Bayesian Programming

Transcription

1 Bayesian Programming Building Jaynes robot Pierre Bessière CNRS - INRIA - Grenoble University Laboratoire LIG - E-Motion Bayesian-Programming.org 1

2 Introducing the Robot In order to direct attention to constructive things and away from controversial irrelevancies, we shall invent an imaginary being. Its brain is to be designed by us, so that it reasons according to certain definite rules. These rules will be deduced from simple desiderata which, it appears to us, would be desirable in human brains; i.e., we think that a rational person, should he discover that he was violating one of these desiderata, would wish to revise his thinking. In principle, we are free to adopt any rules we please; that is our way of defining which robot we shall study. Comparing its reasoning with yours, if you find no resemblance you are in turn free to reject our robot and design a different one more to your liking. But if you find a very strong resemblance, and decide that you want and trust this robot to help you in your own problems of inference, then that will be an accomplishment of the theory, not a premise. Our robot is going to reason about propositions. As already indicated above, we shall denote various propositions by italicized capital letters, A, B, C, etc., and for the time being we must require that any proposition used must have, to the robot, an unambiguous meaning and must be of the simple, definite logical type that must be either true or false. That is, until otherwise stated we shall be concerned only with two valued logic, or Aristotelian logic. We do not require that the truth or falsity of such an Aristotelian proposition be ascertainable by any feasible investigation; indeed, our inability to do this is usually just the reason why we need the robot's help.... E.T. Jaynes, Probability theory: The logic of Science, Page 9 2

3 Incompleteness (1989) Formalism: Bayesian Programming (1994) Industrial Applications: ProBAYES (2003) Probability as an alternative to Logic (1991) Inference: ProBT (1995) Cognitive models: BIBA, BACS (2000) 3

4 Incompleteness (1989) Formalism: Bayesian Programming (1994) Industrial Applications: ProBAYES (2003) Probability as an alternative to Logic Inference: ProBT (1995) (1991) Cognitive models: BIBA, BACS (2000) 3

5 Incompleteness (1989) Formalism: Bayesian Programming (1994) Industrial Applications: ProBAYES (2003) Probability as an alternative to Logic Inference: ProBT (1995) (1991) Cognitive models: BIBA, BACS (2000) 3

6 Incompleteness (1989) Formalism: Bayesian Programming (1994) Industrial Applications: ProBAYES (2003) Probability as an alternative to Logic Inference: ProBT (1995) (1991) Cognitive models: BIBA, BACS (2000) 3

7 Overview Incompleteness (1989) Formalism: Bayesian Programming (1994) Industrial Applications: ProBAYES (2003) Probability as an alternative to Logic (1991) Inference: ProBT (1995) Cognitive models: BIBA, BACS (2000) 4

8 Incompleteness Beam-in-the-Bin experiment (set-up) 5

9 Incompleteness Beam-in-the-Bin experiment (Result 1) 6

10 Incompleteness Beam-in-the-Bin experiment (Result 2) 7

11 Incompleteness Beam-in-the-Bin experiment (Result 3) 8

12 Incompleteness Beam-in-the-Bin experiment (Result 3) 8

13 Incompleteness Logical paradigm Avoid Obstacle Environment 9

14 Incompleteness Logical paradigm Avoid Obstacle O1 AvoidObs(01) begin end Environment P A 9

15 Incompleteness Logical paradigm Avoid Obstacle O1 AvoidObs(01) begin end? = O1 Environment P A 9

16 Incompleteness Logical paradigm Avoid Obstacle O1 AvoidObs(01) begin end Incompletness Environment P A? = O1 9

17 Overview Incompleteness (1989) Formalism: Bayesian Programming (1994) Industrial Applications: ProBAYES (2003) Probability as an alternative to Logic (1991) Inference: ProBT (1995) Cognitive models: BIBA, BACS (2000) 10

18 Bayesian Approach Principle Environment 11

19 Bayesian Approach Principle Avoid Obstacle R ( S, M) Connaissances Préalables Environment 11

20 Bayesian Approach Principle Avoid Obstacle R ( S, M) Connaissances Préalables P(MS DC) M Données Expérimentales S Environment 11

21 Bayesian Approach Principle Avoid Obstacle R ( S, M) =P(M SDC) Connaissances Préalables P(MS DC) M Données Expérimentales S Environment 11

22 Bayesian Approach An alternative to Logic Incompleteness 12

23 Bayesian Approach An alternative to Logic Incompleteness Preliminary Knowledge + Experimental Data = Probabilistic Representation Uncertainty Learning Entropy Principles 12

24 Bayesian Approach An alternative to Logic Incompleteness Preliminary Knowledge + Experimental Data = Probabilistic Representation Learning Entropy Principles Uncertainty P( a) + P ( a) =1 Bayesian inference P( a b) = P(a) P(b a) = P(b) P(a b) Decision 12

25 Overview Incompleteness (1989) Formalism: Bayesian Programming (1994) Industrial Applications: ProBAYES (2003) Probability as an alternative to Logic (1991) Inference: ProBT (1995) Cognitive models: BIBA, BACS (2000) 13

26 Bayesian Programming 14

27 Bayesian Programming Bayesian Program Description Question 14

28 Bayesian Programming Specification Bayesian Program Description Question Identification 14

29 Bayesian Programming Specification Variables Bayesian Program Description Decomposition Parametric Forms Identification Question 14

30 Bayesian Programming Specification Variables Bayesian Program Description Decomposition Parametric Forms Identification Question 14

31 Bayesian Programming Specification Variables Bayesian Program Description Decomposition Parametric Forms Identification Question 14

32 Bayesian Programming Specification Variables Bayesian Program Description Decomposition Parametric Forms Identification Question 14

33 Bayesian Programming Specification Variables Bayesian Program Description Question Decomposition Parametric Forms Identification Learning from instances 14

34 Bayesian Programming Specification Variables Bayesian Program Description Question Decomposition Parametric Forms Identification Learning from instances 14

35 Bayesian Programming Related formalisms 15

36 Overview Incompleteness (1989) Formalism: Bayesian Programming (1994) Industrial Applications: ProBAYES (2003) Probability as an alternative to Logic (1991) Inference: ProBT (1995) Cognitive models: BIBA, BACS (2000) 16

37 ProBT Specification Variables Bayesian Program Description Question Decomposition Parametric Forms Identification Learning from instances 17

38 ProBT main () { //Variables plfloat read_time; plintegertype id_type(0,1); plfloat times[5] = {1,2,3,5,10}; plsparsetype time_type(5,times); plsymbol id("id",id_type); plsymbol time("time",time_type); Bayesian Program Description //Parametrical forms //Construction of P(id) plprobvalue id_dist[2] = {0.75,0.25}; plprobtable P_id(id,id_dist); //Construction of P(time id = john) plprobvalue t_john_dist[5] = {20,30,10,5,2}; plprobtable P_t_john(time,t_john_dist); Bayesian-Programming.org //Construction of P(time id = bill) plprobvalue t_bill_dist[5] = {2,6,10,40,20}; plprobtable P_t_bill(time,t_bill_dist); //Construction de P(time id) plkerneltable Pt_id(time,id); plvalues t_and_id(time^id); t_and_id[id] = 0; Pt_id.push(P_t_john,t_and_id); t_and_id[id] = 1; Question Pt_id.push(P_t_bill,t_and_id); //Decomposition // P(time id) = P(id) P(time id) pljointdistribution jd(time^id,p_id*pt_id); //Question //Getting the question P(id time) plcndkernel Pid_t; jd.ask(pid_t,id,time); //Read a time from the key board cout<<"p(id,time)= "<<Pid_t<<"\n"; cout<<"time? : "; cin>>read_time; //Getting P(id time = read_time) plkernel Pid_readTime; 17

39 Question P( X 1 X 2... X n ) = P( L 1 ) P( L 2 R 2 )... P( L k R k ) 18

40 Question P( X 1 X 2... X n ) = P( L 1 ) P( L 2 R 2 )... P( L k R k ) P( Search Known) 18

41 Question P( X 1 X 2... X n ) = P( L 1 ) P( L 2 R 2 )... P( L k R k ) P( Search Known) ( ) = P Search Free Known Free 18

42 Question P( X 1 X 2... X n ) = P( L 1 ) P( L 2 R 2 )... P( L k R k ) P( Search Known) ( ) = P Search Free Known = Free Free ( ) P( Known) P Search Free Known 18

43 Question P( X 1 X 2... X n ) = P( L 1 ) P( L 2 R 2 )... P( L k R k ) P( Search Known) ( ) = P Search Free Known = = 1 Z Free Free ( ) P( Known) P Search Free Known ( ) P Search Free Known Free 18

44 Question P( X 1 X 2... X n ) = P( L 1 ) P( L 2 R 2 )... P( L k R k ) P( Search Known) ( ) = P Search Free Known = = 1 Z Free Free ( ) P( Known) P Search Free Known ( ) P Search Free Known Free = 1 Z P L1 Free ( ) P( L 2 R 2 )... P( L k R k ) 18

45 Symbolic simplification ( ) = 1 Z P Li R i P Search Known Free k i =0 ( ) 19

46 Symbolic simplification Factorization: ( ) = 1 Z P Li R i P Search Known P Search Known Free k i =0 ( ) = 1 Z P L j R j j J ( ) ( ) P L k R k Free1 k K ( ) 19

47 Symbolic simplification Factorization: Sum to 1: [Bessière et al. 2003] ( ) = 1 Z P Li R i P Search Known P Search Known Free k i =0 ( ) = 1 Z P L j R j P Search Known j J ( ) = 1 Z P L j R j ( ) ( ) P L k R k j J Free1 k K ( ) ( ) P L l R l Free2 l L ( ) 19

48 Symbolic simplification Factorization: Sum to 1: [Bessière et al. 2003] ( ) = 1 Z P Li R i P Search Known P Search Known Free k i =0 ( ) = 1 Z P L j R j P Search Known j J ( ) = 1 Z P L j R j ( ) ( ) P L k R k j J Free1 k K ( ) ( ) P L l R l Free2 l L ( ) Distributivity: Generalized distributive law [Aji & McEliece2000] Restrictions successives [Raoult & Smail2003] ( ) = 1 Z P L j R j P Search Known ( ) P L m R m j J Free3 m M ( ) P( L n R n ) Free4... P( L o R o ) n N FreeX o O 19

49 Using cache memory ( ) = 1 Z P L j R j P Search Known ( ) P L m R m j J Free3 m M ( ) P( L n R n ) Free4... P( L o R o ) n N FreeX o O P 1 Π P 3 P 4 Π P 2 Π Σ Σ Π P 5 P 6 Σ Σ Σ P 8 P 7 Π P 9 20

50 Using cache memory ( ) = 1 Z P L j R j P Search Known ( ) P L m R m j J Free3 m M ( ) P( L n R n ) Free4... P( L o R o ) n N FreeX o O P 1 Π P 3 P 4 Π P 2 Π Σ Σ Σ P 10 Π P 8 P 9 20

51 Using cache memory ( ) = 1 Z P L j R j P Search Known ( ) P L m R m j J Free3 m M ( ) P( L n R n ) Free4... P( L o R o ) n N FreeX o O P 1 P 11 Π P 2 Π Σ Σ P 10 Π P 8 P 9 20

52 Overview Incompleteness (1989) Formalism: Bayesian Programming (1994) Industrial Applications: ProBAYES (2003) Probability as an alternative to Logic (1991) Inference: ProBT (1995) Cognitive models: BIBA, BACS (2000) 21

53 BIBA & BACS Main Questions How to develop better artifacts using Bayesian reasoning? Biological plausibility of Bayesian reasoning at a macroscopic level? Biological plausibility of Bayesian reasoning at a microscopic level? 22

54 Bayesian Robots Programming PhD O. Lebeltel (1999) Program Description >< >: Specification >< >: >< >: Relevant Variables: Dir, P rox, Θ, V rot, H Decomposition: P (Dir P rox Θ V rot H π homing ) = P (Dir P rox Θ π homing ) P (H P rox π homing ) P (V rot Dir P rox Θ H π homing ) Parametric Forms: P (Dir P rox Θ π homing ) Uniform P ([H = avoidance] P rox π homing ) = Sigmoid α,β (P rox) P (V rot Dir P rox Θ [H = avoidance] π homing ) P (V rot Dir P rox π avoidance ) P (V rot Dir P rox Θ [H = phototaxy] π homing ) P (V rot Θ P rox π phototaxy ) Identification: No learning Question: P (V rot dir prox θ π homing ) Fig. 11. Homing BP [Lebeltel99] [Lebeltel04] 23

55 Bayesian Robots Programming PhD O. Lebeltel (1999) [Lebeltel99] [Lebeltel04] 23

56 Bayesian Robots Programming PhD O. Lebeltel (1999) Program Description >< >: Specification >< >: >< >: Relevant Variables: Dir, P rox, Θ, V rot, H Decomposition: P (Dir P rox Θ V rot H π homing ) = P (Dir P rox Θ π homing ) P (H P rox π homing ) P (V rot Dir P rox Θ H π homing ) Parametric Forms: P (Dir P rox Θ π homing ) Uniform P ([H = avoidance] P rox π homing ) = Sigmoid α,β (P rox) P (V rot Dir P rox Θ [H = avoidance] π homing ) P (V rot Dir P rox π avoidance ) P (V rot Dir P rox Θ [H = phototaxy] π homing ) P (V rot Θ P rox π phototaxy ) Identification: No learning Question: P (V rot dir prox θ π homing ) Fig. 11. Homing BP [Lebeltel99] [Lebeltel04] 1 - Calling Bayesian subroutines 2 - Probabilistic if-then-else 24

57 Mobile Robot Navigation PhD C. Pradalier (2004) [Pradalier04] [Pradalier05] 25

58 Mobile Robot Navigation PhD C. Pradalier (2004) [Pradalier04] [Pradalier05] 25

59 Mobile Robot Navigation PhD C. Pradalier (2004) [Pradalier04] [Pradalier05] 3 - Proscriptive programming 26

60 Mobile Robot Navigation PhD C. Pradalier (2004) [Pradalier04] [Pradalier05] 27

61 Mobile Robot Navigation PhD C. Pradalier (2004) [Pradalier04] [Pradalier05] 27

62 Training Video-Games Avatars PhD R. Le Hy (2007) Program Description >< >: Specification >< >: >< >: Relevant Variables: B t, B t 1, H t, W t, OW t, N t, NO t, W P t and HP t Decomposition: P (B t B t 1 H t W t OW t N t NO t W P t HP t ) = P (B t 1 ) P (B t B t 1 ) P (H t B t ) P (W t B t ) P (OW t B t ) P (N t B t ) P (NO t B t ) P (W P t B t ) P (HP t B t ) Parametric Forms: P (B t 1 ): uniform; All other distributions are tables Identification: None Question: P (B t b t 1 h t w t ow t n t no t wp t hp t ) Fig. 2. Sequencing Bot s behaviors by inverse programming [LeHy05] [LeHy07] 28

63 Training Video-Games Avatars PhD R. Le Hy (2007) Program Description >< >: Specification >< >: >< >: Relevant Variables: B t, B t 1, H t, W t, OW t, N t, NO t, W P t and HP t Decomposition: P (B t B t 1 H t W t OW t N t NO t W P t HP t ) = P (B t 1 ) P (B t B t 1 ) P (H t B t ) P (W t B t ) P (OW t B t ) P (N t B t ) P (NO t B t ) P (W P t B t ) P (HP t B t ) Parametric Forms: P (B t 1 ): uniform; All other distributions are tables Identification: None Question: P (B t b t 1 h t w t ow t n t no t wp t hp t ) Fig. 2. Sequencing Bot s behaviors by inverse programming [LeHy05] [LeHy07] 28

64 Training Video-Games Avatars PhD R. Le Hy (2007) Program Description >< >: Specification >< >: >< >: Relevant Variables: B t, B t 1, H t, W t, OW t, N t, NO t, W P t and HP t Decomposition: P (B t B t 1 H t W t OW t N t NO t W P t HP t ) = P (B t 1 ) P (B t B t 1 ) P (H t B t ) P (W t B t ) P (OW t B t ) P (N t B t ) P (NO t B t ) P (W P t B t ) P (HP t B t ) Parametric Forms: P (B t 1 ): uniform; All other distributions are tables Identification: None Question: P (B t b t 1 h t w t ow t n t no t wp t hp t ) Fig. 2. Sequencing Bot s behaviors by inverse programming [LeHy05] [LeHy07] 28

65 Training Video-Games Avatars PhD R. Le Hy (2007) Program Description >< >: Specification >< >: >< >: Relevant Variables: B t, B t 1, H t, W t, OW t, N t, NO t, W P t and HP t Decomposition: P (B t B t 1 H t W t OW t N t NO t W P t HP t ) = P (B t 1 ) P (B t B t 1 ) P (H t B t ) P (W t B t ) P (OW t B t ) P (N t B t ) P (NO t B t ) P (W P t B t ) P (HP t B t ) Parametric Forms: P (B t 1 ): uniform; All other distributions are tables Identification: None Question: P (B t b t 1 h t w t ow t n t no t wp t hp t ) Fig. 2. Sequencing Bot s behaviors by inverse programming [LeHy05] [LeHy07] 4 - Inverse programming 29

66 Bayesian control of robotics arms PhD R. Garcia (2003) Description de l objet Points de référence sur les caméras (VS) Calibration des caméras Localisation d un point avec la V.S. Localisation d un objet Positions apprises (mixture) Positions de référence Positions articulaire et cartésienne Capteurs d état Gestion d activités Action de la pince Position but correspondant à l activité Description du mouvement du bras [Garcia03] 30

67 Bayesian Approach to Action Selection and Attention focusing PhD C. Koike (2005) Program Relevant Variables: Si 0:t, Zi 0:t, α 0:t i, C 0:t, βi 0:t, B 0:t, λ 0:t i, M 0:t Decomposition: P (Si 0:t Zi 0:t C 0:t α 0:t i B 0:t βi 0:t M 0:t λ 0:t i π i ) = 2 P (S j i Sj 1 i M j 1 3 π i ) Q t P (Z j i Sj i Cj π i ) j=1 6 P (C j π i ) P (α i C j B j S j i 4 π i) 7 P (B j π i ) P (β i B j S j i Bj 1 π i ) 5 P (M >< j π i ) P (λ i M j S j i Bj M j 1 π i ) P (Si >< 0 Zi 0 βi 0 B 0 λ 0 i M 0 π i ). Parametric Forms: P (S j i Sj 1 i M j 1 π i ) = Dynamic Model Description >< >: Specification >: >: P (Z j i Sj i Cj π i ) = Sensor model P (C j π i ) = A priori about Attention Variables P (α i C j B j S j i π i)attention model in fusion with coherence form P (B j π i ) = A priori about Behaviour variables P (β i B j S j i Bj 1 π i ) = Behaviour model in fusion with coherence form P (M j π i ) = A priori about motor variables P (λ i M j S j i Bj M j 1 π i ) = Motor model in fusion with coherence form P (S 0 i Z 0 i λ 0 i M 0 π i ) = Initial Conditions Identification: A Priori or Learning Method Question: P (S j i z0:j 1 i m 0:j 1 c 0:j 1 α 0:j 1 i P (C j z 0:j 1 i m 0:j 1 c 0:j 1 α 0:j i P (B j z 0:j i m 0:j 1 c 0:j α 0:j i β 0:j i P (S j i z0:j i m 0:j 1 c 0:j α 0:j i β 0:j i β 0:j 1 i β 0:j 1 i λ 0:j 1 i λ 0:j 1 i P (M j z 0:j i m 0:j 1 b 0:j c 0:j α 0:j i β 0:j i λ 0:j i π i ) - Prediction of States λ 0:j 1 i π i ) - Determination of Attention π i ) - Determination of Behaviour π i ) - Estimation of States π i ) - Motor Commands λ 0:j 1 i Fig. 9. Elementary Filter with Attention Selection Bayesian Program [Koike05] 31

68 Bayesian Approach to Action Selection and Attention focusing PhD C. Koike (2005) Inclusion of intermediary state variables Bayesian time filtering Motor model addition Domains of interest Behaviour selection, synergies and strategies Reduction of perception data pre-processing [Koike05] 31

69 Bayesian Occupancy Filters for ADAS PhD C. Coué (2003) Program Relevant Variables: C :an index that identify each 2D cell of the grid A :an index that identify each possible antecedent of the cell c over all the cells in the 2D grid Z t :sensor measurement relative to the cell c V :The set of velocities for the cell c where V is discretized in n cases; V V = {v 1,..., v n } Description >< >: Specification >< >: >< >: O, O 1 : Taking values from the set O {occ, emp} indicating if the cell c is occupied or empty. O 1 represents the random variable of the state of an antecedent cell of c through the possible motion through c. Decomposition: P (C A Z O O 1 V ) = P (A)P (V A)P (C V, A)P (O 1 A)P (O O 1 )P (Z O, V, C) Parametric Forms: P (A): uniform; P (V A): conditional velocity distribution of antecedent cell; P (C V A): dirac representing reachability (see 2.3); P (O 1 A): conditional occupancy distribution of antecdent cell; P (O O 1 ): occupancy transitional matrix (see 2.3); P (Z O V C): observation model; Identification: None Question: P (O Z C) P (V Z C) [Coué03] [Coué05] Fig. 5. BOF with Velocity Inference 32

70 Bayesian Occupancy Filters for ADAS PhD C. Coué (2003) [Coué03] [Coué05] 32

71 Bayesian Occupancy Filters for ADAS PhD C. Coué (2003) [Coué03] [Coué05] 32

72 Bayesian Occupancy Filters for ADAS PhD C. Coué (2003) Program Relevant Variables: C :an index that identify each 2D cell of the grid A :an index that identify each possible antecedent of the cell c over all the cells in the 2D grid Z t :sensor measurement relative to the cell c V :The set of velocities for the cell c where V is discretized in n cases; V V = {v 1,..., v n } Description >< >: Specification >< >: >< >: O, O 1 : Taking values from the set O {occ, emp} indicating if the cell c is occupied or empty. O 1 represents the random variable of the state of an antecedent cell of c through the possible motion through c. Decomposition: P (C A Z O O 1 V ) = P (A)P (V A)P (C V, A)P (O 1 A)P (O O 1 )P (Z O, V, C) Parametric Forms: P (A): uniform; P (V A): conditional velocity distribution of antecedent cell; P (C V A): dirac representing reachability (see 2.3); P (O 1 A): conditional occupancy distribution of antecdent cell; P (O O 1 ): occupancy transitional matrix (see 2.3); P (Z O V C): observation model; Identification: None Question: P (O Z C) P (V Z C) Fig. 5. BOF with Velocity Inference [Coué03] [Coué05] 5 - Time vs Space dependencies 33

73 Early development of speech: Orofacial imitation PhD S. Serkhane (2005) Building a Talking Baby Robot 359 Program 8 >< >: Description 8 >< >: Specification 8 >< >: Relevant Variables: Lh, T b, T d, Xh, Y h, Al, F 1 and F 2 Decomposition: P (Lh T b T d Xh Y h Al F 1 F 2 ) = P (Xh) P (Y h) P (Al) P (Lh Al) P (T b Xh Y h) P (T d Xh Y h T b) P (F 1 Xh Y h Al) P (F 2 Xh Y h Al) Parametric Forms: P (Xh) Uniform P (Y h) Uniform P (Al) Uniform P (Lh Al) G(µ(Al), σ(al)) P (T b Xh Y h) G(µ(Xh, Y h), σ(xh, Y h)) P (T d Xh Y h T b) G(µ(Xh, Y h, T b), σ(xh, Y h, T b)) P (F 1 Xh Y h Al) G(µ(Xh, Y h, Al), σ(xh, Y h, Al)) P (F 2 Xh Y h Al) G(µ(Xh, Y h, Al), σ(xh, Y h, Al)) Identification: See text (Sections 4.3, 4.5 and 4.6) Question: P (Lh T b T d f 1 f 2 ) or P (Lh T b T d f 1 f 2 al) Fig. 13. Talking baby robot Bayesian model [Serkhane05a] [Serkhane05b] 34

74 Early development of speech: Orofacial imitation PhD S. Serkhane (2005) [Serkhane05a] [Serkhane05b] 34

75 Early development of speech: Orofacial imitation PhD S. Serkhane (2005) [Serkhane05a] [Serkhane05b] 34

76 Early development of speech: Orofacial imitation PhD S. Serkhane (2005) Building a Talking Baby Robot 359 Program 8 >< >: Description 8 >< >: Specification 8 >< >: Relevant Variables: Lh, T b, T d, Xh, Y h, Al, F 1 and F 2 Decomposition: P (Lh T b T d Xh Y h Al F 1 F 2 ) = P (Xh) P (Y h) P (Al) P (Lh Al) P (T b Xh Y h) P (T d Xh Y h T b) P (F 1 Xh Y h Al) P (F 2 Xh Y h Al) Parametric Forms: P (Xh) Uniform P (Y h) Uniform P (Al) Uniform P (Lh Al) G(µ(Al), σ(al)) P (T b Xh Y h) G(µ(Xh, Y h), σ(xh, Y h)) P (T d Xh Y h T b) G(µ(Xh, Y h, T b), σ(xh, Y h, T b)) P (F 1 Xh Y h Al) G(µ(Xh, Y h, Al), σ(xh, Y h, Al)) P (F 2 Xh Y h Al) G(µ(Xh, Y h, Al), σ(xh, Y h, Al)) Identification: See text (Sections 4.3, 4.5 and 4.6) Question: P (Lh T b T d f 1 f 2 ) or P (Lh T b T d f 1 f 2 al) Fig. 13. Talking baby robot Bayesian model [Serkhane05a] [Serkhane05b] 6 - Simplification using intermediary variables 35

77 Shape From Movement PhD F. Colas (2006) Courtesy of Max Plant Institute [Colas06] [Colas07] 36

78 Shape From Movement PhD F. Colas (2006) [Colas06] [Colas07] 7 - Expressing psychological priors 37

79 Perspectives Incompleteness (1989) Formalism: Bayesian Programming (1994) Industrial Applications: ProBAYES (2003) Probability as an alternative to Logic (1991) Inference: ProBT (1995) Cognitive models: BIBA, BACS (2000) Bayesian Neuron 38

80 Probabilistic inference by the biochemical mechanisms of phototransduction 39 Courtesy of A. Houillon, LPPA, Collège de France

81 Probabilistic inference by the biochemical mechanisms of phototransduction Activation of Phosphodiastrase 39 Courtesy of A. Houillon, LPPA, Collège de France

82 Probabilistic inference by the biochemical mechanisms of phototransduction Activation of Phosphodiastrase Hydrolysis of cgmp 39 Courtesy of A. Houillon, LPPA, Collège de France

83 Probabilistic inference by the biochemical mechanisms of phototransduction Activation of Phosphodiastrase Hydrolysis of cgmp Closure of Ca++ ion channels 39 Courtesy of A. Houillon, LPPA, Collège de France

84 Probabilistic inference by the biochemical mechanisms of phototransduction Activation of Phosphodiastrase Hydrolysis of cgmp Closure of Ca++ ion channels Decrease of Ca++ concentration 39 Courtesy of A. Houillon, LPPA, Collège de France

85 Probabilistic inference by the biochemical mechanisms of phototransduction Activation of Phosphodiastrase Hydrolysis of cgmp Closure of Ca++ ion channels Decrease of Ca++ concentration 39 AMPLIFICATION Hyperpolarization of the Membrane Courtesy of A. Houillon, LPPA, Collège de France

86 Probabilistic inference by the biochemical mechanisms of phototransduction Activation of Phosphodiastrase Hydrolysis of cgmp Closure of Ca++ ion channels Decrease of Ca++ concentration 39 AMPLIFICATION Hyperpolarization of the Membrane Ca++ regulates Activation of GC Courtesy of A. Houillon, LPPA, Collège de France

87 Probabilistic inference by the biochemical mechanisms of phototransduction Activation of Phosphodiastrase Hydrolysis of cgmp Closure of Ca++ ion channels Decrease of Ca++ concentration 39 AMPLIFICATION Hyperpolarization of the Membrane Ca++ regulates Activation of GC GC catalyses production of cgmp Courtesy of A. Houillon, LPPA, Collège de France

88 Probabilistic inference by the biochemical mechanisms of phototransduction Activation of Phosphodiastrase Hydrolysis of cgmp Closure of Ca++ ion channels Decrease of Ca++ concentration 39 AMPLIFICATION Hyperpolarization of the Membrane Ca++ regulates Activation of GC GC catalyses production of cgmp Courtesy of A. Houillon, LPPA, Collège de France

89 Biochemical Probability equivalence 40

90 Biochemical Probability equivalence Biochemistry 40

91 Biochemical Probability equivalence Biochemistry 40

92 Biochemical Probability equivalence Biochemistry Probability 40

93 Biochemical Probability equivalence Biochemistry Probability 40

94 Biochemical Probability equivalence 41

95 Overview Incompleteness (1989) Formalism: Bayesian Programming (1994) Industrial Applications: ProBAYES (2003) Probability as an alternative to Logic (1991) Inference: ProBT (1995) Cognitive models: BIBA, BACS (2000) 42

96 Want to know more? Bayesian-Programming.org Bayesian Programming Unfinished Draft 43