In this article, I explore a Bayesian approach to avalanche decision-making. I motivate this perspective by highlighting a version of the base-rate fallacy and show that a similar pattern applies to decision-making in avalanche-terrain. I then draw out three theoretical lessons from adopting a Bayesian approach and discuss these lessons critically. Lastly, I highlight a number of challenges for avalanche educators when incorporating the Bayesian perspective in their curriculum.

This brief annotated bibliography is intended to help students get started with their research. It is not a substitute for personal investigation of the literature, and it is not a comprehensive bibliography on the subject. For those just beginning to study Bayesian reasoning, I suggest the starred items as good places to start your reading.

Delusions are one of the most classical symptoms described in schizophrenia. However, despite delusions are often emotionally charged, they have been investigated using tasks involving non-affective material, such as the Beads task. In this study we compared 30 patients with schizophrenia experiencing delusions with 32 matched controls in their pattern of responses to two versions of the Beads task within a Bayesian framework. The two versions of the Beads task consisted of one emotional and one neutral, both with ratios (...) of beads of 60:40 and 80:20, considered, respectively, as the “difficult” and “easy” variants of the task. Results indicate that patients showed a greater deviation from the normative model, especially in the 60:40 ratio, suggesting that more inaccurate probability estimations are more likely to occur under uncertainty conditions. Additionally, both patients and controls showed a greater deviation in the emotional version of the task, providing evidence of a reasoning bias modulated by the content of the stimuli. Finally, a positive correlation between patients’ deviation and delusional symptomatology was found. Impairments in the 60:40 ratio with emotional content was related to the amount of disruption in life caused by delusions. These results contribute to the understanding of how cognitive mechanisms interact with characteristics of the task in the context of delusional thinking. These findings might be used to inform improved intervention programs in the domain of inferential reasoning. (shrink)

This brief annotated bibliography is intended to help students get started with their research. It is not a substitute for personal investigation of the literature, and it is not a comprehensive bibliography on the subject. For those just beginning to study probabilistic confirmation theory and Bayesian reasoning, I suggest the starred items as good places to start your reading.

Three experiments investigated the effect of rarity on people's selection and interpretation of data in a variant of the pseudodiagnosticity task. For familiar (Experiment 1) but not for arbitrary (Experiment 3) materials, participants were more likely to select evidence so as to complete a likelihood ratio when the initial evidence they received was a single likelihood concerning a rare feature. This rarity effect with familiar materials was replicated in Experiment 2 where it was shown that participants were relatively insensitive to (...) explicit manipulations of the likely diagnosticity of rare evidence. In contrast to the effects for data selection, there was an effect of rarity on confidence ratings after receipt of a single likelihood for arbitrary but not for familiar materials. It is suggested that selecting diagnostic evidence necessitates explicit consideration of the alternative hypothesis and that consideration of the possible consequences of the evidence for the alternative weakens the rarity effect in confidence ratings. Paradoxically, although rarity effects in evidence selection and confidence ratings are in the spirit of Bayesian reasoning, the effect on confidence ratings appears to rely on participants thinking less about the alternative hypothesis. (shrink)

This work examines the influence of reference dependence, including value selection bias and congruence effects, on diagnostic reasoning. Across two studies, we explored how dependence on the initial problem structure influences the ability to solve simplified precursors to the more traditional Bayesian reasoning problems. Analyses evaluated accuracy and types of response errors as a function of congruence between the problem presentation and question of interest, amount of information, need for computation, and individual differences in numerical abilities. Across (...) all problem variations, there was consistent and strong evidence of a value selection bias in that incorrect responses almost always conformed to values that were provided in the problem rather than other errors including those related to computation. The most consistent and unexpected error across all conditions in the first experiment was that people were often more likely to utilize the superordinate value as part of their solution rather than the anticipated reference class values. This resulted in a weakened effect of congruence, with relatively low accuracy even in congruent conditions, and a dominant response error of the superordinate value. Experiment 2 confirmed that the introduction of a new sample drew attention away from the provided reference class, increasing reliance on the overall sample size. This superordinate preference error, along with the benefit of repeating the PPV reference class within the question, demonstrated the importance of reference dependence based on the salience of information within the response prompt. Throughout, higher numerical skills were generally associated with higher accuracy, whether calculations were required or not. (shrink)

Many researchers want to unify probability and logic by defining logical probability or probabilistic logic reasonably. This paper tries to unify statistics and logic so that we can use both statistical probability and logical probability at the same time. For this purpose, this paper proposes the P–T probability framework, which is assembled with Shannon’s statistical probability framework for communication, Kolmogorov’s probability axioms for logical probability, and Zadeh’s membership functions used as truth functions. Two kinds of probabilities are connected by an (...) extended Bayes’ theorem, with which we can convert a likelihood function and a truth function from one to another. Hence, we can train truth functions (in logic) by sampling distributions (in statistics). This probability framework was developed in the author’s long-term studies on semantic information, statistical learning, and color vision. This paper first proposes the P–T probability framework and explains different probabilities in it by its applications to semantic information theory. Then, this framework and the semantic information methods are applied to statistical learning, statistical mechanics, hypothesis evaluation (including falsification), confirmation, and Bayesian reasoning. Theoretical applications illustrate the reasonability and practicability of this framework. This framework is helpful for interpretable AI. To interpret neural networks, we need further study. (shrink)

The Bayesian view of inference has become popular in philosophy in recent years. Scientific Reasoning: a Bayesian Approach, by Colin Howson and Peter Urbach, represents an articulate and persuasive defense of the Bayesian view. We focus on the theme of that book, and argue that there are difficulties with Bayesianism, and alternatives worth considering. One of the most serious drawbacks to Bayesianism is the subjectivity that pervades most versions of it. We argue that this is an (...) instance of a more general contemporary tendency to move away from claims of objectivity, and toward frankly subjective views. This results from a desire to find a deductive, incorrigible, basis for scientific inference. We claim that such a desire is doomed to frustration, but that does not spell the end of efforts to formalize inductive reasoning. (shrink)

For more than 20 years, research has proven the beneficial effect of natural frequencies when it comes to solving Bayesian reasoning tasks (Gigerenzer & Hoffrage, 1995). In a recent meta-analysis, McDowell & Jacobs (2017) showed that presenting a task in natural frequency format increases performance rates to 24% compared to only 4% when the same task is presented in probability format. Nevertheless, on average three quarters of participants in their meta-analysis failed to obtain the correct solution for such (...) a task in frequency format. In this paper, we present an empirical study on what participants typically do wrong when confronted with natural frequencies. We found that many of them did not actually use natural frequencies for their calculations, but translated them back into complicated probabilities instead. This switch from the intuitive presentation format to a less intuitive calculation format will be discussed within the framework of psychological theories (e.g., the Einstellung effect). (shrink)

The Monty Hall Dilemma (MHD) is a two-step decision problem involving counterintuitive conditional probabilities. The first choice is made among three equally probable options, whereas the second choice takes place after the elimination of one of the non-selected options which does not hide the prize. Differing from most Bayesian problems, statistical information in the MHD has to be inferred, either by learning outcome probabilities or by reasoning from the presented sequence of events. This often leads to suboptimal decisions (...) and erroneous probability judgments. Specifically, decision makers commonly develop a wrong intuition that final probabilities are equally distributed, together with a preference for their first choice. Several studies have shown that repeated practice enhances sensitivity to the different reward probabilities, but does not facilitate correct Bayesian reasoning. However, modest improvements in probability judgments have been observed after guided explanations. To explain these dissociations, the present review focuses on two types of causes producing the observed biases: Emotional-based choice biases and cognitive limitations in understanding probabilistic information. Among the latter, we identify a crucial cause for the universal difficulty in overcoming the equiprobability illusion: Incomplete representation of prior and conditional probabilities. We conclude that repeated practice and/or high incentives can be effective for overcoming choice biases, but promoting an adequate partitioning of possibilities seems to be necessary for overcoming cognitive illusions and improving Bayesian reasoning. (shrink)

Bayesian reasoning, defined here as the updating of a posterior probability following new information, has historically been problematic for humans. Classic psychology experiments have tested human Bayesian reasoning through the use of word problems and have evaluated each participant’s performance against the normatively correct answer provided by Bayes’ theorem. The standard finding is of generally poor performance. Over the past two decades, though, progress has been made on how to improve Bayesian reasoning. Most notably, (...) research has demonstrated that the use of frequencies in a natural sampling framework —as opposed to single-event probabilities—can improve participants’ Bayesian estimates. Furthermore, pictorial aids and certain individual difference factors also can play significant roles in Bayesian reasoning success. The mechanics of how to build tasks which show these improvements is not under much debate. The explanations for why naturally-sampled frequencies and pictures help Bayesian reasoning remain hotly contested, however, with many researchers falling into ingrained “camps” organized around two dominant theoretical perspectives. The present paper evaluates the merits of these theoretical perspectives, including the weight of empirical evidence, theoretical coherence, and predictive power. By these criteria, the ecological rationality approach is clearly better than the heuristics and biases view. Progress in the study of Bayesian reasoning will depend on continued research that honestly, vigorously, and consistently engages across these different theoretical accounts rather than staying “siloed” within one particular perspective. The process of science requires an understanding of competing points of view, with the ultimate goal being integration. (shrink)

Previous research suggests that children can infer causal relations from patterns of events. However, what appear to be cases of causal inference may simply reduce to children recognizing relevant associations among events, and responding based on those associations. To examine this claim, in Experiments 1 and 2, children were introduced to a “blicket detector,” a machine that lit up and played music when certain objects were placed upon it. Children observed patterns of contingency between objects and the machine's activation that (...) required them to use indirect evidence to make causal inferences. Critically, associative models either made no predictions, or made incorrect predictions about these inferences. In general, children were able to make these inferences, but some developmental differences between 3- and 4-year-olds were found. We suggest that children's causal inferences are not based on recognizing associations, but rather that children develop a mechanism for Bayesian structure learning. Experiment 3 explicitly tests a prediction of this account. Children were asked to make an inference about ambiguous data based on the base rate of certain events occurring. Four-year-olds, but not 3-year-olds were able to make this inference. (shrink)

Previous research suggests that children can infer causal relations from patterns of events. However, what appear to be cases of causal inference may simply reduce to children recognizing relevant associations among events, and responding based on those associations. To examine this claim, in Experiments 1 and 2, children were introduced to a “blicket detector”, a machine that lit up and played music when certain objects were placed upon it. Children observed patterns of contingency between objects and the machine’s activation that (...) required them to use indirect evidence to make causal inferences. Critically, associative models either made no predictions, or made incorrect predictions about these inferences. In general, children were able to make these inferences, but some developmental differences between 3- and 4- year-olds were found. We suggest that children’s causal inferences are not based on recognizing associations, but rather that children develop a mechanism for Bayesian structure learning. Experiment 3 explicitly tests a prediction of this account. Children were asked to make an inference about ambiguous data based on the base-rate of certain events occurring. Fouryear-olds, but not 3-year-olds were able to make this inference. (shrink)

Scientific reasoning is—and ought to be—conducted in accordance with the axioms of probability. This Bayesian view—so called because of the central role it accords to a theorem first proved by Thomas Bayes in the late eighteenth ...

In contrast to traditional research on base-rate neglect, an ecologically-oriented research program would analyze the correspondence between cognitive algorithms and the nature of information in the environment. Bayesian computations turn out to be simpler when information is represented in frequency formats as opposed to the probability formats used in previous research. Frequency formats often enable even uninstructed subjects to perform Bayesian reasoning.

The terms nested sets, partitive frequencies, inside-outside view, and dual processes add little but confusion to our original analysis (Gigerenzer & Hoffrage 1995; 1999). The idea of nested set was introduced because of an oversight; it simply rephrases two of our equations. Representation in terms of chances, in contrast, is a novel contribution yet consistent with our computational analysis System 1.dual process theory” is: Unless the two processes are defined, this distinction can account post hoc for almost everything. In contrast, (...) an ecological view of cognition helps to explain how insight is elicited from the outside (the external representation of information) and, more generally, how cognitive strategies match with environmental structures. (shrink)

Many Bayesians such as White and Silins have argued that Moorean reasoning is defective because it is a case where probabilistic support fails to transmit across the relevant entailment. In this paper, I argue against their claim. On the Bayesian argument, a skeptical hypothesis is that you are a brain in a vat that appears to have hands. To disclose the defect in Moorean reasoning, the Bayesian argument is supposed to show that its appearing to you (...) as if you have hands does not increase the probability of a non-skeptical hypothesis such that you are not a brain in a vat. But what their argument really shows is rather a trivial fact that acquiring theses handlike experiences eliminates the possibility of not having these experiences, and so your having these experiences lowers the probability of the disjunctive claim that you are not a brain in a vat or it does not appear as if you have hands. Along these lines, I argue that what the Bayesian proof establishes has nothing to do with the defect in Moorean reasoning. (shrink)

Bayesian epistemology tells us with great precision how we should move from prior to posterior beliefs in light of new evidence or information, but says little about where our prior beliefs come from. It offers few resources to describe some prior beliefs as rational or well-justified, and others as irrational or unreasonable. A different strand of epistemology takes the central epistemological question to be not how to change one’s beliefs in light of new evidence, but what reasons justify a (...) given set of beliefs in the first place. We offer an account of rational belief formation that closes some of the gap between Bayesianism and its reason-based alternative, formalizing the idea that an agent can have reasons for his or her (prior) beliefs, in addition to evidence or information in the ordinary Bayesian sense. Our analysis of reasons for belief is part of a larger programme of research on the role of reasons in rational agency (Dietrich and List, Nous, 2012a, in press; Int J Game Theory, 2012b, in press). (shrink)

Are people rational? This question was central to Greek thought and has been at the heart of psychology and philosophy for millennia. This book provides a radical and controversial reappraisal of conventional wisdom in the psychology of reasoning, proposing that the Western conception of the mind as a logical system is flawed at the very outset. It argues that cognition should be understood in terms of probability theory, the calculus of uncertain reasoning, rather than in terms of logic, (...) the calculus of certain reasoning. (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)

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)

Normal 0 0 1 85 487 UBC 4 1 598 11.773 0 0 0 Under what conditions is the failure to have evidence that p evidence that p is false? Absent evidence reasoning is common in many sciences, including astronomy, archeology, biology and medicine. An often-repeated epistemological motto is that “the absence of evidence is not evidence of absence.” Analysis of absent evidence reasoning usually takes place in a deductive or frequentist hypothesis-testing framework. Instead, I develop a (...) class='Hi'>Bayesian analysis of this motto and prove that, under plausible assumptions about the nature of evidence, the absence of evidence is evidence of absence. (shrink)

When unmanned underwater vehicles perform tasks, the marine environment situation information perceived by their sensors is insufficient and cannot be shared; moreover, the reasoning efficiency of the situation information is not high. To deal with these problems, this paper proposes an ontology-based situation awareness information expression method, using the Bayesian network method to reason about situation information. First, the situation awareness information is determined in uncertain events when performing tasks in the marine environment. The core and application ontologies (...) of UUV situation awareness are also established. Subsequently, semantic rules are constructed, and uncertain events are identified through the corresponding semantic rules. The Jess inference engine is used to update the ontology model. Because the ontology cannot reason about uncertainty, it is transformed into the Bayesian network to reason about the impacts of uncertain events on tasks. Simulation experiments verify the effectiveness and accuracy of the situation awareness reasoning method that combines the ontology and the Bayesian network. (shrink)

Bayesian reasoning has been applied formally to statistical inference, machine learning and analysing scientific method. Here I apply it informally to more common forms of inference, namely natural language arguments. I analyse a variety of traditional fallacies, deductive, inductive and causal, and find more merit in them than is generally acknowledged. Bayesian principles provide a framework for understanding ordinary arguments which is well worth developing.

In everyday matters, as well as in law, we allow that someone’s reasons can be causes of her actions, and often are. That correct reasoning accords with Bayesian principles is now so widely held in philosophy, psychology, computer science and elsewhere that the contrary is beginning to seem obtuse, or at best quaint. And that rational agents should learn about the world from energies striking sensory inputs nerves in people—seems beyond question. Even rats seem to recognize the difference (...) between correlation and causation,1 and accordingly make different inferences from passive observation than from interventions. A few statisticians aside,” so do most of us. To square these views with the demands of computability, increasing numbers of psychologists and others have embraced a particular formalization, causal Bayes nets, as an account of human reasoning about and to causal connections.111 Such structures can be used by rational agents, including humans in so far as they are rational, to have degrees of belief in various conceptual contents, which they use to reason to expectations, which are realized or defeated by sensory inputs, which cause them to change their degrees of belief in other contents in accord with Bayes Rule, or some generalization of it. How is all of this supposed to be carried out? l. Representing Causal Structures The causal Bayes net framework adopted by a growing number of psychologists goes like this: Our representations of causal relations are captured in a graphical causal. (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.

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.

A longstanding question is the extent to which "reasonable doubt" may be expressed simply in terms of a threshold degree of belief. In this context, we examine the extent to which learning about possible alternatives may alter one's beliefs about a target hypothesis, even when no new "evidence" linking them to the hypothesis is acquired. Imagine the following scenario: a crime has been committed and Alice, the police's main suspect has been brought to trial. There are several pieces of evidence (...) that raise the probability that Alice committed the crime. Her attorney's defense strategy is not to challenge this evidence, but instead to provide personal details about Alice's neighbour, Jane. While Jane is one of many people the police spoke to, they saw no reason to investigate her further. You now learn that Jane, too, had access to the shed where the murder weapon was stored, just like Alice. To what extent should this alter your beliefs about Alice's guilt? In this paper, we provide a formal description of the problem and a solution indicating circumstances under which learning about Jane will more or less impact beliefs about Alice. (shrink)

By identifying and pursuing analogies between causal and logical influence I show how the Bayesian network formalism can be applied to reasoning about logical deductions.

Subjects are reported as being somewhat Bayesian, but as violating the normative ideal on occasion. To abjure probability altogether is difficult. To use Bayes' Theorem scrupulously when weighing evidence can incur costs without corresponding benefits. The subjects' evident nuanced probabilism appears both realistic and reasonable.

This book analyzes the origins of statistical thinking as well as its related philosophical questions, such as causality, determinism or chance. Bayesian and frequentist approaches are subjected to a historical, cognitive and epistemological analysis, making it possible to not only compare the two competing theories, but to also find a potential solution. The work pursues a naturalistic approach, proceeding from the existence of numerosity in natural environments to the existence of contemporary formulas and methodologies to heuristic pragmatism, a concept (...) introduced in the book's final section. This monograph will be of interest to philosophers and historians of science and students in related fields. Despite the mathematical nature of the topic, no statistical background is required, making the book a valuable read for anyone interested in the history of statistics and human cognition. (shrink)

According to Aristotle, humans are the rational animal. The borderline between rationality and irrationality is fundamental to many aspects of human life including the law, mental health, and language interpretation. But what is it to be rational? One answer, deeply embedded in the Western intellectual tradition since ancient Greece, is that rationality concerns reasoning according to the rules of logic – the formal theory that specifies the inferential connections that hold with certainty between propositions. Piaget viewed logical reasoning (...) as defining the end-point of cognitive development; and contemporary psychology of reasoning has focussed on comparing human reasoning against logical standards. (shrink)