Estimation and Prediction Using Belief Functions: Application to Stochastic Frontier Analysis
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Abstract Bayesian decision theory is a mathematical framework that models reasoning and decision‐making under uncertain conditions. The past few decades have witnessed an explosion of Bayesian modeling within cognitive science. Bayesian models are explanatorily successful for an array of psychological domains. This article gives an opinionated survey of foundational issues raised by Bayesian cognitive science, focusing primarily on Bayesian modeling of perception and motor control. Issues discussed include the normative basis of Bayesian decision theory; explanatory achievements of Bayesian cognitive science; intractability of Bayesian computation; realist versus instrumentalist interpretation of Bayesian models; and neural implementation of Bayesian inference. This article is categorized under: Philosophy > Foundations of Cognitive Science
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Can children solve Bayesian problems, given that these pose great difficulties even for most adults? We present an ecological framework in which Bayesian intuitions emerge from a match between children's numerical competencies and external representations of numerosity. Bayesian intuition is defined here as the ability to determine the exact Bayesian posterior probability by minds untutored in probability theory or in Bayes' rule. As we show, Bayesian intuitions do not require processing of probabilities or Arabic numbers, but basically the ability to count tokens in icon arrays and to understand what to count. A series of experiments demonstrates for the first time that icon arrays elicited Bayesian intuitions in children as young as second-graders for 22% to 32% of all problems; fourth-graders achieved 50% to 60%. Most surprisingly, icon arrays elicited Bayesian intuitions in children with dyscalculia, a specific learning disorder that has been attributed to genetic causes. These children could solve an impressive 50% of Bayesian problems, a level similar to that of children without dyscalculia. By seventh grade, children solved about two thirds of Bayesian problems with natural frequencies alone, without the additional help of icon arrays. We identify four non-Bayesian rules. On the basis of these results, we propose a common solution for the phylogenetic, the ontogenetic, and the 1970s puzzles in the Bayesian literature and argue for a revision of how to teach statistical thinking. In accordance with recent work on infants' numerical abilities, these findings indicate that children have more numerical ability than previously assumed. (PsycInfo Database Record (c) 2021 APA, all rights reserved).
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Abstract The present paper includes a systematic survey of recent studies on the role of statistical models, when the assessment of conditional (predictive) distributions of observables, given objective events, is viewed as the primary object of statistical inference. The paper aims at providing some insight into the nature and status of the conventional assumption of mathematical models involving ‘‘unknown parameters”, within the predictivist setting stressed by de Finetti (1937).
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Bayesian inference comprises of parameter estimation and model selection/comparison. A common approach to solving both of these problems has been to use statistical sampling techniques that are inherently non-Bayesian. This paper presents new Bayesian sampling method that solves both problems by changing the focus of Bayesian inference to the model selection problem first. In the papers [N. Xiang and P. M. Goggins, J. Acoust. Soc. Am. 110, 1415–1424 (2001); 113, 2685–2697 (2003)], the authors developed a model for the decay times and decay modes of acoustically coupled rooms in terms of measured Schroeder’s decay functions. This paper shows how the Bayesian sampling method can be used to evaluate the ‘‘Bayesian evidence’’ term used in model selection as well as determining the decay times along with error estimates.
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In this paper we discuss by examples an atypical approach to statistical inference based on the optimization of the predictive performances of a parametric model. We show how to presented approach permits to deal with several statistical problems in an unified way
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We propose a novel Bayesian formulation for the reconstruction from compressed measurements. We demonstrate that high-sparsity enforcing priors based on l p -norms, with 0 < p ≤ 1, can be used within a Bayesian framework by majorization-minimization methods. By employing a fully Bayesian analysis of the compressed sensing system and a variational Bayesian analysis for inference, the proposed framework provides model parameter estimates along with the unknown signal, as well as the uncertainties of these estimates. We also show that some existing methods can be derived as special cases of the proposed framework. Experimental results demonstrate the high performance of the proposed algorithm in comparison with commonly used methods for compressed sensing recovery.
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We mimic the cognitive ability of Human perception, based on Bayesian hypothesis, to recognize view-based 3D objects. We consider approximate Bayesian (Empirical Bayesian) for perceptual inference for recognition. We essentially handle computation with perception.
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Inferring Subjective Prior Knowledge: An Integrative Bayesian Approach Sean Tauber (sean.tauber@uci.edu) Mark Steyvers (mark.steyvers@uci.edu) Department of Cognitive Sciences, University of California, Irvine Irvine, CA 92697 USA Abstract The standard approach to Bayesian models of Cognition (also known as rational models) requires researchers to make strong assumptions about people’s prior beliefs. For example, it is often assumed that people’s subjective knowledge is best represented by “true” environmental data. We show that an integrative Bayesian approach—combining Bayesian cognitive models with Bayesian data analysis—allows us to relax this assumption. We demonstrate how this approach can be used to estimate people’s subjective prior beliefs based on their responses in a prediction task. Keywords: Bayesian modeling; rational analysis; cognitive models; Bayesian data analysis; Bayesian inference; knowledge representation; prior knowledge Introduction In the standard approach to Bayesian models of Cognition (also referred to as rational models), researchers make strong assumptions about people’s prior beliefs in order to make predictions about their behavior. These models are used to simulate the expected behavior—such as decisions, judgments or predictions—of someone whose computational-level solution to a cognitive task is well described by the model. Analysis of Bayesian models of cognition usually involves a qualitative comparison between human responses and simulated model predictions. For an overview of Bayesian models of cognition see Oaksford and Chater (1998); but also see Mozer, Pashler, and Homaei (2008); and Jones and Love (2011) for a critique. As an alternative to the standard approach, we present an integrative Bayesian approach that allows us to relax the assumptions about people’s prior beliefs. This approach is motivated by previous efforts to infer subjective mental representations (Lewandowsky, Griffiths, & Kalish, 2009; Sanborn & Griffiths, 2008; Sanborn, Griffiths, & Shiffrin, 2010) and more specifically to combine Bayesian models of cognition and Bayesian data analysis (Huszar, Noppeney & Lengyel, 2010; Lee & Sarnecka, 2008). The integrative approach allows us to use people’s responses on a cognitive task to infer posterior distributions over the psychological variables in a Bayesian model of cognition. It also allows us to estimate probabilistic representations of people’s subjective prior beliefs. We recently applied this approach to a Bayesian cognitive model of reconstructive memory (Hemmer, Tauber, & Steyvers, in prep). We estimated individuals’ subjective prior beliefs about the distribution of people’s heights based on their responses in a memory task. The technical requirements for integrated Bayesian inference were simplified because the posterior distribution, based on inference in the cognitive model, had a simple Gaussian form. This made it straight forward to define individuals’ responses as Gaussian distributed random variables in an integrated Bayesian model. In this study, we develop a method for applying integrated Bayesian inference that does not require the posterior of the cognitive model to have a simple parametric form. We apply this method to a Bayesian cognitive model for predictions that was developed by Griffiths and Tenenbaum (2006). Their Bayesian model of cognition was a computational-level description of how people combine prior knowledge with new information to make predictions about real-world phenomena. They asked participants to make a series of predictions about duration or extent that were similar to the following examples: If you were assessing the prospects of a 60-year-old man, how much longer would you expect him to live? If you were an executive evaluating the performance of a movie that had made $40 million at the box office so far, what would you estimate for its total gross? All of the questions used by Griffiths and Tenenbaum (2006) were based on real-world phenomena such as, life spans, box office grosses for movies, movie runtimes, poem lengths and waiting times. Their assumption was that people make predictions about these phenomena based on prior beliefs that reflect their true extents or durations in the real world. Although it is possible that people’s beliefs about these phenomena are tuned to the environment, this assumption cannot be used to explain how people make similar sorts of predictions about counterfactual phenomena that have no true statistics in the environment. For example, consider the following question: Suppose it is the year 2075 and medical science has advanced significantly. You meet a man that is 60 years old. To what age will this man live? There is no “true” answer to this question and therefore no environmental data is available. This creates a problem for a Bayesian model of cognition that requires environmental data in order to make predictions.
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Abstract The contrast sensitivity function (CSF) is crucial in predicting functional vision both in research and clinical areas. Recently, a group of novel strategies, multi-dimensional adaptive methods, were proposed and allowed more rapid measurements when compared to usual methods such as Ψ or staircase. Our study further presents a multi-dimensional Bayesian framework to estimate parameters of the CSF from experimental data obtained by classical sampling. We extensively simulated the framework’s performance as well as validated the results in a psychophysical experiment. The results showed that the Bayesian framework significantly improves the accuracy and precision of parameter estimates from usual strategies, and requires about the same number of observations as the novel methods to obtain reliable inferences. Additionally, the improvement with the Bayesian framework was maintained when the prior poorly matched the observer’s CSFs. The results indicated that the Bayesian framework is flexible and sufficiently precise for estimating CSFs.
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