STATISTICAL DECISION THEORY AND BIOLOGICAL VISION
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STATISTICAL DECISION THEORY AND BIOLOGICAL VISION
Statistical Decision Theory andBiological VisionLaurence T. MaloneyDepartment of Psychology Center for Neural Science New York University New York. NY STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONY 10003Draft: May 6, 2000111 Perception and the Physical World. Heyer. D. & Mausfekl. R. (Eds.J Chichester. UK: Wiley, in press./ know of only one case in mathematics of a doctrine which has been accepted and developed by the most eminent men of their time ... which at the same has appeared Io a suc STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONcession of sound writers Io be fundamentally false and devoid of foundation. Yet this is quite exactly the position in respect of inverse probabilitySTATISTICAL DECISION THEORY AND BIOLOGICAL VISION
fan estimation method based on Hayes theorem I.— Ronald A. Fisher (1930). Inverse probability.Statistical Decision Theory (SDT) emerged in essentiallyStatistical Decision Theory andBiological VisionLaurence T. MaloneyDepartment of Psychology Center for Neural Science New York University New York. NY STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONantedate it. in some cases by centuries, and. as the title indicates, an immediate stimulus to its development was the publication of Theory of Games and Economic Behavior by von Neumann & Morgenstern (1944/1953). Like Game Theory. SDT is normative, a set of principles that tell US how to act so as STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONto maximize gain and minimize loss.1The basic metaphor of SDT is that of a game between an Observer and the World. The Observer has imperfect informatSTATISTICAL DECISION THEORY AND BIOLOGICAL VISION
ion about the state of the World, analogous to sensory information, and must choose an action from among a limited repertoire of possible actions. ThiStatistical Decision Theory andBiological VisionLaurence T. MaloneyDepartment of Psychology Center for Neural Science New York University New York. NY STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONe invitation to (be) lunch, or - most important of all --correctly responded in a psychophysical task. SDT prescribes how the Observer should choose among possible actions, given what information it has. so as to maximize its expected gain.Bayesian Decision Theory (BDT) is a special case of SDT. but STATISTICAL DECISION THEORY AND BIOLOGICAL VISION one of particular relevance to a vision scientist. Recently, a number of authors (see. in particular. Knill. Kersten &. Yuille. 1996; Knill A RichardSTATISTICAL DECISION THEORY AND BIOLOGICAL VISION
s. 1996: Kersten A Schrater, this volume) have argued that BDT and related11 will use rhe terms gain, expected gain, etc. throughout and avoid the terStatistical Decision Theory andBiological VisionLaurence T. MaloneyDepartment of Psychology Center for Neural Science New York University New York. NY STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONons as when we seek to ‘maximize negative least-squares’. You win some, you negative-win some.https://khothuvien.cori!Statistical Decision Theory and Biological Vision10.'T8?'222Baycsian-inspircd techniques form a particularly congenial ’language' for modeling aspects of biological vision. We arc. i STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONn effect, invited to believe that increased familiarity with this 'language' (its concepts, terminology, and theory') will eventually lead to a deeperSTATISTICAL DECISION THEORY AND BIOLOGICAL VISION
understanding of biological vision through better models, better hypotheses and better experiments. To evaluate a claim of this son is very differentStatistical Decision Theory andBiological VisionLaurence T. MaloneyDepartment of Psychology Center for Neural Science New York University New York. NY STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONof SDT/BDT before evaluating, disparaging, or applying it as a framework for modeling biological vision.Yet the presentation of SDT and BDT in research articles is typically brief. Standard texts concerning BDT and Bayesian methods arc directed to statisticians and statistical problems. Consequently STATISTICAL DECISION THEORY AND BIOLOGICAL VISION, it is difficult for the reader to separate important assumptions underlying applications of BDT to biological vision from the computational details;STATISTICAL DECISION THEORY AND BIOLOGICAL VISION
it is precisely these assumptions that need to be understood and tested experimentally. Accordingly, this chapter is intended as an introduction for Statistical Decision Theory andBiological VisionLaurence T. MaloneyDepartment of Psychology Center for Neural Science New York University New York. NY STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONs divided into an introduction, four 'sections', and a conclusion.In the first of the four sections. I present the basic framework of SDT. including BDT. Illis framework is remarkably simple; I have chosen to present it in a way that emphasizes its visual or geometric aspects, although the equations STATISTICAL DECISION THEORY AND BIOLOGICAL VISION are there as well. As the opening quote from Fisher hints, certain Bayesian practices remain controversial. The controversy centers on the representaSTATISTICAL DECISION THEORY AND BIOLOGICAL VISION
tion of belief in human judgment and decision making, and the 'updating' of belief in response to evidence. In the initial presentation of the elementStatistical Decision Theory andBiological VisionLaurence T. MaloneyDepartment of Psychology Center for Neural Science New York University New York. NY STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONmade at a single instant of time Ợinsianlaneous BDT'). where the observer has complete information.SDT comprises a mathematical toolbox’ of techniques, and anyone using it to model decision making in biological vision must, of course, decide how to assemble the elements into a biologically-pcrtincnt STATISTICAL DECISION THEORY AND BIOLOGICAL VISION model: SDT itself is no more a model of visual processing than is the computer language Matlab®. The second section of the article contains a discussSTATISTICAL DECISION THEORY AND BIOLOGICAL VISION
ion of the elements of SDT. how they might be combined into biological models, and the difficulties likely to be encountered Shi mojo & Nakayama (1992Statistical Decision Theory andBiological VisionLaurence T. MaloneyDepartment of Psychology Center for Neural Science New York University New York. NY STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONheir argument seems conclusive. If organisms arc to have accurate estimates of relevant probabilities in moderately complex visual tasks, then they must have the capability to assign probabilities to events they have never encountered, and to estimate gains for actions they have never taken. The imp STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONlications of this claim arc discussed.The third section comprises two ‘challenges’ to the Bayesian approach, the first concerning the status of the viSTATISTICAL DECISION THEORY AND BIOLOGICAL VISION
sual representation in BDT-derived models. To date, essentially all applications of BDT to biological vision have been attempts to model the process oStatistical Decision Theory andBiological VisionLaurence T. MaloneyDepartment of Psychology Center for Neural Science New York University New York. NY STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONdefault' goal is to minimize the least-square error of the estimate. But the consequences of errors in, for example, depth estimation depend on the specific visual task that the organism is engaged in - leaping a chasm, say. versus tossing a stone at a target. BDT is in essence a way to choose among STATISTICAL DECISION THEORY AND BIOLOGICAL VISION actions given know ledge of their consequences: it is equally applicable toleaping chasms, and to tossing stones. What is not obvious is how BDT canSTATISTICAL DECISION THEORY AND BIOLOGICAL VISION
be used to computeStatistical Decision Theory and Biological Vision448524internal estimates when the real consequences of error arc not known. This diStatistical Decision Theory andBiological VisionLaurence T. MaloneyDepartment of Psychology Center for Neural Science New York University New York. NY STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONision across time and what I will call the updating problem. Instantaneous BDT assumes that, in each instant of time, the environment is essentially stochastic. Given full knowledge of the distributions of the possible outcomes, instantaneous BDT prescribes how to choose the optimal action. Across t STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONime, however, the distributional information may itself change, and change deterministically. The amount of light available outdoors in terrestrial enSTATISTICAL DECISION THEORY AND BIOLOGICAL VISION
vironments varies stochastically from day to day but also cycles deterministically over every twenty-four hour period. I describe a class of AugmentedStatistical Decision Theory andBiological VisionLaurence T. MaloneyDepartment of Psychology Center for Neural Science New York University New York. NY STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONonally implausible. Given that we know essentially nothing about the computational resources of the brain, this sort of criticism is premature. Nevertheless, it is instructive to consider possible implementations of BDT. and the fourth section of the article discusses what might be called Bayesian c STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONomputation' and its computational complexity.Blackwell and Girshick’s Theory of Games and Statistical Decisions appeared just 300 years after the 1654STATISTICAL DECISION THEORY AND BIOLOGICAL VISION
correspondence of Pascal and Fermat in which they developed the modern concepts of expectation and decision making guided by expectation maximizationStatistical Decision Theory andBiological VisionLaurence T. MaloneyDepartment of Psychology Center for Neural Science New York University New York. NY STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONnable and reasonably intelligent person would act so as to maximize gain. It is a peculiar fact (hat all of (he ideas underlying SDT and BDT (probabilistic representation of evidence, expectation maximization, etc.) were originally intended to serve as STATISTICAL DECISION THEORY AND BIOLOGICAL VISIONGọi ngay
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