Présentation au sujet: "The most incomprehensible thing about the world"— Transcription de la présentation:
1 The most incomprehensible thing about the world is that it is comprehensibleAlbert Einstein
2 Bayesian Cognition Julien Diard Pierre Bessière Probabilistic models of action, perception, inference,decision and learningJulien DiardCNRS - Laboratoire de Psychologie et NeuroCognitionPierre BessièreCNRS - Laboratoire de Physiologie de la Perception et de l’Action
3 To get more infoBayesian-Programming.orgftp://ftp-serv.inrialpes.fr/pub/emotion/bayesian-programming/Cours
4 Plan / planning Bessière c1 15/11 Diard c2 29/11, c3 13/12, c4 03/01 Incomplétude, incertitude, Programme Bayésien, inférence BayésienneDiard c2 29/11, c3 13/12, c4 03/01Modèles Bayésiens en robotique et sciences cognitivesDiard c5 10/01Sélection de modèles, machine learning, distinguabilité de modèlesBessière c6 17/01Compléments : algorithmes d’inférence, maximum d’entropie
5 Daniel J. Simons & Christopher Chabris Perception testDaniel J. Simons & Christopher ChabrisHarvard University
10 Probability Theory as an alternative to Logic The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man's mind .James Clerk Maxwell
11 Incompleteness and Uncertainty A very small cause which escapes our notice determines a considerableeffect that we cannot fail to see, and then we say that the effect is due tochance.H. Poincaré
13 Shape from Motion DROULEZ COLAS PLOS 6 EXPE PSYCHO Colas, F., Droulez, J., Wexler, M. & Bessiere, P. (2008)Unified probabilistic model of perception of three-dimensional structure from optic flow; in Biological Cybernetics,in pressColas, F. (2006) Perception des objets en mouvement : Composition bayésienne du flux optique et du mouvement de l’observateur, Thèse INPG
14 Illusions: McGurkeffect Courtesy of Masso Arnt, Associate Professor, University of OsloCathiard, M.-A., Schwartz, J.-L. & Abry, C. (2001). Asking a naivequestion to the McGurk effect : why does audio [b] give more [d]percepts with usual [g] than with visual [d] ? In Proceedings of the/Auditory Visual Speech processing, AVSP'2001/, Aalborg, Copenhague,
23 ThesisProbabilistic inference and learning theory, considered as a model of reasoning, is a new paradigm (an alternative to logic) to explain and understand perception, inference, decision, learning and action.La théorie des probabilités n'est rien d'autre que le sens commun fait calcul.Marquis Pierre-Simon de LaplaceThe actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man's mind .James Clerk MaxwellBy inference we mean simply: deductive reasoning whenever enough information is at hand to permit it; inductive or probabilistic reasoning when - as is almost invariably the case in real problems - all the necessary information is not available. Thus the topic of « Probability as Logic » is the optimal processing of uncertain and incomplete knowledge .E.T. JaynesSubjectivist vs Objectivist epistemology of probabilities ?
24 A water treatment unit (1) Complete simulationIncomplete modelObserve the consequences of this incompleteness11 values, 0 the worst
31 Not taking into account the effect of hidden variables may lead to wrong decision (1) C=0,1 or 2 leads to optimal value O*=6With H the “reality” is somewhat more complexThe adequate choice of C is more complex but also more informed
32 Not taking into account the effect of hidden variables may lead to wrong decision (2) C=0,1 or 2 leads to optimal value O*=6With H the “reality” is somewhat more complexThe adequate choice of C is more complex but also more informed
52 Specification = Variables + Decomposition + Parametric Forms Variables: the choice of relevant variables for the problemDecomposition: the expression of the joint probability distribution as the product of simpler distributionParametric Forms: the choice of the mathematical functions of each of these distributions
58 Bayesian Program = Description + Question SpecificationIdentificationDescriptionQuestionProgramVariablesParametrical Forms or Recursive QuestionDecompositionPreliminary Knowledge pExperimental Data dUtilization
62 Logical Proposition Logical Proposition are denoted by lowercase name: Usual logical operators:
63 Probability of Logical Proposition We assume that to assign a probability to a given proposition a,it is necessary to have at least some preliminary knowledge,summed up by a proposition p.Of course, we will be interested in reasoning on the probabilitiesof the conjunctions, disjunctions and negations of propositions,denoted, respectively, by:We will also be interested in the probability of proposition aconditioned by both the preliminary knowledge p and someother proposition b:
64 Normalization and Conjunction Postulates Bayes ruleCox TheoremResolution PrincipleWhy don't you take the disjunction rule as an axiom?
65 Discrete VariableVariable are denoted by name starting with one uppercase letter:By definition a discrete variable is a set of propositionsMutually exclusive:Exhaustive: at least one is trueThe cardinal of X is denoted:
72 DescriptionThe purpose of a description is to specify an effective method to compute a joint distribution on a set of variables:Given some preliminary knowledge p and a set of experimental data d.This joint distribution is denoted as:
73 Decomposition Partion in K subsets: Conjunction rule: Conditional independance:Decomposition:
75 QuestionGiven a description, a question is obtained by partitionning the set of variables into 3 subsets: the searched variables, the known variables and the free variables.We define the Search, Known and Free as the conjunctions of the variables belonging to these three sets.We define the corresponding question as the distribution:
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