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Bayesian Cognition Cours 2: Bayesian Programming

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Bayesian Cognition Cours 2: Bayesian Programming Bayesian Cognition Cours 2: Bayesian programming Julien Diard CNRS - Laboratoire de Psychologie et NeuroCognition Grenoble Pierre Bessière CNRS - Institut des Systèmes Intelligents et de Robotique Paris Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 2 Contents / schedule • Cours 1 30/09/15, C-ADM008-Amphi J. Besson – Incompleteness to uncertainty: Bayesian programming and inference • Cours 2 14/10/15, C-ADM008-Amphi J. Besson – Bayesian programming • Cours 3 21/10/15, C-ADM008-Amphi J. Besson – Bayesian robot programming (part 1) • Cours 4 04/11/15, C-ADM008-Amphi J. Besson – Bayesian robot programming (part 2) – Bayesian cognitive modeling (part 1) • Cours 5 18/11/15, C-ADM008-Amphi J. Besson – Bayesian cognitive modeling (part 2) • Cours 6 20/12/15, C-ADM008-Amphi J. Besson – Bayesian model comparison, bayesian model distinguishability • Examen ?/?/? (pour les M2) Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 3 Plan • Summary & questions! • Bayesian Programming methodology – Variables – Decomposition & conditional independence hypotheses – Parametric forms (demo) – Learning – Inference • Taxonomy of Bayesian models Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 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. Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 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€ Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 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 Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 7 Bayesian Program methodology Specification • Variables • Decomposition • Parametrical Forms or Recursive Question Description Preliminary Knowledge π Identification Program Program Experimental Data δ Inference Question Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 8 Bayesian Program = Spam detection example Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 9 Plan • Summary & questions! • Bayesian Programming methodology – Variables – Decomposition & conditional independence hypotheses – Parametric forms (demo) – Learning – Inference • Taxonomy of Bayesian models Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 10 Bayesian Program Specification • Variables • Decomposition • Parametrical Forms or Recursive Question Description Preliminary Knowledge π Identification Program Program Experimental Data δ Inference Question Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 11 Logical Proposition Logical Propositions are denoted by lowercase names: a Usual logical operators: a∧b a∨b € ¬a € Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 12 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. P(a | π) ∈ [0,1] Probabilities of the conjunctions, disjunctions and negations of propositions: € P(a ∧ b | π) P(a ∨ b | π) P(¬a | π) Probability of proposition a conditioned by both the preliminary knowledge π and some other proposition b: P(a | b ∧ π) Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 13 Normalization and Product rule P(a | π) + P(¬a | π) =1 € P(a∧b | π) = P(a | π) × P(b | a∧π) = P(b | π) × P(a | b∧π) € P(a∨b | π) = P(a | π) + P(b | π) − P(a∧b | π) € Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 14 Discrete Variable • Variables are denoted by names starting with one uppercase letter: X • Definition: a discrete variable X is a set of propositions xi – Mutually exclusive: i ≠ j ⇒ x ∧ x = false € [ i j ] – Exhaustive: at least one€ is true • The cardinal€ of X is denoted: "X# € • Continuous variable: limit case when "! X$#→ ∞ Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 15 € Variable combination • Variable conjunction X ∧Y = {xi ∧ y j } • Variable disjunction € X ∨Y = {xi ∨ y j } Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 16 € Bayesian Program Specification • Variables • Decomposition • Parametrical Forms or Recursive Question Description Preliminary Knowledge π Identification Program Program Experimental Data δ Inference Question Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 17 Description The purpose of a description is to specify an effective method to compute a joint distribution on a set of variables: {X1, X 2 ,..., X n} given some preliminary knowledge π and a set of experimental data δ. € This joint distribution is denoted as: P(X1 ∧ X 2 ∧...∧ X n |δ ∧π) € Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 18 Decomposition Partition in K subsets: Li ≠ ∅ Li = Xi1 ∧ Xi2 ∧... Conjunction rule: 1 2 n € P(X ∧ X ∧...∧ X |δ ∧π) = P(L1 |δ ∧π) × P(L2 | L1 ∧δ ∧π) ×...× P(Lk | Lk−1 ∧...∧L1 ∧δ ∧π) Conditional independence hypotheses: Ri ⊂ Li P Li | Li−1 ∧...∧L1 ∧δ ∧π € ( ) = P(Li | Ri ∧δ ∧π) Decomposition: P(X1 ∧ X 2 ∧...∧ X n |δ ∧π) € = P L1 |δ ∧π × P L2 | R2 ∧δ ∧π ×...× P Lk | Rk ∧δ ∧π ( ) ( ) ( ) Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 19 € Indépendance et indépendance conditionnelle • Indépendance – P(X Y) = P(X) P(Y) – P(X | Y) = P(X) • Indépendance conditionnelle – P(X Y | Z) = P(X | Z) P(Y | Z) – P(X | Y Z) = P(X | Z) – P(Y | X Z) = P(Y | Z) Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 20 Independence vs Conditional Independence Indépendance mais pas indépendance conditionnelle : I0 indépendant de F0 : P(F0 | I0) = P(F0) I0 pas indépendant de F0, conditionnellement à S0 : P(F0 | I0 S0) ≠ P(F0 | S0) Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 21 Independence vs Conditional Independence Indépendance conditionnelle mais pas indépendance : S2 indépendant de C0, conditionnellement à O0 : P(S2 | C0 O0) = P(S2 | O0) S2 pas indépendant de C0 : P(S2 | C0) ≠ P(S2) Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 22 The importance of Conditional Independence • Nombre de probabilités (≠ nb param) – Avant hypothèses d’indépendance conditionnelle P (I I I F ... O O ) 19 63 18 0 ∧ 1 ∧ 3 ∧ 0 ∧ ∧ 2 ∧ 3 11 ≅ 2 ≅ 9.2 x 10 – Après hypothèses P (I I I F ... O O ) 0 ∧ 1 ∧ 3 ∧ 0 ∧ ∧ 2 ∧ 3 = P (I0)P (I1)P (I3) (11x11) + (113x4) + (115x4) 218 P (F0)P (F1)P (F2)P (F3) ≅ 218 = 262 144 P (C0)P (C1)P (C2)P (C3) P (S I F )P (S I F )P (S O F )P (S I F ) 0 | 0 ∧ 0 1 | 0 ∧ 1 2 | 0 ∧ 2 3 | 3 ∧ 3 P (O I I S C )P (O I I S C ) 0 | 0 ∧ 1 ∧ 0 ∧ 0 1 | 0 ∧ 1 ∧ 1 ∧ 1 P (O O O S C )P (O I O S C ) 2 | 0 ∧ 1 ∧ 2 ∧ 2 3 | 3 ∧ 2 ∧ 3 ∧ 3 Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 23 Bayesian Program Specification • Variables • Decomposition • Parametrical Forms or Recursive Question Description Preliminary Knowledge π Identification Program Program Experimental Data δ Inference Question Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 24 Parametrical Forms or Recursive Questions Parametrical form: P Li | Ri ∧δ ∧π = f Li ( ) µ(Ri,δ)( ) Recursive Question: € P(Li | Ri ∧δ ∧π) = P(Li | Ri ∧δ % ∧π %) - modular and hierarchical programs - (or coherence variables) € Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 25 « Vocabulaire probabiliste » • Démo Mathematica Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 26 Bayesian Program Specification • Variables • Decomposition • Parametrical Forms or Recursive Question Description Preliminary Knowledge π Identification Program Program Experimental Data δ Inference Question Julien Diard — LPNC-CNRS Cours EDISCE/EDMSTII - M2R Sciences Cognitives, « Cognition bayésienne » — 2015 27 Translating knowledge
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