The breadth-first search adopted by Bayesian researchers to map out the conceptual space and identify what the framework can do is beneficial for science and reflective of its collaborative and incremental nature. Theoretical pluralism among researchers facilitates refinement of models within various levels of analysis, which ultimately enables effective cross-talk between different levels of analysis.

This paper addresses a common challenge with computational cognitive models: identifying parameter values that are both theoretically plausible and generate predictions that match well with empirical data. While computational models can offer deep explanations of cognition, they are computationally complex and often out of reach of traditional parameter fitting methods. Weak methodology may lead to premature rejection of valid models or to acceptance of models that might otherwise be falsified. Mathematically robust fitting methods are, therefore, essential to the progress of (...) computational modeling in cognitive science. In this article, we investigate the capability and role of modern fitting methods—including Bayesian optimization and approximate Bayesian computation—and contrast them to some more commonly used methods: grid search and Nelder–Mead optimization. Our investigation consists of a reanalysis of the fitting of two previous computational models: an Adaptive Control of Thought—Rational model of skill acquisition and a computational rationality model of visual search. The results contrast the efficiency and informativeness of the methods. A key advantage of the Bayesian methods is the ability to estimate the uncertainty of fitted parameter values. We conclude that approximate Bayesian computation is (a) efficient, (b) informative, and (c) offers a path to reproducible results. (shrink)

Bayesian learning theory and evolutionary theory both formalize adaptive competition dynamics in possibly high‐dimensional, varying, and noisy environments. What do they have in common and how do they differ? In this paper, we discuss structural and dynamical analogies and their limits, both at a computational and an algorithmic‐mechanical level. We point out mathematical equivalences between their basic dynamical equations, generalizing the isomorphism between Bayesian update and replicator dynamics. We discuss how these mechanisms provide analogous answers to the challenge (...) of adapting to stochastically changing environments at multiple timescales. We elucidate an algorithmic equivalence between a sampling approximation, particle filters, and the Wright‐Fisher model of population genetics. These equivalences suggest that the frequency distribution of types in replicator populations optimally encodes regularities of a stochastic environment to predict future environments, without invoking the known mechanisms of multilevel selection and evolvability. A unified view of the theories of learning and evolution comes in sight. (shrink)

Bayesian nets are widely used in artificial intelligence as a calculus for causal reasoning, enabling machines to make predictions, perform diagnoses, take decisions and even to discover causal relationships. This book, aimed at researchers and graduate students in computer science, mathematics and philosophy, brings together two important research topics: how to automate reasoning in artificial intelligence, and the nature of causality and probability in philosophy.

Inspired by recent Bayesian interpretations about the psychology underlying religion, the paper introduces a theory proposing that religious conversion is shaped by three factors: (i) novel relevant information, experienced in perceptual or in social form (e.g., following interaction with missionaries); (ii) changes in the utility (e.g., expressed in an opportunity to raise in social rank) associated with accepting a new religious creed; and (iii) prior beliefs, favouring religious faiths that, although new, still remain consistent with entrenched cultural views (resulting (...) in the phenomenon of syncretism). From the theory, a multifactorial picture of conversion emerges. Based on which factor is primarily engaged in each case, a classification of different types of conversion can be derived, with a remarkable fit with empirical literature. The theory offers a description of the processes underlying religious conversion and, highlighting the links among apparently incompatible previous views, it reconciles these views within a unifying framework. (shrink)

Knauff and Gazzo Castañeda (2022) object to using the term “new paradigm” to describe recent developments in the psychology of reasoning. This paper concedes that the Kuhnian term “paradigm” may be queried. What cannot is that the work subsumed under this heading is part of a new, progressive movement that spans the brain and cognitive sciences: Bayesian cognitive science. Sampling algorithms and Bayes nets used to explain biases in JDM can implement the Bayesian new paradigm approach belying any (...) advantages of mental models theory (MMT) at the algorithmic level. Moreover, this paper argues that new versions of MMT lack a computational level theory and questions the grounds for MMTs much-vaunted generality. The paper then examines common ground on the importance of small-scale models/simulations of the world and the importance of argumentation in the social domain rather than individual reasoning. Finally, the paper concludes that although there may be prospects for moving reasoning research forward in a more collective, collaborative manner, many disagreements remain to be resolved. (shrink)

Humans have long been characterized as poor probabilistic reasoners when presented with explicit numerical information. Bayesian word problems provide a well-known example of this, where even highly educated and cognitively skilled individuals fail to adhere to mathematical norms. It is widely agreed that natural frequencies can facilitate Bayesian reasoning relative to normalized formats (e.g. probabilities, percentages), both by clarifying logical set-subset relations and by simplifying numerical calculations. Nevertheless, between-study performance on “transparent” Bayesian problems varies widely, and generally (...) remains rather unimpressive. We suggest there has been an over-focus on this representational facilitator (i.e. transparent problem structures) at the expense of the specific logical and numerical processing requirements and the corresponding individual abilities and skills necessary for providing Bayesian-like output given specific verbal and numerical input. We further suggest that understanding this task-individual pair could benefit from considerations from the literature on mathematical cognition, which emphasizes text comprehension and problem solving, along with contributions of online executive working memory, metacognitive regulation, and relevant stored knowledge and skills. We conclude by offering avenues for future research aimed at identifying the stages in problem solving at which correct versus incorrect reasoners depart, and how individual difference might influence this time point. (shrink)

Bayesian networks are computer programs which represent probabilitistic relationships graphically as directed acyclic graphs, and which can use those graphs to reason probabilistically , often at relatively low computational cost. Almost every expert system in the past tried to support probabilistic reasoning, but because of the computational difficulties they took approximating short-cuts, such as those afforded by MYCIN's certainty factors. That all changed with the publication of Judea Pearl's Probabilistic Reasoning in Intelligent Systems, in 1988, which synthesized a decade (...) of research making accurate graphical probabilistic reasoning computationally achievable.Bayesian network technology is now one of the fastest growing fields of research in artificial intelligence. That it has become a publication industry in its own right is shown by a search on Google scholar :This development, together with a parallel related growth in the use of causal discovery algorithms which automate the learning of Bayesian networks from sample data, has generated considerable interest, and controversy, within the philosophy-of-science community.Three central questions bringing together AI researchers and philosophers of science are: Are Bayesian networks Bayesian? What is the relation between probability and causality? Are the assumptions behind causal discovery of Bayesian networks realistic or fantastical?Jon Williamson, as a philosopher of science with a keen interest in the technology, asks and answers these questions in his new book. Although it is self-contained, his book is not very likely as an introduction to the technology , nor is it optimal even as an introduction to the philosophical problems in interpreting Bayesian networks . Rather Williamson's book is an attempt to move the debate forward by solving the central problems of the …. (shrink)

A large and growing body of research in computational psychiatry draws on Bayesian modelling to illuminate the dysfunctions and aberrations that underlie psychiatric disorders. After identifying the chief attractions of this research programme, we argue that its typical focus on abstract, domain-general inferential processes is likely to obscure many of the distinctive ways in which the human mind can break down and malfunction. We illustrate this by appeal to psychosis and the social phenomenology of delusions.

Bayesian reverse-engineering is a research strategy for developing three-level explanations of behavior and cognition. Starting from a computational-level analysis of behavior and cognition as optimal probabilistic inference, Bayesian reverse-engineers apply numerous tweaks and heuristics to formulate testable hypotheses at the algorithmic and implementational levels. In so doing, they exploit recent technological advances in Bayesian artificial intelligence, machine learning, and statistics, but also consider established principles from cognitive psychology and neuroscience. Although these tweaks and heuristics are highly pragmatic (...) in character and are often deployed unsystematically, Bayesian reverse-engineering avoids several important worries that have been raised about the explanatory credentials of Bayesian cognitive science: the worry that the lower levels of analysis are being ignored altogether; the challenge that the mathematical models being developed are unfalsifiable; and the charge that the terms ‘optimal’ and ‘rational’ have lost their customary normative force. But while Bayesian reverse-engineering is therefore a viable and productive research strategy, it is also no fool-proof recipe for explanatory success. (shrink)

Bayesian models are often criticized for postulating computations that are computationally intractable (e.g., NP-hard) and therefore implausibly performed by our resource-bounded minds/brains. Our letter is motivated by the observation that Bayesian modelers have been claiming that they can counter this charge of “intractability” by proposing that Bayesian computations can be tractably approximated. We would like to make the cognitive science community aware of the problematic nature of such claims. We cite mathematical proofs from the computer science literature (...) that show intractable Bayesian computations, such as postulated in existing Bayesian models, cannot be tractably approximated. This does not mean that human brains do not (or cannot) implement the type of algorithms that Bayesian modelers are advancing, but it does mean that proposing that they do by itself does nothing to parry the charge of intractability, because the postulated algorithms are as intractable (i.e., require exponential time) as the computations they try to approximate. Besides our negative message for the community, our letter also makes a positive contribution by referring to a methodology that Bayesian modelers can use to try and parry the charge of intractability in a mathematically sound way. (shrink)

The mechanisms by which delusion and anxiety affect the tendency to make hasty decisions (Jumping-to-Conclusions bias) remain unclear. This paper proposes a Bayesian computational model that explores the assignment of evidence weights as a potential explanation of the Jumping-to-Conclusions bias using the Beads Task. We also investigate the Beads Task as a repeated measure by varying the key aspects of the paradigm. The Bayesian model estimations from two online studies showed that higher delusional ideation promoted reduced belief updating (...) but the impact of general and social anxiety on evidence weighting was inconsistent. The altered evidence weighting as a result of a psychopathological trait appeared insufficient in contributing to the Jumping-to-Conclusions bias. Variations in Beads Task aspects significantly affected subjective certainty at the point of decisions but not the number of draws to decisions. Repetitions of the Beads Task are feasible if one assesses the Jumping-to-Conclusions bias using number of draws to decisions. (shrink)

We study the phenomenon of merging of opinions for computationally limited Bayesian agents from the perspective of algorithmic randomness. When they agree on which data streams are algorithmically random, two Bayesian agents beginning the learning process with different priors may be seen as having compatible beliefs about the global uniformity of nature. This is because the algorithmically random data streams are of necessity globally regular: they are precisely the sequences that satisfy certain important statistical laws. By virtue of (...) agreeing on what data streams are algorithmically random, two Bayesian agents can thus be taken to concur on what global regularities they expect to see in the data. We show that this type of compatibility between priors suffices to ensure that two computable Bayesian agents will reach inter-subjective agreement with increasing information. In other words, it guarantees that their respective probability assignments will almost surely become arbitrarily close to each other as the number of observations increases. Thus, when shared by computable Bayesian learners with different subjective priors, the beliefs about uniformity captured by algorithmic randomness provably lead to merging of opinions. (shrink)

The exponential growth of social data both in volume and complexity has increasingly exposed many of the shortcomings of the conventional frequentist approach to statistics. The scientific community has called for careful usage of the approach and its inference. Meanwhile, the alternative method, Bayesian statistics, still faces considerable barriers toward a more widespread application. The bayesvl R package is an open program, designed for implementing Bayesian modeling and analysis using the Stan language’s no-U-turn (NUTS) sampler. The package combines (...) the ability to construct Bayesian network models using directed acyclic graphs (DAGs), the Markov chain Monte Carlo (MCMC) simulation technique, and the graphic capability of the ggplot2 package. As a result, it can improve the user experience and intuitive understanding when constructing and analyzing Bayesian network models. A case example is offered to illustrate the usefulness of the package for Big Data analytics and cognitive computing. (shrink)

The full Bayesian signi/cance test (FBST) for precise hypotheses is presented, with some illustrative applications. In the FBST we compute the evidence against the precise hypothesis. We discuss some of the theoretical properties of the FBST, and provide an invariant formulation for coordinate transformations, provided a reference density has been established. This evidence is the probability of the highest relative surprise set, “tangential” to the sub-manifold (of the parameter space) that defines the null hypothesis.

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as the Riemann hypothesis, have had to be considered in terms of the evidence for and against them. In recent decades, massive increases in computer power have permitted the gathering of huge amounts of numerical evidence, both for conjectures in pure mathematics and (...) for the behavior of complex applied mathematical models and statistical algorithms. Mathematics has therefore become (among other things) an experimental science (though that has not diminished the importance of proof in the traditional style). We examine how the evaluation of evidence for conjectures works in mathematical practice. We explain the (objective) Bayesian view of probability, which gives a theoretical framework for unifying evidence evaluation in science and law as well as in mathematics. Numerical evidence in mathematics is related to the problem of induction; the occurrence of straightforward inductive reasoning in the purely logical material of pure mathematics casts light on the nature of induction as well as of mathematical reasoning. (shrink)

Bayesian cognitive science is a research programme that relies on modelling resources from Bayesian statistics for studying and understanding mind, brain, and behaviour. Conceiving of mental capacities as computing solutions to inductive problems, Bayesian cognitive scientists develop probabilistic models of mental capacities and evaluate their adequacy based on behavioural and neural data generated by humans (or other cognitive agents) performing a pertinent task. The overarching goal is to identify the mathematical principles, algorithmic procedures, and causal mechanisms that (...) enable cognitive agents to take uncertainty into account and weigh it appropriately in producing adaptive behaviour. -/- The appeal of Bayesian cognitive science derives from its transparency, normativity, and unifying power. Resources from Bayesian statistics allow cognitive scientists to characterise transparently, in terms of random variables and probabilistic dependencies between them, the inductive problems that many mental capacities are presumed to solve. Given this characterisation, researchers can define an upper bound on one’s performance on that problem and use it as a benchmark for interpreting experimental results, formulating hypotheses about why and when deviations from optimal performance should be expected, and seeking explanations of mental capacities within one encompassing theoretical framework. -/- Although critics question the testability, explanatory power, and plausibility of Bayesian models of mental capacities, Bayesian cognitive science has proved itself to be an incredibly fruitful research programme. Fuelled by empirical and theoretical results from psychology, neuroscience, philosophy, computer science, artificial intelligence, and evolution, Bayesian cognitive science has: advanced our understanding of why cognitive systems should keep track of uncertainty and use it to facilitate adaptive behaviour; clarified how the brain might encode information about the uncertainty of its stimuli and perform computations with probability distributions; and crystallised the insight that cognitive agents’ management of uncertainty might be grounded in predictive processes aimed at avoiding surprising exchanges with the environment. (shrink)

Two key research issues in the field of causal learning are how people acquire causal knowledge when observing data that are presented sequentially, and the level of abstraction at which learning takes place. Does sequential causal learning solely involve the acquisition of specific cause-effect links, or do learners also acquire knowledge about abstract causal constraints? Recent empirical studies have revealed that experience with one set of causal cues can dramatically alter subsequent learning and performance with entirely different cues, suggesting that (...) learning involves abstract transfer, and such transfer effects involve sequential presentation of distinct sets of causal cues. It has been demonstrated that pre-training can modulate classic causal learning phenomena such as forward and backward blocking. To account for these effects, we propose a Bayesian theory of sequential causal learning. The theory assumes that humans are able to consider and use several alternative causal generative models, each instantiating a different causal integration rule. Model selection is used to decide which integration rule to use in a given learning environment in order to infer causal knowledge from sequential data. Detailed computer simulations demonstrate that humans rely on the abstract characteristics of outcome variables to select a causal integration rule, which in turn alters causal learning in a variety of blocking and overshadowing paradigms. When the nature of the outcome variable is ambiguous, humans select the model that yields the best fit with the recent environment, and then apply it to subsequent learning tasks. Based on sequential patterns of cue-outcome co-occurrence, the theory can account for a range of phenomena in sequential causal learning, including various blocking effects, primacy effects in some experimental conditions, and apparently abstract transfer of causal knowledge. (shrink)

This article proposes a formal model that integrates cognitive and psychodynamic psychotherapeutic models of psychopathy to show how two major psychopathic traits called lacks remorse and self-aggrandizing can be understood as a form of abnormal Bayesian inference about the self. This model draws on the predictive coding (i.e., active inference) framework, a neurobiologically plausible explanatory framework for message passing in the brain that is formalized in terms of hierarchical Bayesian inference. In summary, this model proposes that these two (...) cardinal psychopathic traits reflect entrenched maladaptive Bayesian inferences about the self, which defend against the experience of deep-seated, self-related negative emotions, specifically shame and worthlessness. Support for the model in extant research on the neurobiology of psychopathy and quantitative simulations are provided. Finally, we offer a preliminary overview of a novel treatment for psychopathy that rests on our Bayesian formulation. (shrink)

Infant language learners are faced with the difficult inductive problem of determining how new words map to novel or known objects in their environment. Bayesian inference models have been successful at using the sparse information available in natural child-directed speech to build candidate lexicons and infer speakers’ referential intentions. We begin by asking how a Bayesian model optimized for monolingual input generalizes to new monolingual or bilingual corpora and find that, especially in the case of the bilingual input, (...) the model shows a significant decrease in performance. In the next experiment, we propose the ME Model, a modified Bayesian model, which approximates infants’ mutual exclusivity bias to support the differential demands of monolingual and bilingual learning situations. The extended model is assessed using the same corpora of real child-directed speech, showing that its performance is more robust against varying input and less dependent than the Intentional Model on optimization of its parsimony parameter. We argue that both monolingual and bilingual demands on word learning are important considerations for a computational model, as they can yield significantly different results than when only one such context is considered. (shrink)

The idea that socially responsible investments can be viewed in terms of real options is relatively new. We expand on this notion by demonstrating how real option theory, within a Bayesian decision-making framework, can be used by managers to help when making green technology investment decisions. The Bayesian decision framework provides a more flexible approach to investment decision making because it adjusts for new information. Responding to a call for multidisciplinary and multifaceted research in environmental sustainability, this paper (...) integrates ethics, finance, and information technology by viewing investments in environmentally friendly technology as a profit-driven CSR real option. Our model provides managers with the analytic tool needed to make the business case for CSR initiatives providing an opportunity for firms to create economic, social, and ecological solutions that benefit all stakeholders. The practical applicability of our model is demonstrated in an illustrative case scenario based on real data. (shrink)

Abstract: The Pull Bayesian Significance Test (FBST) for precise hy- potheses is applied to a Multivariate Normal Structure (MNS) model. In the FBST we compute the evidence against the precise hypothesis. This evi- dence is the probability of the Highest Relative Surprise Set (HRSS) tangent to the sub-manifold (of the parameter space) that defines the null hypothesis. The MNS model we present appears when testing equivalence conditions for genetic expression measurements, using micro-array technology.

This article reveals one general scheme for creating counter examples to Bayesian confirmation theory. The reason of the problems is that: in daily life the degree of confirmation is affected not only by probability but also by some non-probabilistic factors, e.g., structural similarity, quantity of evidence, and marginal utility, while Bayesian confirmation theory considers only probabilities to measure the degree of confirmation. This article resolves these problems after some detail analyses, and proposes a new confirmation measure based on (...) these factors. (shrink)

The causal view of categories assumes that categories are represented by features and their causal relations. To study the effect of causal knowledge on categorization, researchers have used Bayesian causal models. Within that framework, categorization may be viewed as dependent on a likelihood computation (i.e., the likelihood of an exemplar with a certain combination of features, given the category's causal model) or as a posterior computation (i.e., the probability that the exemplar belongs to the category, given its (...) features). Across three experiments, in combination with computational modeling, we offer evidence that categorization is better accounted for by assuming that people compute posteriors and not likelihoods, though both probabilities are closely related. This result contrasts with existing analyses of causal-based categorization, which assume that likelihood computations give a good approximation of human judgments. We also find that people are able to compute likelihoods in a closely related task that elicits judgments of consistency rather than category membership judgments. Our analyses show that people do use causal probabilistic information as prescribed by a Bayesian model but that they flexibly compute likelihoods or posteriors depending on the task. We discuss our results in relation to the relevant literature on the topic. (shrink)

Event segmentation theory posits that people segment continuous experience into discrete events and that event boundaries occur when there are large transient increases in prediction error. Here, we set out to test this theory in the context of story listening, by using a deep learning language model (GPT‐2) to compute the predicted probability distribution of the next word, at each point in the story. For three stories, we used the probability distributions generated by GPT‐2 to compute the time series of (...) prediction error. We also asked participants to listen to these stories while marking event boundaries. We used regression models to relate the GPT‐2 measures to the human segmentation data. We found that event boundaries are associated with transient increases in Bayesian surprise but not with a simpler measure of prediction error (surprisal) that tracks, for each word in the story, how strongly that word was predicted at the previous time point. These results support the hypothesis that prediction error serves as a control mechanism governing event segmentation and point to important differences between operational definitions of prediction error. (shrink)

One of the most lasting contributions of Marr's posthumous book is his articulation of the different “levels of analysis” that are needed to understand vision. Although a variety of work has examined how these different levels are related, there is comparatively little examination of the assumptions on which his proposed levels rest, or the plausibility of the approach Marr articulated given those assumptions. Marr placed particular significance on computational level theory, which specifies the “goal” of a computation, its appropriateness (...) for solving a particular problem, and the logic by which it can be carried out. The structure of computational level theory is inherently teleological: What the brain does is described in terms of its purpose. I argue that computational level theory, and the reverse-engineering approach it inspires, requires understanding the historical trajectory that gave rise to functional capacities that can be meaningfully attributed with some sense of purpose or goal, that is, a reconstruction of the fitness function on which natural selection acted in shaping our visual abilities. I argue that this reconstruction is required to distinguish abilities shaped by natural selection—“natural tasks” —from evolutionary “by-products”, rather than merely demonstrating that computational goals can be embedded in a Bayesian model that renders a particular behavior or process rational. (shrink)

Computational-level models proposed in recent Bayesian cognitive science predict both the “biased” and correct responses on many tasks. So, rather than possessing two reasoning systems, people can generate both possible responses within a single system. Consequently, although an account of why people make one response rather than another is required, dual processes of reasoning may not be.

This paper spells out the attention schema theory of consciousness in terms of the predictive processing framework. As it stands, the attention schema theory lacks a plausible computational formalization that could be used for developing possible mechanistic models of how it is realized in the brain. The predictive processing framework, on the other hand, fails to provide a plausible explanation of the subjective quality or the phenomenal aspect of conscious experience. The aim of this work is to apply the formal (...) tools of predictive processing to outline a possible model of cognition that details a plausible way in which organisms may acquire or construct the kinds of schemas postulated by the attention schema theory. This is done in terms of Bayesian control models, which tend to favour representing information in a frugal or 'sparse' manner that strips away detail in precisely the way that the attention schema requires. Once integrated, these two accounts can offer a plausible explanation of the supposedly irreducible and mysterious 'feel' of conscious experience, in terms of the frugal manner in which perceptual mechanisms are represented by the attention schema. (shrink)

How children go about learning the general regularities that govern language, as well as keeping track of the exceptions to them, remains one of the challenging open questions in the cognitive science of language. Computational modeling is an important methodology in research aimed at addressing this issue. We must determine appropriate learning mechanisms that can grasp generalizations from examples of specific usages, and that exhibit patterns of behavior over the course of learning similar to those in children. Early learning of (...) verb argument structure is an area of language acquisition that provides an interesting testbed for such approaches due to the complexity of verb usages. A range of linguistic factors interact in determining the felicitous use of a verb in various constructions—associations between syntactic forms and properties of meaning that form the basis for a number of linguistic and psycholinguistic theories of language. This article presents a computational model for the representation, acquisition, and use of verbs and constructions. The Bayesian framework is founded on a novel view of constructions as a probabilistic association between syntactic and semantic features. The computational experiments reported here demonstrate the feasibility of learning general constructions, and their exceptions, from individual usages of verbs. The behavior of the model over the timecourse of acquisition mimics, in relevant aspects, the stages of learning exhibited by children. Therefore, this proposal sheds light on the possible mechanisms at work in forming linguistic generalizations and maintaining knowledge of exceptions. (shrink)

Bayesian epistemologists support the norms of probabilism and conditionalization using Dutch book and accuracy arguments. These arguments assume that rationality requires agents to maximize practical or epistemic value in every doxastic state, which is evaluated from a subjective point of view (e.g., the agent’s expectancy of value). The accuracy arguments also presuppose that agents are opinionated. The goal of this paper is to discuss the assumptions of these arguments, including the measure of epistemic value. I have designed AI agents (...) based on the Bayesian model and a nonmonotonic framework and tested how they achieve practical and epistemic value in conditions in which an alternative set of assumptions holds. In one of the tested conditions, the nonmonotonic agent, which is not opinionated and fulfills neither probabilism nor conditionalization, outperforms the Bayesian in the measure of epistemic value that I argue for in the paper (α-value). I discuss the consequences of these results for the epistemology of rationality. (shrink)

Bayesian approaches to legal reasoning propose causal models of the relation between evidence, the credibility of evidence, and ultimate hypotheses, or verdicts. They assume that legal reasoning is the process whereby one infers the posterior probability of a verdict based on observed evidence, or facts. In practice, legal reasoning does not operate quite that way. Legal reasoning is also an attempt at inferring applicable rules derived from legal precedents or statutes based on the facts at hand. To make such (...) an inference, legal reasoning follows syllogistic logic and first order transitivity. This paper proposes a Bayesian model of legal syllogistic reasoning that complements existing Bayesian models of legal reasoning using a Bayesian network whose variables correspond to legal precedents, statutes, and facts. We suggest that legal reasoning should be modelled as a process of finding the posterior probability of precedents and statutes based on available facts. (shrink)

Consider two agents who want to be Bayesians with a common prior, but who cannot due to computational limitations. If these agents agree that their estimates are consistent with certain easy-to-compute consistency constraints, then they can agree to disagree about any random variable only if they also agree to disagree, to a similar degree and in a stronger sense, about an average error. Yet average error is a state-independent random variable, and one agent's estimate of it is also agreed to (...) be state-independent. Thus suggests that disagreements are not fundamentally due to differing information about the state of the world. (shrink)

This article considers the extent to which Bayesian networks with imprecise probabilities, which are used in statistics and computer science for predictive purposes, can be used to represent causal structure. It is argued that the adequacy conditions for causal representation in the precise context—the Causal Markov Condition and Minimality—do not readily translate into the imprecise context. Crucial to this argument is the fact that the independence relation between random variables can be understood in several different ways when the joint (...) probability distribution over those variables is imprecise, none of which provides a compelling basis for the causal interpretation of imprecise Bayes nets. I conclude that there are serious limits to the use of imprecise Bayesian networks to represent causal structure. (shrink)

(2013). Hierarchical Bayesian models as formal models of causal reasoning. Argument & Computation: Vol. 4, Formal Models of Reasoning in Cognitive Psychology, pp. 36-45. doi: 10.1080/19462166.2012.700321.

In criminal trials, evidence often involves a degree of uncertainty and decision-making includes moving from the initial presumption of innocence to inference about guilt based on that evidence. The jurors’ ability to combine evidence and make accurate intuitive probabilistic judgments underpins this process. Previous research has shown that errors in probabilistic reasoning can be explained by a misalignment of the evidence presented with the intuitive causal models that people construct. This has been explored in abstract and context-free situations. However, less (...) is known about how people interpret evidence in context-rich situations such as legal cases. The present study examined participants’ intuitive probabilistic reasoning in legal contexts and assessed how people’s causal models underlie the process of belief updating in the light of new evidence. The study assessed whether participants update beliefs in line with Bayesian norms and if errors in belief updating can be explained by the causal structures underpinning the evidence integration process. The study was based on a recent case in England where a couple was accused of intentionally harming their baby but was eventually exonerated because the child’s symptoms were found to be caused by a rare blood disorder. Participants were presented with a range of evidence, one piece at a time, including physical evidence and reports from experts. Participants made probability judgments about the abuse and disorder as causes of the child’s symptoms. Subjective probability judgments were compared against Bayesian norms. The causal models constructed by participants were also elicited. Results showed that overall participants revised their beliefs appropriately in the right direction based on evidence. However, this revision was done without exact Bayesian computation and errors were observed in estimating the weight of evidence. Errors in probabilistic judgments were partly accounted for, by differences in the causal models representing the evidence. Our findings suggest that understanding causal models that guide people’s judgments may help shed light on errors made in evidence integration and potentially identify ways to address accuracy in judgment. (shrink)

The contributions in this volume focus on the Bayesian interpretation of natural languages, which is widely used in areas of artificial intelligence, cognitive science, and computational linguistics. This is the first volume to take up topics in Bayesian Natural Language Interpretation and make proposals based on information theory, probability theory, and related fields. The methodologies offered here extend to the target semantic and pragmatic analyses of computational natural language interpretation. Bayesian approaches to natural language semantics and pragmatics (...) are based on methods from signal processing and the causal Bayesian models pioneered by especially Pearl. In signal processing, the Bayesian method finds the most probable interpretation by finding the one that maximizes the product of the prior probability and the likelihood of the interpretation. It thus stresses the importance of a production model for interpretation as in Grice’s contributions to pragmatics or in interpretation by abduction. (shrink)

This work provides a conceptual foundation for a Bayesian approach to artificial inference and learning. I argue that Bayesian confirmation theory provides a general normative theory of inductive learning and therefore should have a role in any artificially intelligent system that is to learn inductively about its world. I modify the usual Bayesian theory in three ways directly pertinent to an eventual research program in artificial intelligence. First, I construe Bayesian inference rules as defeasible, allowing them (...) to be overridden in certain contexts and therefore allowing them to play a part in a hybrid system, coexisting with non-Bayesian modes of inference. I take seriously the need to find meaningful prior probabilities for hypotheses, and elaborate means for supplying an artificial intelligence with such. And I address the computational complexity of Bayesian inference by reference to simplifications using causal networks and by allowing the probabilistic acceptance of hypotheses and subsequent qualitative inference to supplement Bayesian reasoning. The result is a Pragmatic Bayesian model of induction. (shrink)

The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameters can be computed from these samples. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same as those (...) based on the maximum likelihood solution, e.g., output from LISREL or EQS. In small samples, however, the likelihood surface is not Gaussian and in some cases contains local maxima. Nevertheless, the Gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface. With an informative prior distribution over the parameters, the posterior can be used to make inferences about the parameters of underidentified models, as we illustrate on a simple errors-in-variables model. (shrink)

Conditional independence tests have received special attention lately in machine learning and computational intelligence related literature as an important indicator of the relationship among the variables used by their models. In the field of probabilistic graphical models, which includes Bayesian network models, conditional independence tests are especially important for the task of learning the probabilistic graphical model structure from data. In this paper, we propose the full Bayesian significance test for tests of conditional independence for discrete datasets. The (...) full Bayesian significance test is a powerful Bayesian test for precise hypothesis, as an alternative to the frequentist’s significance tests (characterized by the calculation of the p-value). (shrink)