Facilitating Normative Judgments of Conditional Probability: Frequency or Nested Sets?

2003 ◽  
Vol 50 (2) ◽  
pp. 97-106 ◽  
Author(s):  
Kimihiko Yamagishi
Keyword(s):  
Conditional Probability ◽  
Probabilistic Reasoning ◽  
Probability Judgment ◽  
Probability Judgments ◽  
Normative Reasoning ◽  
Roulette Wheel ◽  
Probability Task ◽  
Nested Sets ◽  
Probabilistic Structure ◽  
Inspection Machine

Abstract. Recent probability judgment research contrasts two opposing views. Some theorists have emphasized the role of frequency representations in facilitating probabilistic correctness; opponents have noted that visualizing the probabilistic structure of the task sufficiently facilitates normative reasoning. In the current experiment, the following conditional probability task, an isomorph of the “Problem of Three Prisoners” was tested. “A factory manufactures artificial gemstones. Each gemstone has a 1/3 chance of being blurred, a 1/3 chance of being cracked, and a 1/3 chance of being clear. An inspection machine removes all cracked gemstones, and retains all clear gemstones. However, the machine removes ½ of the blurred gemstones. What is the chance that a gemstone is blurred after the inspection?” A 2 × 2 design was administered. The first variable was the use of frequency instruction. The second manipulation was the use of a roulette-wheel diagram that illustrated a “nested-sets” relationship between the prior and the posterior probabilities. Results from two experiments showed that frequency alone had modest effects, while the nested-sets instruction achieved a superior facilitation of normative reasoning. The third experiment compared the roulette-wheel diagram to tree diagrams that also showed the nested-sets relationship. The roulette-wheel diagram outperformed the tree diagrams in facilitation of probabilistic reasoning. Implications for understanding the nature of intuitive probability judgments are discussed.

scholarly journals The Bayesian Sampler: Generic Bayesian Inference Causes Incoherence in Human Probability Judgments

2018 ◽  
Author(s):  
Jian-Qiao Zhu ◽  
Adam N Sanborn ◽  
Nick Chater
Keyword(s):  
Probabilistic Models ◽  
Conjunction Fallacy ◽  
Probability Judgment ◽  
Noise Model ◽  
Probability Judgments ◽  
Probability Estimates ◽  
Bayesian Models Of Cognition ◽  
Improved Accuracy ◽  
Biases And Heuristics ◽  
The Brain

Human probability judgments are systematically biased, in apparent tension with Bayesian models of cognition. But perhaps the brain does not represent probabilities explicitly, but approximates probabilistic calculations through a process of sampling, as used in computational probabilistic models in statistics. Naïve probability estimates can be obtained by calculating the relative frequency of an event within a sample, but these estimates tend to be extreme when the sample size is small. We propose instead that people use a generic prior to improve the accuracy of their probability estimates based on samples, and we call this model the Bayesian sampler. The Bayesian sampler trades off the coherence of probabilistic judgments for improved accuracy, and provides a single framework for explaining phenomena associated with diverse biases and heuristics such as conservatism and the conjunction fallacy. The approach turns out to provide a rational reinterpretation of “noise” in an important recent model of probability judgment, the probability theory plus noise model (Costello & Watts, 2014, 2016a, 2017, 2019; Costello, Watts, & Fisher, 2018), making equivalent average predictions for simple events, conjunctions, and disjunctions. The Bayesian sampler does, however, make distinct predictions for conditional probabilities, and we show in a new experiment that this model better captures these judgments both qualitatively and quantitatively.

Towards a New Model for Causal Reasoning in Expert Systems

2017 ◽  
Vol 7 (1) ◽  
pp. 32-63 ◽  
Author(s):  
M. Keith Wright
Keyword(s):  
Expert Systems ◽  
Conditional Probability ◽  
Causal Reasoning ◽  
Causal Theory ◽  
Probability Judgment ◽  
Probability Assessment ◽  
Conditional Probabilities ◽  
Long Run ◽  
Causal Strength ◽  
As If

This paper presents ideas for improved conditional probability assessment and improved expert systems consultations. It cautions that knowledge engineers may sometimes be imprecise when capturing causal information from experts: their elicitation questions may not distinguish between causal and correlational expertise. This paper shows why and how such models cannot support normative inferencing over conditional probabilities as if they were all based on frequencies in the long run. In some cases, these probabilities are instead causal theory-based judgments, and therefore are not traditional conditional probabilities. This paper argues that these should be processed as if they were causal strength probabilities or causal propensity probabilities. This paper reviews the literature on causal and probability judgment, and then presents a probabilistic inferencing model that integrates theory-based causal probabilities with frequency-based conditional probabilities. The paper also proposes guidelines for elicitation questions that knowledge engineers may use to avoid conflating causal theory-based judgment with frequency based judgment.

Pseudo Independent Models

2011 ◽  
pp. 935-940
Author(s):  
Yang Xiang
Keyword(s):  
Probability Distribution ◽  
Computer System ◽  
Conditional Probability ◽  
Graphical Models ◽  
Intelligent Systems ◽  
Probabilistic Reasoning ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Conditional Probability Distribution ◽  
Binary Variables

Graphical models such as Bayesian networks (BNs) (Pearl, 1988) and decomposable Markov networks (DMNs) (Xiang, Wong & Cercone, 1997) have been applied widely to probabilistic reasoning in intelligent systems. Figure1 illustrates a BN and a DMN on a trivial uncertain domain: A virus can damage computer files, and so can a power glitch. A power glitch also causes a VCR to reset. The BN in (a) has four nodes, corresponding to four binary variables taking values from {true, false}. The graph structure encodes a set of dependence and independence assumptions (e.g., that f is directly dependent on v, and p but is independent of r, once the value of p is known). Each node is associated with a conditional probability distribution conditioned on its parent nodes (e.g., P(f | v, p)). The joint probability distribution is the product P(v, p, f, r) = P(f | v, p) P(r | p) P(v) P(p). The DMN in (b) has two groups of nodes that are maximally pair-wise connected, called cliques. Each clique is associated with a probability distribution (e.g., clique {v, p, f} is assigned P(v, p, f)). The joint probability distribution is P(v, p, f, r) = P(v, p, f) P(r, p) / P(p), where P(p) can be derived from one of the clique distributions. The networks, for instance, can be used to reason about whether there are viruses in the computer system, after observations on f and r are made.

Evidence for the Jeffrey Table

2019 ◽  
Vol 66 (3) ◽  
pp. 187-194
Author(s):  
Moyun Wang ◽  
Mingyi Zhu
Keyword(s):  
Conditional Probability ◽  
Fundamental Question ◽  
The Other ◽  
Adequate Account ◽  
Natural Languages ◽  
Probability Judgments ◽  
Low Level ◽  
De Finetti

Abstract. Conditionals statements are a common and necessary component in natural languages. The research reported in this paper is on a fundamental question about singular conditionals. Is there an adequate account of people’s truth, falsity, and credibility (probability) judgments about these conditionals when their antecedents are false? Two experiments examined people’s quantitative credibility ratings and qualitative truth and falsity judgments for singular conditionals, if p then q, given false antecedent, not-p, cases. The results demonstrate that, when relevant knowledge about the conditional probability of q given p, P( q|p), is available to participants in not-p cases, they tend to make credibility ratings based on P( q|p), and to make “true” (or “false”) judgments at a high (or low) level of these credibility ratings. These findings favor the Jeffrey table account of these conditionals over the other existing accounts, including that of the de Finetti table.

scholarly journals Induction as conditional probability judgment

2007 ◽  
Vol 35 (6) ◽  
pp. 1353-1364 ◽  
Author(s):  
Sergey V. Blok ◽  
Douglas L. Medin ◽  
Daniel N. Osherson
Keyword(s):  
Conditional Probability ◽  
Probability Judgment

Double Conjunction Fallacies in Physicians’ Probability Judgment

2018 ◽  
Vol 38 (6) ◽  
pp. 756-760 ◽  
Author(s):  
Vincenzo Crupi ◽  
Fabrizio Elia ◽  
Franco Aprà ◽  
Katya Tentori
Keyword(s):  
Clinical Practice ◽  
Empirical Data ◽  
Clinical Signs ◽  
Probability Judgment ◽  
Uncertain Reasoning ◽  
Probability Judgments ◽  
Reasoning Bias

We report the first empirical data showing a significant amount of double conjunction fallacies in physicians’ probability judgments concerning prognosis and diagnosis. Our results support the hypothesis that physicians’ probability judgments are guided by assessments of evidential impact between diagnostic conditions and clinical signs. Moreover, we show that, contrary to some influential views, double conjunction fallacies represent an experimentally replicable reasoning bias. We discuss how the phenomenon eludes major current accounts of uncertain reasoning in medicine and beyond and how it relates to clinical practice.

Sources of Bias in Judgment and Decision Making

2015 ◽  
Author(s):  
Joe W. Tidwell ◽  
Daniel Buttaccio ◽  
Jeffrey S. Chrabaszcz ◽  
Michael R. Dougherty ◽  
Rick P. Thomas
Keyword(s):  
Decision Making ◽  
Judgment And Decision Making ◽  
External Factors ◽  
Probability Judgment ◽  
Memory Model ◽  
Memory Process ◽  
Memory Processes ◽  
Probability Judgments ◽  
Elicitation Format ◽  
P Sources

Sources of bias in confidence and probability judgments, for example conservatism, overconfidence, and subadditivity, are some of the most important and rigorously researched topics within judgment and decision making. However, despite the seemingly obvious importance of memory processes on these types of judgments, much of this research has focused on external factors independent of memory processes, such as the effects of various types of elicitation format. In this chapter, we review the research relevant to commonly observed effects related to confidence and probability judgment, and then provide a memory-process account of these phenomena based on two models: Minerva-DM, a multiple-trace memory model; and HyGene, an extension of Minerva-DM that incorporates hypothesis generation. We contend that accounting for the dependence of judgments on memory provides a unifying theoretical framework for these various phenomena, as well as cognitive models that accurately reflect real-world behavior.

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