Applications of Probability, Misc

Inductive logic would be the logic of arguments that are not valid, but nevertheless justify belief in something like the way in which valid arguments would. Maybe we could describe it as the logic of “almost valid” arguments. There is a sort of transitivity to valid arguments. Valid arguments can be chained together to form arguments and such arguments are themselves valid. One wants to distinguish the “almost valid” arguments by noting that chains of “almost valid” arguments are weaker than (...) the links that form them. But it is not clear that this is so. I have an apparent counterexample the claim. Though: as is typical in these sorts of situations, it is hard to tell where the problem lies. (shrink)

The disjunction problem and the distality problem each presents a challenge that any theory of mental content must address. Here we consider their bearing on purely probabilistic causal (ppc) theories. In addition to considering these problems separately, we consider a third challenge – that a theory must solve both. We call this “the hard problem.” We consider 8 basic ppc theories along with 240 hybrids of them, and show that some can handle the disjunction problem and some can handle the (...) distality problem, but none can handle the hard problem. This is our main result. We then discuss three possible responses to that result, and argue that though the first two fail, the third has some promise. (shrink)

A popular account of luck, with a firm basis in common sense, holds that a necessary condition for an event to be lucky, is that it was suitably improbable. It has recently been proposed that this improbability condition is best understood in epistemic terms. Two different versions of this proposal have been advanced. According to my own proposal :361–377, 2010), whether an event is lucky for some agent depends on whether the agent was in a position to know that the (...) event would occur. And according to Stoutenburg :319–334, 2015, Synthese, 1–15, 2018), whether an event is lucky for an agent depends on whether the event was guaranteed or certain to occur in light of the agent’s evidence. In this paper, I argue that we should prefer the account in terms of knowledge over that in terms of evidential certainty. (shrink)

The new evidence scholarship addresses three distinct approaches: legal probabilism, Bayesian decision theory and relative plausibility theory. Each has major insights to offer, but none seems satisfactory as it stands. This paper proposes that relative plausibility theory be modified in two substantial ways. The first is by defining its key concept of plausibility, hitherto treated as primitive, by generalising the standard axioms of probability. The second is by complementing the descriptive component of the theory with a normative decision theory adapted (...) to legal process. Because this version of decision theory is based on plausibilities rather than probabilities, it generates plausibilistic expectations as outputs. Because these outputs are comparable, they function as relative plausibilities. Hence the resulting framework is an extension of relative plausibility theory, but it retains deep ties to legal probabilism, through the proposed definition of plausibility, and to Bayesian decision theory, through the normative use of decision theory. (shrink)

Bayesian epistemologyEpistemology offers a powerful framework for characterizing scientific inference. Its basic idea is that rational belief comes in degrees that can be measured in terms of probabilities. The axioms of the probability calculus and a rule for updatingUpdating emerge as constraints on the formation of rational belief. Bayesian epistemologyEpistemology has led to useful explications of notions such asConfirmation confirmation. It thus is natural to ask whether Bayesian epistemologyEpistemology offers a useful framework for thinking about the inferences implicit in the (...) validation of computer simulations. The aim of this chapter is to answer this question. Bayesian epistemologyEpistemology is briefly summarized and then applied to validation. UpdatingUpdating is shown to form a viable method for data-driven validation. Bayesians can also express how a simulation obtains prior credibilityCredibility because the underlying conceptual modelConceptual model is credible. But the impact of this prior credibilityCredibility is indirect since simulations at best provide partial and approximate solutions to theConceptual model conceptual model. Fortunately, this gap between the simulations and the conceptual model can be addressed using what we call Bayesian verification. The final part of the chapter systematically evaluates the use of Bayesian epistemologyEpistemology in validation, e.g., by comparing it to a falsificationist approach. It is argued that Bayesian epistemologyEpistemology goes beyond mere calibrationCalibration and that it can provide the foundations for a sound evaluationEvaluation of computer simulations. (shrink)

In this article applied and theoretical epistemologies benefit each other in a study of the British legal case of R. vs. Clark. Clark's first infant died at 11 weeks of age, in December 1996. About a year later, Clark had a second child. After that child died at eight weeks of age, Clark was tried for murdering both infants. Statisticians and philosophers have disputed how to apply Bayesian analyses to this case, and thereby arrived at different judgments about it. By (...) dwelling on this applied case, I make theoretical gains: clarifying and defending pragmatic principles of inference that are important for estimating key probabilities in a range of cases. Then, partly by drawing on such principles, and uncovering overlooked data on post-partum psychosis, I make applied gains: improving the rationality of judgments about the Sally Clark case in particular, judgments important to future similar cases. (shrink)

William James’s moral and political thought was remarkably well adapted to its historical context, in particular to the emergence in the late nineteenth century of a generalized culture of uncertainty, contingency, and probability that called into question traditional conceptions of sovereign selfhood and autonomous freedom. Facing the solidification of numerous apparatus of chance, James developed a strenuous ethics rooted in a conception of freedom as self-transformation. That this ethics was attuned to the pressing problematics of his day is shown by (...) sideways glances at his contemporaries Edith Wharton, William Dean Howells, Charles Sanders Peirce, Josiah Royce, as well as then-emergent forms of everydayness such as personal health insurance. Such an ethics remains a valuable source of freedom today insofar as we remain bound to chance as an ongoing problematic conditioning our moral possibilities. (shrink)

How can the objectivity of an argument’s conclusion be determined? To propose an answer, this paper builds on Betz’s view of premises as hedged hypotheses. If an argument’s premises are hedged, its conclusion must be hedged as well. But how? The paper first introduces a two-dimensional critical grid. The grid’s vertical dimension is inductive, reflecting the argument’s downward flow from premises to conclusion. It specifies the inductive probability of the conclusion given the premises. The grid’s horizontal dimension is epistemic, focusing (...) on the premises without dropping down to the conclusion. It evaluates the epistemic probability of the premises when conjoined. This two-dimensional grid is then applied to three kinds of cases: vertical and horizontal evaluations rely on point-valued probabilities; vertical and horizontal evaluations rely on interval probabilities; vertical and horizontal evaluations rely on non-numeric plausibilities. The result is that, in each case, the argument’s conclusion can be assigned a credence tag, as it were, that reflects a critical appraisal of its objectivity. Reference Betz, Gregor. 2013. “In defence of the value free ideal.” European Journal for Philosophy of Science 3, 207–220. (shrink)

It is now generally recognized that Earth is at risk of a devastating collision with an asteroid or a comet. Impressive strides in our understanding of this threat have been made in recent decades, and various efforts to deal with it have been undertaken. However, the pace of government action hasn’t kept up with the advance of our knowledge. Despite the daunting dimensions of planetary defense, one intrepid NGO has stepped up to the plate: The B612 Foundation has embarked on (...) a half-billion-dollar project, called Sentinel, to map the estimated one million near-Earth objects that could wipe out a city or even end civilization. This article offers an explanation of how it came to be that private citizens have assumed what is arguably government’s primary mandate to "provide for the common defence." It then examines some of the practical – or, if you will, ethical – risks that may be attendant on such a shift of responsibility. Finally, a new policy emphasis is proposed. (shrink)

In this note, we consider a Merton model for default risk, where the firm’s value is driven by a Brownian motion and a compound Poisson process.

In his recent book Error Theory: History, Critique, Defence, Jonas Olson attempts to revive the argument from queerness originally made famous by J.L. Mackie. In this paper, we do three things. First, we eliminate four untenable formulations of the argument. Second, we argue that the most plausible formulation is one that depends crucially upon considerations of parsimony. Finally, we evaluate this formulation of the argument. We conclude that it is unproblematic for proponents of moral non-naturalism—the target of the argument from (...) queerness. (shrink)

This volume recreates the received notion of reflective equilibrium. It reconfigures reflective equilibrium as both a cognitive ideal and a method for approximating this ideal. The ideal of reflective equilibrium is restructured using the concept of discursive strata, which are formed by sentences and differentiated by function. Sentences that perform the same kind of linguistic function constitute a stratum. The book shows how moral discourse can be analyzed into phenomenal, instrumental, and teleological strata, and the ideal of reflective equilibrium reworked (...) in these terms. In addition, the work strengthens the method of reflective equilibrium by harnessing the resources of decision theory and inductive logic. It launches a comparative version of decision theory and employs this framework as a guide to moral theory choice. It also recruits quantitative inductive logic to inform a standard of inductive cogency. When used in tandem with comparative decision theory, this standard can aid in the effort to turn the undesirable condition of reflective disequilibrium into reflective equilibrium. (shrink)

Some propositions add more information to bodies of propositions than do others. We start with intuitive considerations on qualitative comparisons of information added . Central to these are considerations bearing on conjunctions and on negations. We find that we can discern two distinct, incompatible, notions of information added. From the comparative notions we pass to quantitative measurement of information added. In this we borrow heavily from the literature on quantitative representations of qualitative, comparative conditional probability. We look at two ways (...) to obtain a quantitative conception of information added. One, the most direct, mirrors Bernard Koopman’s construction of conditional probability: by making a strong structural assumption, it leads to a measure that is, transparently, some function of a function P which is, formally, an assignment of conditional probability (in fact, a Popper function). P reverses the information added order and mislocates the natural zero of the scale so some transformation of this scale is needed but the derivation of P falls out so readily that no particular transformation suggests itself. The Cox–Good–Aczél method assumes the existence of a quantitative measure matching the qualitative relation, and builds on the structural constraints to obtain a measure of information that can be rescaled as, formally, an assignment of conditional probability. A classical result of Cantor’s, subsequently strengthened by Debreu, goes some way towards justifying the assumption of the existence of a quantitative scale. What the two approaches give us is a pointer towards a novel interpretation of probability as a rescaling of a measure of information added. (shrink)

Objective: The aim of this study was to examine logical positivist statistical probability statements used to support and justify “evidence-based” prescribing rules in psychiatry when viewed from the major philosophical theories of probability, and to propose “phenomenological probability” based on Maurice Merleau-Ponty's philosophy of “phenomenological positivism” as a better clinical and ethical basis for psychiatric prescribing. -/- Conclusions: The logical positivist statistical probability statements which are currently used to support “evidence-based” prescribing rules in psychiatry have little clinical or ethical justification (...) when subjected to critical analysis from any of the major theories of probability and represent dangerous “spin” because they necessarily exclude the individual , intersubjective and ambiguous meaning of mental illness. A concept of “phenomenological probability” founded on Merleau-Ponty's philosophy of “phenomenological positivism” overcomes the clinically destructive “objectivist” and “subjectivist” consequences of logical positivist statistical probability and allows psychopharmacological treatments to be appropriately integrated into psychiatric treatment. -/- Read More: http://informahealthcare.com/doi/abs/10.3109/10398562.2011.603333. (shrink)

To clarify and illuminate the place of probability in science Ellery Eells and James H. Fetzer have brought together some of the most distinguished philosophers ...

Some risks have extremely high stakes. For example, a worldwide pandemic or asteroid impact could potentially kill more than a billion people. Comfortingly, scientific calculations often put very low probabilities on the occurrence of such catastrophes. In this paper, we argue that there are important new methodological problems which arise when assessing global catastrophic risks and we focus on a problem regarding probability estimation. When an expert provides a calculation of the probability of an outcome, they are really providing the (...) probability of the outcome occurring, given that their argument is watertight. However, their argument may fail for a number of reasons such as a flaw in the underlying theory, a flaw in the modeling of the problem, or a mistake in the calculations. If the probability estimate given by an argument is dwarfed by the chance that the argument itself is flawed, then the estimate is suspect. We develop this idea formally, explaining how it differs from the related distinctions of model and parameter uncertainty. Using the risk estimates from the Large Hadron Collider as a test case, we show how serious the problem can be when it comes to catastrophic risks and how best to address it. (shrink)

Like Glenn Shafer, we are nostalgic for the time when “philosophers, mathematicians, and scientists interested in probability, induction, and scientific methodology talked with each other more than they do now”, [p.10]. 1 Shafer goes on to mention other relevant contemporary communities. He himself has been at the interface of many of these communities while at the same time making major contributions to them and this very symposium represents something of that desired discussion. We begin with a couple of general points (...) about issues several commentators have raised and then discuss other more particular issues. (shrink)

Bayesian methods have become among the most popular methods in phylogenetics, but theoretical opposition to this methodology remains. After providing an introduction to Bayesian theory in this context, I attempt to tackle the problem mentioned most often in the literature: the “problem of the priors”—how to assign prior probabilities to tree hypotheses. I first argue that a recent objection—that an appropriate assignment of priors is impossible—is based on a misunderstanding of what ignorance and bias are. I then consider different methods (...) of assigning prior probabilities to trees. I argue that priors need to be derived from an understanding of how distinct taxa have evolved and that the appropriate evolutionary model is captured by the Yule birth–death process. This process leads to a well-known statistical distribution over trees. Though further modifications may be necessary to model more complex aspects of the branching process, they must be modifications to parameters in an underlying Yule model. Ignoring these Yule priors commits a fallacy leading to mistaken inferences both about the trees themselves and about macroevolutionary processes more generally. (shrink)

Evolution is a time process. It proceeds in steps of definite length. The probability of each step is relatively high, so self organization of complex systems will be possible in finite time. Prerequisite for such a process is a selection rule, which certainly exists in evolution. Therefore, it would be wrong to calculate the probability of the formation of a complex system solely on the basis of the number of its components and as a momentary event.

In this article I criticize the recommendations of some prominent statisticians about how to estimate and compare probabilities of the repeated sudden infant death and repeated murder. The issue has drawn considerable public attention in connection with several recent court cases in the UK. I try to show that when the three components of the Bayesian inference are carefully analyzed in this context, the advice of the statisticians turns out to be problematic in each of the steps.

Vagueness is epistemic, according to some. Vagueness is ontological, according to others. This article deploys what I take to be a compromise position. Predicates are coined in specific contexts for specific purposes, but these limited practices do not automatically fix the extensions of predicates over the domain of all objects. The linguistic community using the predicate has rarely considered, much less decided, all questions that might arise about the predicate’s extension. To this extent, the ontological view is correct. But a (...) predicate that applies in some contexts can be reasonably extended to other contexts where it is initially vague. This process of development approximates the cognitive remedy for vagueness that the epistemic view prescribes. The process is piecemeal and inductive, akin to what von Wright described as the molding of concepts. Vagueness cannot be understood apart from the backdrop of classification, for vagueness is classification gone awry. Hence these pages explore the classification of particulars, both its clear successes and vague failures. How we classify unique particulars is the theme of Sections 2 and 3, which are primarily descriptive. Section 2 identifies a way of classifying particulars that pervades discourse of all sorts, and Section 3 illustrates its use in a field notorious for vagueness: ethics. Why a certain particular should (or should not) be classified in a certain way is a normative question, however, and it occupies Sections 4 and 5. Section 4 proposes a norm for strong arguments by analogy, and Section 5 illustrates how the norm might resolve vagueness in one kind of ethical dispute. This norm, which has a strong probabilistic component, is one way of affirming that probability is a guide to life. (shrink)

The term ’risk’ is used in a wide range of situations, but there is no real consensus of what it means. ‘Risk ‘is often stipulatively defined as “a probability for the occurrence of a negative event” or something similar. This formulation is however not very informative, and it fails to capture many of our intuitions about the concept or risk. One way of trying to find a common definition of a term within a group is to use a Socratic Dialogue (...) (SD). This method is fairly new, and it is rather different from the original Socratic dialogues (at least if we are to judge from how they are described by Plato). The best explanation for the name ought to be that it is inspired by the original Socratic dialogues. The SD in its modern form was originally developed as a tool for enabling laymen to perform rather advanced concept analyses under the supervision of a professional philosopher. The formal goal of the method is to find a common way of perceiving of a particular term, or at least to find out exactly how the members of the group differ in their understandings of the term, and why. The largest gain from the process has in practice turned out to be a higher awareness among the participants of different ways of understanding the term, and the ideas and intuitions behind it. This has turned out to be very useful in educational settings, but the method has also been used with great success both in research, and in e.g. business, public administration and nongovernmental organisations. In the present case, a Socratic dialogue on the concept of risk was performed within the framework of a Ph D-course about risk and uncertainty at the Swedish University of Agriculture in Alnarp, Sweden. The participants on the course where all quite familiar with practical issues relating to risks. Both from the course work, and from their own research. (shrink)

Some conditions are given to ensure that for a jump homogeneous Markov process $\{X(t),t\ge 0\}$ the law of the integral functional of the process $T^{-1/2} \int^T_0\varphi(X(t))dt$ converges to the normal law $N(0,\sigma^2)$ as $T\to \infty$, where $\varphi$ is a mapping from the state space $E$ into $\bbfR$.

There is a long philosophical tradition of addressing questions in philosophy of science and epistemology by means of the tools of Bayesian probability theory (see Earman (1992) and Howson and Urbach (1993)). In the late '70s, an axiomatic approach to conditional independence was developed within a Bayesian framework. This approach in conjunction with developments in graph theory are the two pillars of the theory of Bayesian Networks, which is a theory of probabilistic reasoning in artificial intelligence. The theory has been (...) very successful over the last two decades and has found a wide array of applications ranging from medical diagnosis to safety systems for hazardous industries. (shrink)

Bonjour (1985: 101 and 1999: 124) and other coherence theorists of justification before him (e.g. Ewing, 1934: 246) have complained that we do not have a satisfactory analysis of the notion of coherence. The problem with existing accounts of coherence is that they try to bring precision to our intuitive notion of coherence independently of the particular role that it is meant to play within the coherence theory of justification (e.g Lewis, 1946: 338). This is a mistake: it does not (...) make any sense to ask what precisely makes for a more coherent information set independently of the particular role that coherence is supposed to play within the context in question. What is this context and what is this role? The coherence theory of justification rides on a particular common sense intuition: when we gather information from less than fully reliable sources, then the more coherent the story that materializes is, the more confident we may be, ceteris paribus. Within the context of information gathering from certain types of sources, coherence is a property of stories which plays a confidence boosting role. But what features should the information sources have, so that the coherence of the information set is indeed a determinant of our degree of confidence in question? And what goes into the ceteris paribus clause? In other words, what other factors affect our confidence in the information set in question? (shrink)

This chapter identifies two types of moral dilemma. The first type is described as ethical clash: whether affirmative action is just or unjust, for example, or whether withholding information from an inquisitive relative is honest or dishonest. In these cases the dilemma takes the form of conflict between an ethical predicate and its complement. The second type of moral dilemma is ethical overlap. Instead of a clash between a single predicate and its complement, here two or more predicates apply. Dilemmas (...) associated with white lies, for example, often depart from the recognition that such acts are both dishonest and avoid inflicting pain. Similarly, social dilemmas over progressive taxation may arise despite agreement that progressive systems both decrease liberty and increase equality. Which predicate should take precedence? Strategies for dealing with both types of dilemma are proposed. (shrink)

There is a substantial literature on the Bayesian approach, and the application of Bayes'' theorem, to legal matters. However, I have found no discussion that explores fully the issue of how a Bayesian juror might be led from an initial "presumption of innocence" to the judgment (required for conviction in criminal cases) that the suspect is "guilty beyond a reasonable doubt". I shall argue here that a Bayesian juror, if she acts in accord with what the law prescribes, will virtually (...) never reach such a judgment. In section I, I discuss Bayesianism, Bayes'' rule and the relation between them. Section II addresses the two legal notions key to my worries about Bayesian jurors: the presumption of innocence and the reasonable doubt criterion. Section III explores what a Bayesian is to make of these notions, and how the legal system requires her to reason in their light. If I am right, there emerges a conflict between current legal prescriptions and the Bayesian approach. Section IV explores the import of this conflict and how it might be resolved. (shrink)

The paper explores the handling of singular analogy in quantitative inductive logics. It concentrates on two analogical patterns coextensive with the traditional argument from analogy: perfect and imperfect analogy. Each is examined within Carnap’s λ-continuum, Carnap’s and Stegmüller’s λ-η continuum, Carnap’s Basic System, Hintikka’s α-λ continuum, and Hintikka’s and Niiniluoto’s K-dimensional system. Itis argued that these logics handle perfect analogies with ease, and that imperfect analogies, while unmanageable in some logics, are quite manageable in others. The paper concludes with a (...) modification of the K-dimensional system that synthesizes independent proposals by Kuipers and Niiniluoto. (shrink)

This chapter explores arguments from analogy containing ethical predicates like 'just', 'courageous', and 'honest'. The approach is Wittgensteinian in a double sense. The role of paradigm cases in ethical discourse is emphasized, first of all, and the inductive logics to be employed spring from Wittgenstein's remarks on probability (1922). Although these logics rely on a semantic concept of range, they yield results for the ethical problems treated here only if grounded in certain kinds of pragmatic consensus.

How do we distinguish good and bad analogies? Luis A. Camacho proposed that false analogies be construed as false material conditionals. This article offers a counter-proposal: analogies of all sorts can be understood as singular inductive inferences. For the sake of simplicity, this proposal is illustrated with reference to Carnap's favorite inductive method c*.

Rarely does philosophy produce empirical predictions. The Doomsday argument is an important exception. From seemingly trivial premises it seeks to show that the risk that humankind will go extinct soon has been systematically underestimated. Nearly everybody's first reaction is that there must be something wrong with such an argument. Yet despite being subjected to intense scrutiny by a growing number of philosophers, no simple flaw in the argument has been identified.

The purpose of this paper is to point the way to the money tree. Currently, almost all investment professionals think that outperformance requires an “edge,” that is, the ability to predict the future to some degree. In this paper, I suggest that money can be made in a 0, or even slightly negative, expected value environment by carefully choosing investment/bet sizes. Philosophical considerations are found mainly in Section 4.

Building on work that we reported at ISIPTA 2005 we revisit claims made by Fox and Tversky concerning their "comparative ignorance" hypothesis for decision making under uncertainty.