Philosophy of Probability

This paper begins with a problem stemming from Hume regarding credences about credences. Suppose one has a credence of .95 in p, and suppose one assesses the credence to be such. But suppose one’s second-order credence in this assessment is less than 1. Then, by a standard conditionalization rule, one’s credence in p becomes less than .95. Moreover, such “erosion” can iterate by considering one’s, third-, fourth-, fifth-order credences, etc. (In light of this, some have rejected higher-order credences; however, it (...) is argued that Lewis’ “Principal Principle” forbids this.) The paper then offers a partial solution by describing circumstances under which a person has reason to assign credence 1 to a first-order credence. The solution takes the form of a “transparency” view of self-knowledge from Parent (2017, chapter 8). Briefly, it is argued that when one “reflex-like” utters a sentence “The probability of p is n”, then as a matter of psychological law, the utterance is an expression of one’s own judgment. The Principal Principle then necessitates that one’s credence in p is n, assuming one is being rational. Roughly, upon perceiving such an utterance, one can thereby have a reason to assign a second-order credence of 1. (shrink)

This paper begins with a problem stemming from Hume regarding credences about credences. Suppose one has a credence of .95 in p, and suppose one assesses the credence to be such. But suppose one’s second-order credence in this assessment is less than 1. Then, by a standard conditionalization rule, one’s credence in p becomes less than .95. Moreover, such “erosion” can iterate by considering one’s, third-, fourth-, fifth-order credences, etc. (In light of this, some have rejected higher-order credences; however, it (...) is argued that Lewis’ “Principal Principle” forbids this.) The paper then offers a partial solution by describing circumstances under which a person has reason to assign credence 1 to a first-order credence. The solution takes the form of a “transparency” view of self-knowledge from Parent (2017, chapter 8). Briefly, it is argued that when one “reflex-like” utters a sentence “The probability of p is n”, then as a matter of psychological law, the utterance is an expression of one’s own judgment. The Principal Principle then necessitates that one’s credence in p is n, assuming one is being rational. Thus, upon perceiving such an utterance, one can thereby have a reason to assign a second-order credence of 1. (shrink)

According to libertarianism about free will, at least some of the choices we make are free and undetermined. Many libertarians also accept the thesis that, before we make an undetermined choice, there is a nontrivial objective probability that we will make that choice. In the literature on free will, the ascription of objective probabilities is sometimes justified via an “Argument from Motivation,” which adverts to the fact that typically, in situations of choice, we are more motivated to choose some options (...) over others. In this paper, I will examine this argument and I will argue that it is unsound, as one of its premises is at odds with a widely accepted principle governing the evolution of objective probabilities over time. (shrink)

This paper introduces and discusses three models of future: a determinist model, a stochastic model, and the model of True Prophetic Knowledge. All three models coexist in natural languages and are represented both in their grammatical systems and in the text-building discourse strategies speakers and authors apply to.

Ang Diyos ay hindi naglalaro ng dado", inulit ni Einstein mula sa taas ng kanyang determinismo, ngunit sa katunayan ang kosmos ay naghahagis ng mga buto nito nang sadyang mapagpasya: ang mga dado nito ay laruin. Hindi sa pag-iisip na tayo ay lumikha ng mga mundo. Sa pamamagitan ng pag-unawa sa mundo natututo tayong mag-isip. Ang Cosmovision ay isang termino na dapat ay nangangahulugang isang hanay ng mga pundasyon kung saan lumalabas ang isang sistematikong pag-unawa sa Uniberso, ang mga bahagi (...) nito bilang buhay, ang mundong ating ginagalawan, kalikasan, mga phenomena ng tao, at ang kanilang mga relasyon. Ito ay, samakatuwid, isang larangan ng analytical na pilosopiya na pinapakain ng mga agham, na ang layunin ay ang pinagsama-samang at epistemologically sustainable na kaalaman tungkol sa lahat ng kung ano tayo at nilalaman, na nakapaligid sa atin, at nauugnay sa atin sa anumang paraan. Ito ay isang bagay na kasingtanda ng pag-iisip ng tao, at, bilang karagdagan sa paggamit ng mga elemento ng siyentipikong kosmolohiya, sinasaklaw nito ang lahat ng bagay sa pilosopiya at agham na tumutukoy sa uniberso at buhay. Ang cosmovision ay hindi isang set ng mga ideya, hypotheses, at assumptions kundi isang sistemang batay sa obserbasyon, pagsusuri, ebidensya, at pagpapakita. Walang cosmovision na naglalayong tukuyin, itatag, o imungkahi ngunit para lamang maunawaan, suriin, at bigyang-kahulugan. Bawat isa sa atin ay nagtatayo at naghahatid ng kanyang kosmobisyon sa buong buhay, nang hindi nagtatatag ng mga anyo, bilang isang background para sa ating pag-iisip at pag-uugali. Sa lingguwistika, ang terminong "cosmovision" ay magmumula sa Aleman, katumbas ng konsepto ng "Weltanschauung," na ginamit ng ilang mga pilosopo. Gayunpaman, ang relasyong pangwika na ito ay hindi naaangkop dahil sumasalungat ito sa aming iminumungkahi bilang isang cosmovision. Ang salitang Aleman na ito ay tumutukoy sa isang pre-logical o proto- experimental na pananaw ng realidad, na may intuitive na konteksto at malayo sa kritikal na kaalaman na wala pa rin sa oras ng pagbabalangkas nito. Walang alinlangan, ang mga cosmovision, sa diwa kung saan natin nauunawaan ang mga ito, ay nagtataglay at gumagamit ng mga proto-experimental o pre-logical na elementong ito na kinabibilangan ng kasaysayan, ang sama-samang walang malay, at lahat ng mga archetype na dala natin. Gayunpaman, sa konseptong inilalapat natin dito, ang cosmovision ay higit na lumalampas sa nilalamang ito, una sa pamamagitan ng patuloy na pagsusumite nito upang ipakita ang kritikal na pag-iisip, at sa wakas sa pamamagitan ng paggawa ng analitikong karanasan (at hindi ang kaisipan mismo o intuwisyon) ang aktwal nitong uniberso. Inilantad ni António Lopes ang lawak ng nilalamang ito: "Ang mga cosmovision ay hindi produkto ng pag-iisip. Hindi sila nagmumula sa simpleng pagnanais na malaman. Ang pagkaunawa sa katotohanan ay isang mahalagang sandali sa pagsasaayos nito, ngunit, gayunpaman, ito ay isa lamang. Ito ay nagmumula sa mahahalagang pag-uugali, mula sa karanasan ng pagsusuri sa buhay, at mula sa istruktura ng ating saykiko na kabuuan Ang pagtaas ng buhay sa kamalayan sa kaalaman sa realidad, sa pagpapahalaga, at sa kusang katotohanan ay ang mabagal at mahirap na gawain. na ginawa ng sangkatauhan sa pagbuo ng mga konsepto ng buhay (W. Dilthey, 1992 [1911]: 120). Sa gawaing ito, hinahangad nating balangkasin ang isang kosmobisyon batay sa mga realidad na inaalok natin ngayon hindi nagmumungkahi, sa anumang oras, na gumawa ng agham; o magteorya ng pilosopiya, ngunit lagi nating hahanapin na suportahan sila o, hindi bababa sa, protektahan nila mula sa mga pang-unawang distorsyon na karaniwan nating dala. (shrink)

This book grew out of previously published papers of mine composed over a period of years; they have been reworked (sometimes beyond recognition) so as to form a reasonably coherent whole. Part One treats of informative inference. I argue (Chapter 2) that the traditional principle of induction in its clearest formulation (that laws are confirmed by their positive cases) is clearly false. Other formulations in terms of the 'uniformity of nature' or the 'resemblance of the future to the past' seem (...) to me hopelessly unclear. From a Bayesian point of view, 'learning from experience' goes by conditionalization (Bayes' rule). The traditional stum bling block for Bayesians has been to fmd objective probability inputs to conditionalize upon. Subjective Bayesians allow any probability inputs that do not violate the usual axioms of probability. Many subjectivists grant that this liberality seems prodigal but own themselves unable to think of additional constraints that might plausibly be imposed. To be sure, if we could agree on the correct probabilistic representation of 'ignorance' (or absence of pertinent data), then all probabilities obtained by applying Bayes' rule to an 'informationless' prior would be objective. But familiar contra dictions, like the Bertrand paradox, are thought to vitiate all attempts to objectify 'ignorance'. BuUding on the earlier work of Sir Harold Jeffreys, E. T. Jaynes, and the more recent work ofG. E. P. Box and G. E. Tiao, I have elected to bite this bullet. In Chapter 3, I develop and defend an objectivist Bayesian approach. (shrink)

Mới đây, chương trình máy tính ‘bayesvl’ chạy trên môi trường R do TS. Vương Quân Hoàng và kĩ sư Lã Việt Phương (Trung tâm Nghiên cứu Xã hội Liên ngành ISR, Đại học Phenikaa) thiết kế và phát triển chính thức được ra mắt trên CRAN - hệ thống thư viện chuẩn của R...

My dissertation is about Bayesian rationality for non-ideal agents. I show how to derive subjective probabilities from preferences using much weaker rationality assumptions than other standard representation theorems. I argue that non-ideal agents might be uncertain about how they will update on new information and consider two consequences of this uncertainty: such agents should sometimes reject free information and make choices which, taken together, yield sure loss. The upshot is that Bayesian rationality for non-ideal agents makes very different normative demands (...) than ideal Bayesian rationality. (shrink)

When can a shift from a prior to a posterior be represented by conditionalization? A well-known result, known as “superconditioning” and going back to work by Diaconis and Zabell, gives a sharp answer. This paper extends the result and connects it to the reflection principle and common priors. I show that a shift from a prior to a set of posteriors can be represented within a conditioning model if and only if the prior and the posteriors are connected via a (...) general form of the reflection principle. Common priors can be characterized by principles that require a certain kind of coherence between distinct sets of posteriors. I discuss the implications these results have for diachronic and synchronic modes of updating, learning experiences, the common prior assumption of game theory, and time-slice epistemology. (shrink)

According to many philosophers, belief is a settling state. On this view, someone who believes p is disposed to take p for granted in practical and theoretical reasoning. This paper presents a simple objection to this settling conception of belief: it conflicts with our ordinary patterns of belief ascription. I show that ascriptions of unsettled beliefs are commonplace, and that they pose problems for all of the most promising ways of developing the settling conception. I proceed to explore the implications (...) of my argument for the relation between belief and credence, and for the relative importance of belief in psychological explanation. (shrink)

Imprecise probabilities are often modelled with representors, or sets of probability functions. In the recent literature, two ways of interpreting representors have emerged as especially prominent: vagueness interpretations, according to which each probability function in the set represents how the agent's beliefs would be if any vagueness were precisified away; and comparativist interpretations, according to which the set represents those comparative confidence relations that are common to all probability functions therein. I argue that these interpretations have some important limitations. I (...) also propose an alternative—the functional interpretation—according to which representors are best interpreted by reference to the roles they play in the theories that make use of them. (shrink)

Beliefs come in degrees, and we often represent those degrees with numbers. We might say, for example, that we are 90% confident in the truth of some scientific hypothesis, or only 30% confident in the success of some risky endeavour. But what do these numbers mean? What, in other words, is the underlying psychological reality to which the numbers correspond? And what constitutes a meaningful difference between numerically distinct representations of belief? In this Element, we discuss the main approaches to (...) the measurement of belief. These fall into two broad categories—epistemic and decision-theoretic—with divergent foundations in the theory of measurement. Epistemic approaches explain the measurement of belief by appeal to relations between belief states themselves, whereas decision-theoretic approaches appeal to relations between beliefs and desires in the production of choice and preferences. -/- (This book is part of the Cambridge Elements in Decision Theory and Philosophy series.). (shrink)

This article aims to provide an a posteriori argument for the veracity of the Christian conception of the Abrahamic religion that centres on God’s action of sending a divine and atoning prophet—the ‘Messianic Prophet’—into the world, who we can identify as the person of Jesus of Nazareth. This specific argument will be presented through Richard Swinburne’s (modified) explanatory framework, which focuses on assessing the prior and posterior evidence in support of this identification. This, however, will be done in light of (...) the work of Historical Jesus and NT scholars John P. Meier, N.T. Wright, Richard Bauckham and Larry Hurtado—which, in combination, will provide a means to ward off two important objections against Swinburne's methodology, and ultimately allow one to establish the veracity of the ‘Christian Position’, on firm historical grounds. (shrink)

In Chapter 6 of Objects of Credence, Anna Mahtani argues that the opacity of credence raises difficulties for the Principal Principle and proposes a revised principle relating credence and chance that avoids it. In this comment on her book, I both defend Mahtani’s proposed principle against a charge of triviality and argue that the opacity of belief does not threaten the role of chance in guiding credence.

First, Mahtani argues that both in the game The Mug and in the Sleeping Beauty we should not defer to a trusted person under a particular designation if they do not self-identify under this designation. This invites a more complex Reflection Principle. I respond that there are more parsimonious ways to avoid the challenges posed to the Reflection Principle. Second, Mahtani argues that preferences create a hyperintensional context, which poses a challenge to the Ex-Ante Pareto Principle that can be averted (...) by supervaluation. I respond that such an appeal to supervaluation would block randomization as a fair allocation device. (shrink)

Real human agents, even when they are rational by everyday standards, sometimes assign different credences to objectively equivalent statements, such as “George Orwell is a writer” and “Eric Arthur Blair is a writer”, or credences less than 1 to necessarily true statements, such as not-yet-proven theorems of arithmetic. Anna Mahtani calls this the phenomenon of “opacity” (a form of hyperintensionality). Opaque credences seem probabilistically incoherent, which goes against a key modelling assumption of probability theory. I sketch a modelling strategy for (...) capturing opaque credence assignments without abandoning probabilistic coherence. I draw on ideas from judgment-aggregation theory, where we face similar challenges of defining the “objects of judgment”. (shrink)

Abstract. In this article is presented a model of the change of the probability of the global catastrophic risks in the world with exponentially evolving technologies. Increasingly cheaper technologies become accessible to a larger number of agents. Also, the technologies become more capable to cause a global catastrophe. Examples of such dangerous technologies are artificial viruses constructed by the means of synthetic biology, non-aligned AI and, to less extent, nanotech and nuclear proliferation. The model shows at least double exponential growth (...) of the probability of the global catastrophe which means that the accumulated probability of the catastrophe will grow from negligible to overwhelming at the period of a few doubling times of the technological capabilities. For biotech and AI, such doubling time roughly corresponds to the doubling period of Moore’s law and its analogues in other technologies and is around 2 years. Thus the global catastrophe in the exponential technologies world will most likely happen during a “dangerous decade”. Such a dangerous decade could start as early as in the 2020s. We also found that the double exponential growth in the model makes the model less sensitive to its initial conditions, like the number of dangerous agents or initial probabilities constants. The model also shows that smaller catastrophes could happen earlier than larger ones, and such a smaller catastrophe may be able to stop the technological growth before larger ones become possible, thus global risks will be self-limiting. However, if the growth of the number of dangerous agents will be very quick, the multiple smaller catastrophes could happen simultaneously and will be equal to a global catastrophe. (shrink)

This paper is a discussion note on Isaacs et al. (2022), who have claimed to offer a new motivation for imprecise probabilities, based on the mathematical phenomenon of non-measurability. In this note, I clarify some consequences of their proposal. In particular, I show that if their proposal is applied to a bounded 3-dimensional space, then they have to reject at least one of the following: (i) If A is at most as probable as B and B is at most as (...) probable as C, then A is at most as probable as C. (ii) Let A ∩ C = B ∩ C = ∅. A is at most as probable as B iff (A ∪ C) is at most as probable as (B ∪ C). But rejecting either statement seems unattractive. (shrink)

This book takes purism about knowledge as the default position and defends it from the challenges of pragmatic encroachment. The book is divided into two parts, a negative and a positive one. The negative part critically examines existing purist strategies in response to pragmatic encroachment. The positive part provides a new theory of how practical factors can systematically influence our confidence and explores some implications of such influence.

This essay gives an account of epistemic deference for agents with imprecise credences. I look at the two main imprecise deference principles in the literature, known as Identity Reflection and Pointwise Reflection (Moss, 2021). I show that Pointwise Reflection is strictly weaker than Identity Reflection, and argue that, if you are certain you will update by conditionalisation, you should defer to your future self according to Identity Reflection. Then I give a more general justification for Pointwise and Identity Reflection from (...) the assumption that you defer to someone whenever you consider their doxastic state to be better than yours, in the sense of leading to better decision-making. (shrink)

This work demonstrates the implementation and use of an encoder-decoder model to perform a many-to-many mapping of video data to text captions. The many-to-many mapping occurs via an input temporal sequence of video frames to an output sequence of words to form a caption sentence. Data preprocessing, model construction, and model training are discussed. Caption correctness is evaluated using 2-gram BLEU scores across the different splits of the dataset. Specific examples of output captions were shown to demonstrate model generality over (...) the video temporal dimension. Predicted captions were shown to generalize over video action, even in instances where the video scene changed dramatically. Model architecture changes are discussed to improve sentence grammar and correctness. (shrink)

Text classification is one of the most important task in natural language processing, In this research, we carried out several experimental research on three (3) of the most popular Text classification NLP classifier in Convolutional Neural Network (CNN), Multinomial Naive Bayes (MNB), and Support Vector Machine (SVN). In the presence of enough training data, Deep Learning CNN work best in all parameters for evaluation with 77% accuracy, followed by SVM with accuracy of 76%, and multinomial Bayes with least performance of (...) 69% accuracy. CNN has the best performance in the presence of large enough training dataset because of the presence of filter/ kernels which help to indentify patterns in text data regardless of their position in the sentence. We repeated the training again with just one-third of our data, at this point SVM comes with the best performance, the performance of CNN noticeably drops but still better than Multinomial Naive Bayes, the reason why SVM works best when we reduce the training data was because of its ability to look for a hyper-plane that creates a boundary between different classes of data so as to properly classify them, so we believed that getting the hyper-plane was more efficient when we reduce the dataset, hence reason for the good performance. Multinomial Naive Bayes have the least performance which we attributed to its assumption of independency between the features which sometimes does not hold true. We concluded that availability of data should be an important factor when choosing classifier for Natural Language Processing Text Classification task. CNN should be use in the presence of enough dataset, and SVM should be use when data is not enough. Multinomial Naive Bayes must not be trusted with state of the art NLP task due to its assumption of independency between the features . (shrink)

Integrating medical imaging with computing technologies, such as Artificial Intelligence (AI) and its subsets: Machine learning (ML) and Deep Learning (DL) has advanced into an essential facet of present-day medicine, signaling a pivotal role in diagnostic decision-making and treatment plans (Huang et al., 2023). The significance of medical imaging is escalated by its sustained growth within the realm of modern healthcare (Varoquaux and Cheplygina, 2022). Nevertheless, the ever-increasing volume of medical images compared to the availability of imaging experts. Biomedical experts (...) and radiologists have resulted in a widening disparity, causing an excess and overwhelming workload on these healthcare professionals (Chen et al., 2021). Several studies indicate that the present-day biomedical radiologist is now saddled with the daunting task of interpreting an image almost every 10 seconds to keep pace with the burgeoning clinical demands (McDonald et al., 2015; Hosny et al., 2018; Lantsman et al., 2022). This cognitive drain has invariably led to inevitable consequences such as delays in diagnosis and an amplified risk of diagnostic errors – thus, the biomedical imaging aspect is in dire need of methods that would aid accurate diagnostics and analytics for improved decision making. In this review, the importance of AI-related technologies such as ML and/or DL methods are reviewed in relation to the processing of medical or biomedical images along with their potentials, challenges, and possible suggestions for future studies in the health landscape. The focus will be on machine learning methods associated with the medical field of image classification. (shrink)

Events between which we have no epistemic reason to discriminate have equal epistemic probabilities. Bertrand’s chord paradox, however, appears to show this to be false, and thereby poses a general threat to probabilities for continuum sized state spaces. Articulating the nature of such spaces involves some deep mathematics and that is perhaps why the recent literature on Bertrand’s Paradox has been almost entirely from mathematicians and physicists, who have often deployed elegant mathematics of considerable sophistication. At the same time, the (...) philosophy of probability has been left out. In particular, left out entirely are the philosophical ground of the principle of indifference, the nature of the principle itself, the stringent constraint this places on the mathematical representation of the principle needed for its application to continuum sized event spaces, and what these entail for rigour in developing the paradox itself. This book puts the philosophy and its entailments back in and in so doing casts a new light on the paradox, giving original analyses of the paradox, its possible solutions, the source of the paradox, the philosophical errors we make in attempting to solve it and what the paradox proves for the philosophy of probability. The book finishes with the author’s proposed solution—a solution in the spirit of Bertrand’s, indeed—in which an epistemic principle more general than the principle of indifference offers a principled restriction of the domain of the principle of indifference. (shrink)

This paper is dedicated to numerical computation of higher order derivatives in Simulink. In this paper, a new module has been implemented to achieve this purpose within the Simulink-based Infinity Computer solution, recently introduced by the authors. This module offers several blocks to calculate higher order derivatives of a function given by the arithmetic operations and elementary functions. Traditionally, this can be done in Simulink using finite differences only, for which it is well-known that they can be characterized by instability (...) and low accuracy. Moreover, the proposed module allows to calculate higher order Lie derivatives embedded in the numerical solution to Ordinary Differential Equations (ODEs). Traditionally, Simulink does not offer any practical solution for this case without using difficult external libraries and methodologies, which are domain-specific, not general-purpose and have their own limitations. The proposed differentiation module bridges this gap, is simple and does not require any additional knowledge or skills except basic knowledge of the Simulink programming language. Finally, the block for constructing the Taylor expansion of the differentiated function is also proposed, adding so another efficient numerical method for solving ODEs and for polynomial approximation of the functions. Numerical experiments on several classes of test problems confirm advantages of the proposed solution. (shrink)

This paper considers hybrid systems — dynamical systems that exhibit both continuous and discrete behavior. Usually, in these systems, interactions between the continuous and discrete dynamics occur when a pre-defined function becomes equal to zero, i.e., in the system occurs a zero-crossing (the situation where the function only “touches” zero is considered as the zero-crossing, as well). Determination of zero-crossings plays a crucial role in the correct simulation of the system in this case. However, for models of many real-life hybrid (...) systems, such interactions may lead to the so-called Zeno executions, i.e., situations where the system undergoes an unbounded number of discrete transitions in a finite and bounded length of time. In this case, standard numerical methods of simulating the systems may fail, since the time between two transitions can decrease significantly leading to ill-conditioning of the simulation. Correct determination of zero-crossings for a complex real-life system can require a lot of computational resources and, as a consequence, slow down the simulation significantly. This paper presents a new way to execute the simulation generating time observations of the hybrid system dynamically using numerical infinitesimals introduced recently, allowing thus to determine zero-crossings more accurately. The proposed method allows to automatically detect zero-crossings with predefined accuracy and to analyze better the behavior of the system around the zero-crossings generating observations more densely, where it is necessary. Moreover, the search for zero-crossings is performed efficiently without re-evaluation of the whole system at each observation. To show the validity of the proposed algorithm, the well-known Bouncing Ball hybrid system has been studied and the obtained simulation results were compared with the standard method. (shrink)

Numerical computing is a key part of the traditional computer architecture. Almost all traditional computers implement the IEEE 754-1985 binary floating point standard to represent and work with numbers. The architectural limitations of traditional computers make impossible to work with infinite and infinitesimal quantities numerically. This paper is dedicated to the Infinity Computer, a new kind of a supercomputer that allows one to perform numerical computations with finite, infinite, and infinitesimal numbers. The already available software simulator of the Infinity Computer (...) is used in different research domains for solving important real-world problems, where precision represents a key aspect. However, the software simulator is not suitable for solving problems in control theory and dynamics, where visual programming tools like Simulink are used frequently. In this context, the paper presents an innovative solution that allows one to use the Infinity Computer arithmetic within the Simulink environment. It is shown that the proposed solution is user-friendly, general purpose, and domain independent. (shrink)

This work demonstrates the implementation and use of an encoder-decoder model to perform a many-to-many mapping of video data to text captions. The many-to-many mapping occurs via an input temporal sequence of video frames to an output sequence of words to form a caption sentence. Data preprocessing, model construction, and model training are discussed. Caption correctness is evaluated using 2-gram BLEU scores across the different splits of the dataset. Specific examples of output captions were shown to demonstrate model generality over (...) the video temporal dimension. Predicted captions were shown to generalize over video action, even in instances where the video scene changed dramatically. Model architecture changes are discussed to improve sentence grammar and correctness. (shrink)

By and large, the conditional connective in three-valued logic has two different functions. First, by means of a deduction theorem, it can express a specific relation of logical consequence in the logical language itself. Second, it can represent natural language structures such as "if/then'" or "implies''. This chapter surveys both approaches, shows why none of them will typically end up with a three-valued material conditional, and elaborates on connections to probabilistic reasoning.

We give two new ways of calculating the probability of a chord of circumference randomly selected being larger than the side of an equilateral triangle inscribed in the circumference (this problem is known as the Bertrand paradox). The first one employs an immersion in R4, and the second one uses a direct probability measure over the set of chords.

Bertrand’s paradox is a problem in geometric probability that has resisted resolution for more than one hundred years. Bertrand provided three seemingly reasonable solutions to his problem — hence the paradox. Bertrand’s paradox has also been influential in philosophical debates about frequentist versus Bayesian approaches to statistical inference. In this paper, the paradox is resolved (1) by the clarification of the primary variate upon which the principle of maximum entropy is employed and (2) by imposing constraints, based on a mathematical (...) analysis, on the random process for any subsequent nonlinear transformation to a secondary variable. These steps result in a unique solution to Bertrand’s problem, and this solution differs from the classic answers that Bertrand proposed. It is shown that the solutions proposed by Bertrand and others reflected sampling processes that are not purely random. It is also shown that the same two steps result in the resolution of the Bing–Fisher problem, which has to do with the selection of a consistent prior for Bayesian inference. The resolution of Bertrand’s paradox and the Bing–Fisher problem rebuts philosophical arguments against the Bayesian approach to statistical inference, which were based on those two ostensible problems. (shrink)

This paper corrects three widely held misunderstandings about Maxent when used in common sense reasoning: That it is language dependent; That it produces objective facts; That it subsumes, and so is at least as untenable as, the paradox-ridden Principle of Insufficient Reason.

This paper contributes to the project of articulating and defending the supra-Bayesian approach to judgment aggregation. I discuss three cases where a person is disposed to defer to two different experts, and ask how they should respond when they learn about the opinion of each. The guiding principles are that this learning should go by conditionalisation, and that they should aim to update on the evidence that the expert had updated on. But this doesn’t settle how the update on pairs (...) of experts should go, because we also need to know how the experts are related. I work through three examples showing how the results change given different prior beliefs about this relationship. (shrink)

This dissertation looks at a set of interconnected questions concerning the foundations of probability, and gives a series of interconnected answers. At its core is a piece of old-fashioned philosophical analysis, working out what probability is. Or equivalently, investigating the semantic question of what is the meaning of ‘probability’? Like Keynes and Carnap, I say that probability is degree of reasonable belief. This immediately raises an epistemological question, which degrees count as reasonable? To solve that in its full generality would (...) mean the end of human inquiry, so that won’t be attempted here. Rather I will follow tradition and merely investigate which sets of partial beliefs are coherent. -/- The standard answer to this question, what is commonly called the Bayesian answer, says that degrees of belief are coherent iff they form a probability function. I disagree with the way this is usually justified, but subject to an important qualification I accept the answer. The important qualification is that degrees of belief may be imprecise, or vague. Part one of the dissertation, chapters 1 to 6, looks largely at the consequences of this qualification for the semantic and epistemological questions already mentioned. It turns out that when we allow degrees of belief to be imprecise, we can discharge potentially fatal objections to some philosophically attractive theses. Two of these, that probability is degree of reasonable belief and that the probability calculus provides coherence constraints on partial beliefs, have been mentioned. Others include the claim, defended in chapter 4, that chance is probability given total history. -/- As well as these semantic and epistemological questions, studies of the foundations of probability usually include a detailed discussion of decision theory. For reasons set out in chapter 2, I deny we can gain epistemological insights from decision theory. Nevertheless, it is an interesting field to study on its own, and it might be expected that there would be decision theoretic consequences of allowing imprecise degrees of belief. As I show in part two, this expectation seems to be mistaken. Chapter 9 shows that there aren’t interesting consequences of this theory for decision theory proper, and chapters 10 and 11 show that Keynes’s attempt to use imprecision in degrees of belief to derive a distinctive theory of interest rates is unsound. -/- Chapters 7 and 8 provide a link between these two parts. In chapter 7 I look at some previous philosophical investigations into the effects of imprecision. In chapter 8 I develop what I take to be the best competitor to the theory defended here – a constructivist theory of probability. On this view degrees of belief are precise, but the relevant coherence constraint is a constructivist probability calculus. This view is, I think, mistaken, but the calculus has some intrinsic interest, and there are at least enough arguments for it to warrant a chapter-length examination. (shrink)

When P(E) > 0, conditional probabilities P(H|E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(H|E)$$\end{document} are given by the ratio formula. An agent engages in ratio conditionalization when she updates her credences using conditional probabilities dictated by the ratio formula. Ratio conditionalization cannot eradicate certainties, including certainties gained through prior exercises of ratio conditionalization. An agent who updates her credences only through ratio conditionalization risks permanent certainty in propositions against which she has overwhelming evidence. To avoid this undesirable consequence, (...) I argue that we should supplement ratio conditionalization with Kolmogorov conditionalization, a strategy for updating credences based on propositions E such that P(E) = 0. Kolmogorov conditionalization can eradicate certainties, including certainties gained through prior exercises of conditionalization. Adducing general theorems and detailed examples, I show that Kolmogorov conditionalization helps us model epistemic defeat across a wide range of circumstances. (shrink)

Causal decision theory is often motivated as a ‘revision of [decision theory] intended to solve Newcomb’s problem’ (Bacon 2022). In this paper, I give a direct argument for CDT by deriving it from a venerable decision-theoretic principle: the value of knowledge principle. The general framework that I use to deliver this result also supports a new argument for conditionalization.

Anticiper les épidémies est le souhait le plus vif de toutes les institutions qui veillent sur la santé publique. Un raisonnement naïf permet de penser qu’il suffit de faire évoluer en laboratoire un organisme potentiellement pathogène pour savoir comment sa descendance pourra devenir plus virulente pour l’homme. Il est alors facile de faire croître cet organisme sur des cellules humaines, qu’il infecte mal pour commencer, puis de retenir ses descendants au fur et à mesure qu’ils deviennent plus infectieux. La vision (...) optimiste affirme alors que les caractériser permettra de construire des défenses appropriées. Bien des variations sur ce thème sont possibles, y compris avec l’aide de la synthèse artificielle de variants dont on saurait imaginer la pathogénicité pour l’homme. Depuis plus de deux décennies, des chercheurs développent des expériences de ce type, dite de « gain de fonction » (GdF). Leur interprétation repose sur l’illusion que nous savons prévoir l’évolution des êtres vivants. En réalité, ceux que nous observons aujourd’hui ont été sélectionnés au cours du temps précisément parce qu’ils conduisent toujours à des comportements imprévisibles. Une règle clé de l’évolution est que la vie, pour durer dans des circonstances qui ne peuvent être prévues, doit sans cesse faire émerger des formes innovantes. À ce caractère propre à la vie s’ajoute un imprévisible lié à l’impéritie humaine, cause de l’omniprésence des accidents de laboratoire. Or, en virologie spécialement, l’accident est la règle et non l’exception. Il faut ajouter que le caractère public de ces expériences les rend utilisables par toutes les structures malveillantes qui sont hélas répandues dans le monde. Il n’est pas difficile alors d’imaginer les conséquences catastrophiques des expériences de GdF. Nous proposons donc qu’elles soient purement et simplement interdites et que toute transgression soit sévèrement punie, en frappant d’abord au porte-monnaie, ce qui semble être la meilleure façon de faire dans nos sociétés devenues universellement vénales. (shrink)

In situations where we ignore everything but the space of possibilities, we ought to suspend judgment—that is, remain agnostic—about which of these possibilities is the case. This means that we cannot sum our degrees of belief in different possibilities, something that has been formalized as an axiom of non-additivity. Consistent with this way of representing our ignorance, I defend a doxastic norm that recommends that we should nevertheless follow a certain additivity of possibilities: even if we cannot sum degrees of (...) belief in different possibilities, we should be more confident in larger groups of possibilities. It is thus shown that, in the type of situation considered (in so-called ‘classical ignorance’, i.e. “behind a thin veil of ignorance”), it is epistemically rational for advocates of suspending judgment to endorse this comparative confidence; while on the other hand it is shown that, even in classical ignorance, no stronger belief—such as a precise uniform probability distribution—is warranted. (shrink)

There is a theorem that shows that it is impossible for an algorithm to jointly satisfy the statistical fairness criteria of Calibration and Equalised Odds non-trivially. But what about the recently advocated alternative to Calibration, Base Rate Tracking? Here, we show that Base Rate Tracking is strictly weaker than Calibration, and then take up the question of whether it is possible to jointly satisfy Base Rate Tracking and Equalised Odds in non-trivial scenarios. We show that it is not, thereby establishing (...) an even more general impossibility theorem. (shrink)

Cuốn tài liệu học thuật trình bày phương pháp “BMF analytics” hiện đã bước sang năm thứ ba phục vụ cộng đồng nghiên cứu. Cuốn sách đã góp phần hỗ trợ nhiều nhà nghiên cứu trẻ triển khai và hoàn thành công việc.

The methodology title on BMF analytics is now in its third year of service to our research community, enabling many researchers to implement their research pursuits.

Đến khúc này, 3 ông bói cá cũng bí nên chuyển chủ đề. Hướng sang vấn đề ăn cá vậy. Đây là vấn đề truyền thống lâu đời. Có chuyện gì lớn?

Credences are beliefs about evidential probabilities. We give the view an assessment-sensitive formulation, show how it evades the standard objections, and give several arguments in support.

This paper generalizes the (epistemic) preface paradox beyond the principle of belief aggregation and constructs a similar paradox for deontic reasoning. The analysis of the deontic case yields a solution strategy---restricting belief/obligation aggregation rather than giving it up altogether---that can be transferred to the epistemic case. Our proposal amounts to a reasonable compromise between two goals: (i) sticking to bridge principles between evidence and belief, such as the Lockean Thesis, and (ii) obtaining a sufficiently strong logic of doxastic and deontic (...) reasoning. (shrink)