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Bayesian Cognition Probabilistic Models of Action, Perception, Inference, Decision and Learning

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Bayesian Cognition Probabilistic Models of Action, Perception, Inference, Decision and Learning Bayesian Cognition Probabilistic models of action, perception, inference, decision and learning Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #1 Bayesian Cognition Cours 2 : Bayesian Programming Julien Diard http://diard.wordpress.com [email protected] CNRS - Laboratoire de Psychologie et NeuroCognition, Grenoble Pierre Bessière CNRS - Institut des Systèmes Intelligents et de Robotique, Paris Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #2 Contents / Schedule • c1 : fondements théoriques, • c1 : Mercredi 11 Octobre définition du formalisme de la • c2 : Mercredi 18 Octobre programmation bayésienne • c3 : Mercredi 25 Octobre • *pas de cours la semaine du 30 • c2 : programmation Octobre* bayésienne des robots • c4 : Mercredi 8 Novembre • c5 : Mercredi 15 Novembre • c3 : modélisation bayésienne • *pas de cours la semaine du 20 cognitive Novembre* • c6 : Mercredi 29 Novembre • c4 : comparaison bayésienne de modèles • Examen ?/?/? (pour les M2) Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #3 Plan • Summary & questions! • Basic concepts: minimal example and spam detection example • Bayesian Programming methodology – Variables – Decomposition & conditional independence hypotheses – Parametric forms (demo) – Learning – Inference • Taxonomy of Bayesian models Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #4 Probability Theory As Extended Logic • Probabilités • Probabilités « fréquentistes » « subjectives » E.T. Jaynes (1922-1998) – Une probabilité est une – Référence à un état de propriété physique d'un connaissance d'un sujet • P(« il pleut » | Jean), objet P(« il pleut » | Pierre) – Axiomatique de • Pas de référence à la limite Kolmogorov, théorie d’occurrence d’un des ensembles événement (fréquence) • Probabilités conditionnelles N – P (A)=fA = limN – P(A | π) et jamais P(A) ⇥ NΩ – Statistiques classiques – Statistiques bayésiennes • Population parente, etc. Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #5 Principle Incompleteness Preliminary Knowledge + Bayesian Learning Experimental Data = Probabilistic Representation Uncertainty P(a) P( a) 1 Bayesian Inference + ¬ = P(a∧b) = P(a)P(b | a) = P(b)P(a | b) Decision€ Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #6 Règles de calcul • Règle du produit P (AB C)=P (A C)P (B AC) | | | = P (B C)P (A BC) | | Reverend Thomas Bayes è Théorème de Bayes (~1702-1761) P (B C)P (A BC) P (B AC)= | | , si P (A C) =0 | P (A C) | • Règle de la somme | P (A C)+P (A¯ C)=1 P ([A = a] C)=1 | | | a A è Règle de marginalisation∈ P (AB C)=P (B C) | | A Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #7 Programme Bayésien • Variables Spécification • Décomposition Description PB • Formes paramétriques Identification Question Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #8 Mon premier Programme Bayésien • Variables A = {il pleut, il ne pleut pas} B = {Jean a son p., Jean n'a pas son p.} Sp. • Décomposition Desc. P(B A) = P(A) P(B | A) PB • Formes paramétriques Tables de probabilités conditionnelles P(B | A) A=il pleut A=il ne pleut P(A) A=il pleut A=il ne pleut pas pas B=Jean n'a pas 0,05 0,9 Identification : 0,4 0,6 son parapluie B=Jean a son 0,95 0,1 parapluie Question : P(A | B) = P(A) P(B | A) / P(B) Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #9 Plan • Summary & questions! • Basic concepts: minimal example and spam detection example • Bayesian Programming methodology – Variables – Decomposition & conditional independence hypotheses – Parametric forms (demo) – Learning – Inference • Taxonomy of Bayesian models Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #10 Bayesian Spam Detection • Classify texts in 2 categories “spam” or “nonspam” – Only available information: a set of words • Adapt to the user and learn from experience Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #11 Variables • Spam – Boolean variable: {True, False} • W0, W1, ..., WN-1 – Wi is the presence or absence of word i in a text – Boolean variables: {True, False} Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #12 Decomposition P(Spam ∧W0 ∧ ... ∧WN−1) = P(Spam)× P(W0 | Spam)× P(W1 |W0 ∧Spam) × ... × P(WN−1 |WN−2 ∧ ... ∧W0 ∧Spam) € P(Spam W0 ... WN 1) ^ ^ ^ − N 1 − P(Spam) P(W Spam) ⇡ ⇥ n | Yn=0 Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #13 Importance des hypothèses d’indépendance conditionnelle • Nombre de probabilités – Attention : nb proba ≠ nb param libres – Avant hypothèses d’indépendance conditionnelle N+1 P(Spam W0 ... W N 1) 2 probabilités ^ ^ ^ − – Après hypothèses N 1 − P(Spam) P(W Spam) ⇥ n | 2 + 4N probabilités n=0 Diard – LPNC/CNRSY Cognition Bayésienne – 2017-2018 #14 Graphical representation Bayesian Network P(Spam ∧W ∧ ... ∧W ) P(Spam W0 ... WN 1) 0 N−1 ^ ^ ^ − N 1 = P(Spam)× P(W0 | Spam)× P(W1 |W0 ∧Spam) − P(Spam) P(Wn Spam) × ... × P(WN−1 |WN−2 ∧ ... ∧W0 ∧Spam) ⇡ ⇥ | Yn=0 Spam Spam € W0 W1 W2 … WN-1 W0 W1 W2 … WN-1 Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #15 P(Spam ∧W0 ∧ ... ∧WN−1) N−1 = P(Spam)× ∏ P(Wn | Spam) Parametric forms n=0 and identification € Could also be P([Spam = false]) = 0.25 computed from a learning database P Spam = true = 0.75 ([ ]) P([Spam = false]) =θ € P([Spam = true]) =1−θ € Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #16 P(Spam ∧W0 ∧ ... ∧WN−1) N−1 = P(Spam)× ∏ P(Wn | Spam) Parametric forms n=0 and identification € n Nbref P([Wn = true]| [Spam = false]) = Nbref P([Wn = false]| [Spam = false]) =1− P([Wn = true]| [Spam = false]) n Nbret P([Wn = true]| [Spam = true]) = Nbret P([Wn = false]| [Spam = true]) =1− P([Wn = true]| [Spam = true]) Attention, si un mot Wn n'est jamais vu dans un spam, alors si on le voit dans un mail m, m ne peut pas être un spam Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #17 P(Spam ∧W0 ∧ ... ∧WN−1) N−1 = P(Spam)× ∏ P(Wn | Spam) Parametric forms n=0 and identification • Loi de succession de€ Laplace n 1+ Nbref P([Wn = true]| [Spam = false]) = 2 + Nbref P([Wn = false]| [Spam = false]) =1− P([Wn = true]| [Spam = false]) n 1+ Nbret P([Wn = true]| [Spam = true]) = 2 + Nbret P([Wn = false]| [Spam = true]) =1− P([Wn = true]| [Spam = true]) Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #18 Identification n 1+ Nbref P([Wn = true]| [Spam = false]) = 2 + Nbref n 1+ Nbret P([Wn = true]| [Spam = true]) = 2 + Nbret Notion de paramètre libre, de base de données d’apprentissage, d’algorithme d’identification de paramètres Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #19 P(Spam ∧W0 ∧ ... ∧WN−1) Joint distribution and N−1 = P(Spam)× ∏ P(Wn | Spam) questions (1) n=0 € P(Spam) = ∑P(Spam ∧W0 ∧ ... ∧ Wn ∧ ... ∧WN−1) W0∧ ... ∧WN-1 N−1 P(Spam) = ∑ P(Spam)× ∏ P(Wn | Spam) € W0∧ ... ∧WN-1 n=0 = P(Spam) € Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #20 P(Spam ∧W0 ∧ ... ∧WN−1) Joint distribution and N−1 = P(Spam)× ∏ P(Wn | Spam) questions (2) n=0 € P(Wn ) = ∑P(Spam ∧W0 ∧ ... ∧ Wn ∧ ... ∧WN−1) Spam∧Wi≠n N−1 P(Wn ) = ∑ P(Spam)× ∏ P(Wn | Spam) € Spam∧Wi≠n n=0 = ∑P(Spam)× P(Wn | Spam) Spam # n & # n & 1+ Nbref 1+ Nbret P([Wn = true]) = % 0.25 × ( +% 0.75 × ( % ( 2 + Nbre $ 2 + Nbref ' $ t ' € Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #21 € P(Spam ∧W0 ∧ ... ∧WN−1) Joint distribution and N−1 = P(Spam)× ∏ P(Wn | Spam) questions (3) n=0 ∑P(Spam ∧W0 ∧ ... ∧ Wn ∧ ... ∧WN−1) Wi≠n P(Wn | [Spam = true]) = € ∑P(Spam ∧W0 ∧ ... ∧ Wn ∧ ... ∧WN−1) W0∧ ... ∧WN-1 n 1+ Nbret P(Wn | [Spam = true]) = 2 + Nbret € € P(Spam)× P([Wn = true] | Spam) P(Spam | [Wn = true]) = ∑ P(Spam)× P([Wn = true] | Spam) Spam Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #22 P(Spam ∧W0 ∧ ... ∧WN−1) Joint distribution and N−1 = P(Spam)× ∏ P(Wn | Spam) questions (4) n=0 € P(Spam ∧W0 ∧ ... ∧ Wn ∧ ... ∧WN−1) P(Spam |W0 ∧ ... ∧ WN-1) = ∑P(Spam ∧W0 ∧ ... ∧ Wn ∧ ... ∧WN−1) Spam N−1 P(Spam)× ∏ P(Wn | Spam) P Spam |W ∧ ... W = n=0 € ( 0 N−1) N−1 ∑ P(Spam)× ∏ P(Wn | Spam) Spam n=0 Diard – LPNC/CNRS € Cognition Bayésienne – 2017-2018 #23 Bayesian Program Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #24 Résultat : exemple numérique • 5 mots considérés : fortune, next, programming, money, you • 1 000 mails dans la base de données d’apprentissage (250 non spams, 750 spams) n P(Wn | [Spam=false]) P(Wn | [Spam=true]) W =false W =true W =false W =true n n n n n n n Word n af at 0 fortune 0 375 0 0.996032 0.00396825 0.5 0.5 1 next 125 0 1 0.5 0.5 0.99867 0.00132979 2 programming 250 0 2 0.00396825 0.996032 0.99867 0.00132979 3 money 0 750 3 0.996032 0.00396825 0.00132979 0.99867 4 you 125 375 4 0.5 0.5 0.5 0.5 Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #25 Résultat : exemple numérique • 25=32 mails possibles Subset Words present P(Spam | w0 … w4) number [Spam = false] [Spam = true] 3 {money} 5.24907e-06 0.999995 11 {next, money} 0.00392659 0.996073 12 {next, money, you} 0.00392659 0.996073 15 {next, programming, money} 0.998656 0.00134393 27 {fortune, next, money} 1.57052e-05 0.999984 Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #26 Results SpamSieve http://c-command.com/spamsieve/ Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #27 Plan • Summary & questions! • Basic concepts: minimal example and spam detection example • Bayesian Programming methodology – Variables – Decomposition & conditional independence hypotheses – Parametric forms (demo) – Learning – Inference • Taxonomy of Bayesian models Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #28 Programme Bayésien • Variables Spécification • Décomposition Description PB • Formes paramétriques Identification Question Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #29 Logical Proposition Logical Propositions are denoted by lowercase names: a Usual logical operators: a∧b a∨b € ¬a € Diard – LPNC/CNRS Cognition Bayésienne – 2017-2018 #30 Probability of Logical Proposition To assign a probability to a given proposition a, it is necessary to have at least some preliminary knowledge, summed up by a proposition π.
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