconstructivity of countably infinite lotteries: even if we impose very weak conditions on the assignment of probabilities to subsets of natural numbers we cannot prove the existence of such assignments constructively, i.e., without something such as the axiom of choice. This is a corollary to a more general theorem about large-small filters, a concept that extends the concept of free ultrafilters. The main theorem is that proving the existence of large- (...) class='Hi'>small filters requires a nonconstructive axiom like AC. (shrink)

There is something genuinely puzzling about large-scale simplicity emerging in systems that are complex at the small scale. Consider, for example, a population of hares. Clearly, the number of hares at any given time depends on hare fertility rates, the weather, the number of predators, the health of the predators, availability of hare resources, motor vehicle traffic, individual hare locations, colour of individual hares, and so on. Indeed, given the incredibly complexity of the hares’ environment at the small-scale, (...) it is amazing that anything can be said about hare abundances. But not only can we say something about hare abundances, we can formulate equations for hare abundance as a function of time that are remarkably accurate. But most amazing of all is that such equations have very few parameters—in the simplest cases, just the growth rate and an initial population abundance. How can this be? How can we ignore all the small-scale factors when they clearly play major roles in determining abundance? Put somewhat more grandiosely: How is population ecology possible? Of course it’s not just population ecology that exhibits such small-scale complexity and large-scale simplicity. Other important systems include the weather and gases in equilibrium. The examples can easily be multiplied. The general problem is the same: How can seemingly unpredictable and complex microbehaviour of complex systems result in predictable and simple macrobehaviour? Providing an answer to this question is the central task of Michael Strevens’ excellent book Bigger than Chaos : Understanding Complexity through Probability. (shrink)

Our understanding of the decisions made under scenarios where both descriptive and experience-based information are available is very limited. Underweighting of small probabilities was observed in the gain domain when both description and experience were provided. The divergence observed from the prospect theory suggests a need for a separate or modified theory of decision making under risk. Recent studies suggest a possible role of probability weighting in the choice behaviour under risk. We investigated both gain and loss domains (...) with and without feedback for small and large probability conditions. We characterized the shape of the probability weighting function by a two-parameter functional form representing discriminability (concave-convex shape) and attractiveness (level of absolute weights relative to objective probability). We replicated four-fold pattern of risk attitude on non-WEIRD population. We find that feedback leads to underweighting of small probabilities and overweighting of large probabilities in the gain domain and overall underweighting of probabilities in the loss domain. We find that underweighting of small probabilities is driven by changes in discriminability and attractiveness components in the gain domain and changes in the attractiveness component in the loss domain. We have interpreted the results by proposing an updated belief-based account of decisions under uncertainty in which feedback, when available, influences the probability weighting mediating the choice behaviour. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:137. PROBABILITY IN HUME'S SCIENCE OF MAN This paper is an attempt to make sense of a fragment of Hume's positive philosophy, namely his theory of how we apportion belief on the basis of ambiguous evidence. The topic is one that has received little critical attention from philosophers. One reason for this neglect is the belief that Hume's discussion of probable reasoning is not addressed to philosophical questions, but (...) rather is concerned merely to give a psychological theory of why we tend to make the inferences we do. Another is the view that Hume's psychology of probability is too obscure to merit serious study. I hope to show, however, that Hume's discussion of probable reasoning contains more philosophy, and more interesting psychology, than the prevalence of these attitudes would suggest. The main philosophical content that I see in Hume's account of probable reasoning is that it embodies a certain theory of belief, worked out in some detail. Here the central notion is the "belief-feeling", vivacity. After an initial discussion of Hume's use of this notion, I hazard the opinion that vivacities are related to probabilities by a simple subtraction formula (§1). The investigation of Hume's account of probable inference (§2) gives strong grounds for thinking that this conjecture is indeed correct. Finally, an examination of what Hume says about the influence of probability on the passions unearths a bold theory of the specific psychological mechanism by which the subtraction of probabilities is effected (§3).§1 Degrees of belief Hume's philosophy makes heavy use of the quality of perceptions which he referred to variously as "force", "liveliness", "violence", "vivacity", "strength", "firmness" and so on. As the variety of these terms indicates, Hume was not entirely happy with any of them. He says it is 138. impossible by words to describe this feeling, which every 2 one must be conscious of in his own breast (A19). But we need to use some word, and rather than emulate Hume's diverse usage it will be convenient to adhere uniformly to the single term 'vivacity'. The first use to which Hume puts vivacity is in distinguishing impressions from ideas. Intuitively, impressions are either sensations, passions or emotions, whereas ideas are the perceptions involved in thinking or reasoning. And Hume maintains that we can distinguish these two kinds of perceptions by their differing degrees of vivacity: all impressions have greater vivacity than any idea. Apparently Hume thinks that the difference in vivacity between impressions and ideas is normally quite considerable, although in a few instances there is only a small differential (T2). Vivacity is again used by Hume to distinguish between belief and conception. Although to have an idea of God, say, is not the same thing as to believe in God, Hume argues that what distinguishes the latter is only the manner of conception. In belief, the idea feels different to an idea that is entertained incredulously. And this difference in feeling is identified by Hume with a difference in vivacity (T96).3 In general, Hume thinks of beliefs as being ideas, not impressions (T86 being an exception). This is the natural view, if one thinks of the distinction between impressions and ideas as a distinction between feeling and thinking (T2, A9 ). What distinguishes beliefs then is a somewhat intermediate vivacity, inferior to the vivacity of an impression but superior to the vivacity of a mere conception. A third use to which Hume puts vivacity is in making the distinction between memory and other beliefs. Just as beliefs are distinguished from other conceptions by their greater vivacity, so likewise memories are distinguished from other beliefs by their greater vivacity (T153). 139. Each of these three applications of vivacity seems to be implausib^e. Introspection inevitably suggests that impressions are not always more vivacious or forceful, or whatever, than ideas. Similarly it seems that our beliefs, and even our memories, are often less vivid than our fancies, contrary to what Hume claims. A considerable part of Hume's appendix to the Treatise is concerned with such objections. Hume appears to allow that what we would naturally call the vivacity of ideas is not a quality... (shrink)

The field of of imprecise probability has matured, in no small part because of Teddy Seidenfeld’s decades of original scholarship and essential contributions to building and sustaining the ISIPTA community. Although the basic idea behind imprecise probability is (at least) 150 years old, a mature mathematical theory has only taken full form in the last 30 years. Interest in imprecise probability during this period has also grown, but many of the ideas that the mature theory serves can be difficult (...) to apprehend to those new to the subject. Although these fundamental ideas are common knowledge in the ISIPTA community, they are expressed, when they are expressed at all, obliquely, over the course of years with students and colleagues. -/- A single essay cannot convey the store of common knowledge from any research community, let alone the ISIPTA community. But, this essay nevertheless is an attempt to guide those familiar with the basic Bayesian framework to appreciate some of the elegant and powerful ideas that underpin the contemporary theory of lower previsions, which is the theory that most people associate with the term ‘imprecise probabilities’. (shrink)

The topic of this paper is our knowledge of the natural numbers, and in particular, our knowledge of the basic axioms for the natural numbers, namely the Peano axioms. The thesis defended in this paper is that knowledge of these axioms may be gained by recourse to judgements of probability. While considerations of probability have come to the forefront in recent epistemology, it seems safe to say that the thesis defended here is heterodox from the vantage point of traditional philosophy (...) of mathematics. So this paper focuses on providing a preliminary defense of this thesis, in that it focuses on responding to several objections. Some of these objections are from the classical literature, such as Frege's concern about indiscernibility and circularity, while other are more recent, such as Baker's concern about the unreliability of small samplings in the setting of arithmetic. Another family of objections suggests that we simply do not have access to probability assignments in the setting of arithmetic, either due to issues related to the~$\omega$-rule or to the non-computability and non-continuity of probability assignments. Articulating these objections and the responses to them involves developing some non-trivial results on probability assignments, such as a forcing argument to establish the existence of continuous probability assignments that may be computably approximated. In the concluding section, two problems for future work are discussed: developing the source of arithmetical confirmation and responding to the probabilistic liar. (shrink)

Moritz Schulz explores counterfactual thought and language: what would have happened if things had gone a different way. Counterfactual questions may concern large scale derivations or small scale evaluations of minor derivations. A common impression, which receives a thorough defence in the book, is that oftentimes we find it impossible to know what would have happened. However, this does not mean that we are completely at a loss: we are typically capable of evaluating counterfactual questions probabilistically: we can say (...) what would have been likely or unlikely to happen. Schulz describes these probabilistic ways of evaluating counterfactual questions and turns the data into a novel account of the workings of counterfactual thought. (shrink)

We propose a discrete time branching process to model the appearance of drug resistance under treatment. Under our assumptions at every discrete time a pathogen may die with probability 1−p or divide in two with probability p. Each newborn pathogen is drug resistant with probability μ. We start with N drug sensitive pathogens and with no drug resistant pathogens. We declare the treatment successful if all pathogens are eradicated before drug resistance appears. The model predicts that success is possible only (...) if p<1/2. Even in this case the probability of success decreases exponentially with the parameter m=μN. In particular, even with a very potent drug (i.e. p very small) drug resistance is likely if m is large. (shrink)

We consider a proportional-prize contest to forecast future events, and show that, in equilibrium, this mechanism possesses perfect forecasting ability for any group size when the contestants share common knowledge about the probabilities by which future events realize. Data gathered in a laboratory experiment confirm the performance invariance to group size. By contrast, when realization probabilities are not common knowledge, there are some differences across group sizes. The mechanism operates marginally better with three or four compared to two (...) players. However, this effect is mainly driven by players’ behavior rather than by differences in the beliefs they form about the realization probabilities. (shrink)

Accounts of big data practices often assume that they target individuals. Personalization, with all the risks of discrimination and bias it entails, has been the critical focus in accounts of consumption, government, social media, and health. This paper argues that personalization through models using large-scale data is part of a more expansive change in probabilization that, in principle, is not reducible to individual or ‘personal’ attributes and actions. It describes the ‘personalization’ of an online grocery shopping recommender system to list (...) a small number of grocery items of personal relevance for each of the millions of online grocery shoppers at a major UK supermarket chain. Drawing on a theory of probability proposed by the philosopher of science Karl Popper and anthropological work on shopping, it suggests that the attempt to generate personalized predictions necessarily incorporates impersonal relations to others and things. Using a mixture of discourse analysis and code-based reconstruction of key elements of the recommender system, it suggests that personalization is one facet of an open-ended weave of propensities associated with people and things in contemporary big data configurations. The paper explores how, in the context of recommender systems, the constitutive incompleteness of shopping lists, their propensity to expand or change, might be more important than their capacity to be personalized. (shrink)

One of the tasks faced by young children is the segmentation of a continuous stream of speech into discrete linguistic units. Early in development, syllables emerge as perceptual primitives, and the wholesale storage of syllable chunks is one possible strategy for bootstrapping the segmentation process. Here, we investigate what types of chunks children store. Our method involves selecting syllabified utterances from corpora of child-directed speech, which we vary according to (a) their length in syllables, (b) the mutual predictability of their (...) syllables, and (c) their frequency. We then use the number of utterances within which words are contained to predict the time course of word learning, arguing that utterances which perform well at this task are also more likely to be stored, by young children, as undersegmented chunks. Our results show that short utterances are best-suited for predicting when children acquire the words contained within them, although the effect is rather small. Beyond this, we also find that short utterances are the most likely to correspond to words. Together, the two findings suggest that children may not store many complete utterances as undersegmented chunks, with most of the units that children store as hypothesized words corresponding to \textit{actual} words. However, dovetailing with an item-based account of language-acquisition, when children do store undersegmented chunks, these are likely to be short sequences -- not frequent or internally predictable multi-word chunks. We end by discussing implications for work on formulaic multi-word sequences. (shrink)

If sociopathy is a biological adaptation, it probably evolved in small social groups in which individuals lacked the social mobility required for a con-man strategy to work. On the other hand, conflicts between groups may have provided a large niche for sociopathy throughout human history.

In lieu of an abstract, here is a brief excerpt of the content:125, HUME'S PROBABILITY ARGUMENT OF?,??,? In the Treatise,?,??,?, Hume presents an follows:' argument which, in the barest of outlines, goes as 1 (Pl) Every proposition has a probability less than one. (P2) If reason were the basis of our beliefs, then we would have no beliefs. (follows from (Pl)) (P3) We in fact do have beliefs. Hence, (P4) Reason is not the basis of our beliefs. The argument has (...) not been particularly well received. D. C. Stove, for example, refers to it as being "not merely defective, but one of the worst 2 arguments ever to impose itself on a man of genius." While not everyone is as unsympathetic as Stove, it nonetheless is difficult to find commentators favorably disposed toward the argument. Various sections of the Treatise are notoriously unclear.?,??,? is such a section; as such, it is difficult to say just what Hume intended his argument to be. My contention in the present paper is that there are two reasonable ways of reconstructing Hume's argument, both consistent with what Hume writes in?,??,?. The first reconstruction is clearly unsound; the second reconstruction fares somewhat better. In particular, my contention will be that this second reconstruction, if not sound, is at least valid 4 and contains no obviously false premises. Allow me to make some preliminary comments before presenting these reconstructions. The reconstructions to follow will primarily be concerned with premise (P2) above; in particular, the 126. reconstructions will focus on how Hume can claim that (P2) follows from (Pl). Premise (Pl) is a common 5 sceptical claim, not obviously false, and certainly not unique to Hume. As such, it strikes me that the more interesting part of the argument is Hume's claim that (P2) follows from (Pl), and such will be the focus of the reconstructions (although (Pl) will be discussed toward the end of this paper). Concerning (P2), why does Hume think this follows from (Pl)? His reasoning is contained in the following passage; Having thus found in every probability, beside the original uncertainty inherent in the subject, a new uncertainty deriv'd from the weakness of that faculty, which judges, and having adjusted these two together, we are oblig'd by our reason to add a new doubt deriv'd from the possibility of error in the estimation we make of the truth and fidelity of our faculties. This is a doubt, which immediately occurs to us, and of which, if we wou'd closely pursue our reason, we cannot avoid giving a decision. But this decision, tho' it shou'd be favourable to our preceeding judgment, being founded only on probability, must weaken still further our first evidence, and must itself be weaken'd by a fourth doubt of the same kind, and so on in infinitum; till at last there remain nothing of the original probability, however great we may suppose it to have been, and however small the diminution by every new uncertainty. No finite object can subsist under a decrease repeated in infinitum; and even the vastest quantity, which can enter into human imagination, must in this manner be reduc'd to nothing. (T182) Not for nothing did I claim that this section of the Treatise is unclear. At best, I can discern only the outline of Hume's reasoning, which seems to go as follows: first, when we assign a probability to some 127, proposition, reason dictates that we re-evaluate this probability. In particular, reason dictates that we "add a new doubt deriv'd from the possibility of error in the estimation we make of the truth and fidelity of our faculties." Note that Hume never specifies the object of this 'new doubt.' He might mean a) that our doubt concerns whether the probability we assigned the proposition is correct, or b) that our doubt concerns the evidence on which we based the assignment of that probability. Whatever the case, this 'new doubt' must "weaken still further our first evidence," presumably resulting in a lowering of the probability we originally assigned to the proposition. We then repeat this doubting process, adding new doubts of the same kind in infinitum, "till at last there remain nothing of the original... (shrink)

We consider the question of randomness of the probability ΩU[X] that an optimal Turing machine U halts and outputs a string in a fixed set X. The main results are as follows: ΩU[X] is random whenever X is $\Sigma _{n}^{0}$-complete or $\Pi _{n}^{0}$-complete for some n ≥ 2. However, for n ≥ 2, ΩU[X] is not n-random when X is $\Sigma _{n}^{0}$ or $\Pi _{n}^{0}$ Nevertheless, there exists $\Delta _{n+1}^{0}$ sets such that ΩU[X] is n-random. There are $\Delta _{2}^{0}$ sets (...) X such that ΩU[X] is rational. Also, for every n ≥ 1, there exists a set X which is $\Delta _{n+1}^{0}$ and $\Sigma _{n}^{0}$-hard such that ΩU[X] is not random. We also look at the range of ΩU as an operator. We prove that the set {ΩU[X]: X ⊆ 2<ω} is a finite union of closed intervals. It follows that for any optimal machine U and any sufficiently small real r, there is a set X ⊆ 2<ω recursive in ∅′ ⊕ r, such that ΩU[X] = r. The same questions are also considered in the context of infinite computations, and lead to similar results. (shrink)

Objective interpretations of probability are usually discussed in two varieties: frequency and propensity accounts. But there is a third, neglected possibility, namely, probabilities as deriving from ranges in suitably structured initial state spaces. Roughly, the probability of an event is the proportion of initial states that lead to this event in the space of all possible initial states, provided that this proportion is approximately the same in any not too small interval of the initial state space. This idea (...) can also be expressed by saying that in the types of situations that give rise to probabilistic phenomena we may expect to find an initial state space such that any "reasonable" density function over this space leads to the same probabilities for the possible outcomes. This "method of arbitrary functions" was introduced by Poincaré, studied and extended by Hopf and more recently by Eduardo Engel, Jan von Plato and Michael Strevens. The natural-range, or method-of-arbitrary-functions approach to probabilities is usually treated as an explanation for the occurrence of probabilistic patterns, whereas I examine its prospects for an objective interpretation of probability, in the sense of providing truth conditions for probability statements that do not depend on our state of mind or information. The main objection to such a proposal is that it is circular, i.e. presupposes the concept of probability, because a measure on the initial state has to be introduced, and density functions over the space are considered. I try to argue that this objection can be successfully met. (shrink)

Why do people enjoy watching scary movies? Should we bet on the existence of God? Why is pleasure better than pain? And when is a duck not a duck? Mini Philosophy is a fascinating journey into what some of the greatest minds of the last 2500 years have to say about the big questions in life, and why they are relevant to us today. Covering everything from Sun Tzu's strategy for winning at board games to Freud's insights into our 'death (...) drive'; why De Beauvoir believed the mothering instinct is a myth to why Schopenhauer probably wasn't much fun at parties, these mini meditations will expand your mind (and bend it too). (shrink)

Assuming it rational to treat other persons' probabilities as epistemically significant, how shall their judgements be weighted (Barnes [1998])? Several plausible methods exist, but theorems in classical psychometrics greatly reduce the importance of the problem. If scientists' judgements tend to be positively correlated, the difference between two randomly weighted composites shrinks as the number of judges rises. Since, for reasons such as representative coverage, minimizing bias, and avoiding elitism, we would rarely employ small numbers of judges (e.g. less (...) than 10), the difference between two weighting systems becomes negligible. Suggestions are made for quantifying verisimilitude, identifying 'types' of scientists or theories (taxometrics), inferring latent factors, and estimating reliability of pooled judgements. (shrink)

Coevolution between two antagonistic species has been widely studied theoretically for both ecologically- and genetically-driven Red Queen dynamics. A typical outcome of these systems is an oscillatory behavior causing an endless series of one species adaptation and others counter-adaptation. More recently, a mathematical model combining a three-species food chain system with an adaptive dynamics approach revealed genetically driven chaotic Red Queen coevolution. In the present article, we analyze this mathematical model mainly focusing on the impact of species rates of evolution (...) (mutation rates) in the dynamics. Firstly, we analytically proof the boundedness of the trajectories of the chaotic attractor. The complexity of the coupling between the dynamical variables is quantified using observability indices. By using symbolic dynamics theory, we quantify the complexity of genetically driven Red Queen chaos computing the topological entropy of existing one-dimensional iterated maps using Markov partitions. Co-dimensional two bifurcation diagrams are also built from the period ordering of the orbits of the maps. Then, we study the predictability of the Red Queen chaos, found in narrow regions of mutation rates. To extend the previous analyses, we also computed the likeliness of finding chaos in a given region of the parameter space varying other model parameters simultaneously. Such analyses allowed us to compute a mean predictability measure for the system in the explored region of the parameter space. We found that genetically driven Red Queen chaos, although being restricted to small regions of the analyzed parameter space, might be highly unpredictable. (shrink)

In Everett's many worlds interpretation, quantum measurements are considered to be decoherence events. If so, then inexact decoherence may allow large worlds to mangle the memory of observers in small worlds, creating a cutoff in observable world size. Smaller world are mangled and so not observed. If this cutoff is much closer to the median measure size than to the median world size, the distribution of outcomes seen in unmangled worlds follows the Born rule. Thus deviations from exact decoherence (...) can allow the Born rule to be derived via world counting, with a finite number of worlds and no new fundamental physics. (shrink)

The present study tests a gestalt explanation for the gambler's fallacy which posits that runs in random events will be expected to reverse only when the run is open or ongoing. This is contrasted with the law of small numbers explanation suggesting that people expect random outcomes to balance out generally. Sixty-one university students placed hypothetical guesses and bets on a series of coin tosses. Either heads or tails were dominant . In a closed run condition the run ended (...) prior to the critical trial , and in an open run condition the run remained open . As hypothesised, participants showed the gambler's fallacy in the open run condition, but not in the closed run condition. This difference is not due to differential memory for the outcomes. Men, and people with more previous experience gambling, were also found to be more prone to the gambler's fallacy. It is argued that the gestalt explanation best explains the results. (shrink)

Kendall's (1956) approach to the general epidemic is generalized by dropping the assumptions of constant infectivity and random recovery or death of ill individuals. A great deal of attention is paid to the biological background and the heuristics of the model formulation. Some new results are: (l) the derivation of Kermack's and McKendrick's integral equation from what seems to be the most general set of assumptions in section 2.2, (2) the use of Kermack's and McKendrick's final value equation to arrive (...) at a finite time version of the threshold theorem for the general case, comparable to that for the case of only one Markovian state of illness in section 2.5, (3) the analysis of the behaviour of the solutions of the integral equation when the starting infection approaches zero in section 2.7, (4) the derivation of the probability structure of a general branching process, after conditioning on extinction in section 3.6, (5) the statement of the generalized versions of Kendall's ideas in the form of precise limit conjectures in section 4, (6) the derivation of a closed expression for the limit epidemic resulting from (3) in appendix 4. (shrink)

Imagine the following situation: an act of violent assault has been committed. And there are only two possible suspects, of which one is a small and weak man and the other a big and strong man. The weak man will plead that he is not strong enough and therefore not likely to have committed the crime, which seems reasonable straight away. But there will also be a loophole for the strong man, as Aristotle tells us, who reports exactly that (...) story in book 2, chapter 24, of his Rhetoric. And he also has a name to assign to the inventor of that kind of argument (1402a17–20):The Art of Corax is composed of this topic. For if a man is not likely to be guilty of what he is accused of, for instance if, being weak, he is .. (shrink)

Which is better: a guarantee of a modest amount of moral value, or a tiny probability of arbitrarily large value? To prefer the latter seems fanatical. But, as I argue, avoiding such fanaticism brings severe problems. To do so, we must decline intuitively attractive trade-offs; rank structurally identical pairs of lotteries inconsistently, or else admit absurd sensitivity to tiny probability differences; have rankings depend on remote, unaffected events ; and often neglect to rank lotteries as we already know we would (...) if we learned more. Compared to these implications, fanaticism is highly plausible. (shrink)