The Coin Algebra and Conditional Independence

Transcription

1 The Coin lgebra and Conditional Independence Jinfang Wang Graduate School of Science and Technology, Chiba University conditional in dependence Kyoto University, p.1/81

2 Multidisciplinary Field Statisticians:.P. Dawid (1979, 1980, 2001), S.L. Lauritzen (1996), etc. rtificial intelligence: J. Pearl and. Paz (1987), J. Pearl (1988, 2000), etc. nalytic philosophers: W. Spohn (1980, 1988), etc. nd more... Kyoto University, p.2/81

3 ! Dd 1 = f! 1; ;! d 1 g What Is a Statistical Model? Random vector:! D = (! 1 ;! 2 ; ;! d ) Joint pdf: f (! D ) = f (! 1 ) f (! 2 j! D1 ) f (! 3 j! D2 ) f (! d j! Dd 1 ) (1) where D1 =! 1! D2 = f! 1 ;! 2 g!. Kyoto University, p.3/81

4 >< >:! 2! 1 Independent Model Joint pdf: f (! D ) = f (! 1 ) f (! 2 ) f (! d ) (2) Underlying assumption: f (! i j! Di 1 ) = f (! i), i.e. 3 (! 1 ;! 2 )!. (3) 8! d (! 1 ;! 2 ; ;! d 1 ) Equivalent condition Kyoto University, p.4/81!! B (4)

5 ! d (! 1 ; ;! d 2 )j! d 1 Markov Chain Joint pdf: f (! D ) = f (! 1 ) f (! 2 j! 1 ) f (! 3 j! 2 ) f (! d j! d 1 ) (5) Markov property: f (! i j! Di 1 ) = f (! ij! i 1 ), i.e. 3 (! 1 ;! 2 )j! 2!. Kyoto University, p.5/81

6 >< >:! 2! 1 j! 0! d (! 1 ;! 2 ; ;! d 1 )j! 0 Bayesian Model Joint pdf: f (! D ) = f (! 0 ) f (! 1 j! 0 ) f (! 2 j! 0 ) f (! d j! 0 ) (6) Underlying assumption: 3 (! 1 ;! 2 )j! 0!. (7) 8 Equivalent conditions:!! B j! 0 (8) Kyoto University, p.6/81

7 1 Φ Ω ff! Ω Ωffi Ψ Ω 3 ff! Φ Ψ Ω JJ] ff! 2 ny 4 Φ Ω ff! Ω Ωffi Ψ Ω 6 ff! Φ Ψ Ω JJ] ff! 5 3n ff! Φ Ψ Ω Ω Ω Ωffi JJ] Regression model Regression model: Joint pdf: y i = fi x i + ffl i ; i = 1; ; n f (! D ) = ff (! 3i 1 )f (! 3i 2 )f (! 3n j! 3i 1 ;! 3i 2 )g i=1 Graphical representation: J J J Kyoto University, p.7/81 3n 1 ff! ff!3n 2 Ω Φ Ψ Ω Φ Ψ Φ ΨΩ Φ Ψ Ω

8 ff! 1 Φ Ψ -! 3 ff Ω Φ Ψ ff! Ω 2 Φ Ψ Ω ny -!5 ff Φ Ψ - ff! Ω 4 -!2n 1 ff R(1) Model R(1) model: Joint pdf: y t = y t 1 + ffl t ; i = 2; ; n f (! D ) = f (! 1 ) Φ f (!2(i 1) )f (! 2i 1 j! 2i 3 ;! 2(i 1) )Ψ i=1 Graphical representation: 6 6 Φ Ψ 6 Ω Ω Φ Ψ 2(n 1)! Kyoto University, p.8/81

9 ny State Space Model Random walk plus noise model: Joint pdf: t + t = ffl t y t 1; = ; n (9) t + t 1 = t f (! D ) = f (! 4t 3 )f (! 4t 2 )f (! 4t 1 j! 4t 5 ;! 4t 3 )f (! 4t j! 4t 2 ;! t=1 f (! 0 ) Kyoto University, p.9/81

10 0 ff! Φ Ψ - Ω 2 ff! Φ Ψ Ω ff! 4 ff! 3 1 ff! Φ Ψ Ω 6 ff! Φ Ψ Ω ff! 8 7 ff! Φ Ψ Ω 5 ff! Φ Ψ Ω 4n 2 ff! Φ Ψ Ω 4n ff! Ω 4n 1 ff! Ω 4n 3 ff! Φ Ψ Ω State Space Model Graphical representation:??? Φ Ψ Ω Φ Ψ - Φ Ψ Ω Φ Ψ 6 Ω Φ Ψ 6 6 ff - - Ω Φ Ψ Kyoto University, p.10/81

11 RM Model The reticular action model (Mcrdle 1980) : v = v + μ (10) where = (a ij ) is a p p matrix of coefficients Notation: i = fj j 8j; a ij 6= 0g Note that by i 62 definition, i. Join pdf: py f (! i j! i )f (! p+i ) (11) f (! D ) = i=1 Kyoto University, p.11/81

12 The Coin conditional in dependence Kyoto University, p.12/81

13 Notations Reference set: D = f1; 2; ; dg, d 1 = (! i1 ; ;! ir ) for any = fi 1 ; ; i r g ρ D.In!! particular, =! i fig! and = (! 1 ; ;! d D ). (! ), f (! B ), f (! j! B ), f (! B j! ) are well defined for f any ; B ρ D disjoint. D = fdjd ρ D g: power set of D D Ω D: exclusive direct product of D and D, that is D Ω D = f(r; L) j R; L 2 D and R L = ;g Kyoto University, p.13/81

14 The Coin Operator DEFINITION 1(COIN OPERTOR). The coin operator, denoted by, is a binary operator defined on the exclusive direct product D Ω D space to the posive real : D Ω D! R line,.for + L) 2 D Ω D, we shall write R L (reads as coin-r-l) instead (R; (R; L) of to denote the image (R; L) of by. The coin operator satisfies the following axioms. ; Normalization xiom: (12) 1 = ; R L Coditionality xiom: L R 6= ; (13) Inversion xiom: = RL = 1 (14) L L where R R ; and L ; L Note that ; = ; = 1 : and RL = R [ L. Kyoto University, p.14/81

15 tom Coins DEFINITION 2(TOM COINS). For (R; L) 2 D Ω any D, we shall R L call (reads as coin-r-l) the atom coin with raising R index and lowering index L. DEFINITION 3(RISING, LOWERING, MIXED COINS). We classify the atom coins into three types. (i) Raising coin: is called a raising coin with raising index R R. (ii) L Lowering coin: is called a lowering coin with lowering index L. R (iii) L Mixed coin: is called a mixed coin with R raising index and lowering index L. Kyoto University, p.15/81

16 1 ; 2 ; 12 ; 1; 2; 12; 1 ; Example EXMPLE 1. d = 2 Let D so They are = f1; 2g. There are 9 atom coins. where we have used the shorthand notations 1 ; f1g ; f1;2g ; 1;, etc. f1g f2g 12 ; 1 for 2 Kyoto University, p.16/81

17 L R R L = R = L L R = RL = RL L R R = R L : Kyoto University, p.17/81 Bayes Theorem In terms of pdf: f (! R j! L ) = f (! L j! R )f (! R )f 1 (! L ) In terms of coins: If R 6= ;; L 6= ; and R L = ;, then R L (15) Proof. RL R R L L

18 R R 2 1 = 2 L L 1 Definition of Coin DEFINITION 4(COIN). coin,, is a product of an arbitrary finite sequence of the atom coins. That is, there exist i ; L i ) 2 D Ω D; i = 1; ; r such that (R r R r (16) L (D) D The set of all coins will be denoted by or simply by when is clear from the context. Kyoto University, p.18/81

19 Coin Group D = f1; = R n ( ) n n.ifn ( ) n jnj n n = = THEOREM 1(COIN GROUP). The set of all coins forms a commutative (belian) group with respect to the usual multiplication of real numbers. We call the coin group. EXMPLE 2. Let 2g. Let be the set of all coins. Let be a raising coin. If is a positive integer then we denote by the product of copies of is a negative integer then we denote by the product of copies of 0, welet 1. Then can be written as R. When Φ( 1 ) n1 ( 2 ) n2 ( 12 ) n 3 j n 1 ; n 2 ; n 3 = 0; ±1; ±2; Ψ = Kyoto University, p.19/81

20 B = BC C R L The Raising-up Law (L-Law) THEOREM 2(RISING-UP LW). (i) For (R; L) 2 D Ω any D, we have 6= ;) (17) (R RL = L (ii) If ; B; C are mutually exclusive then we have B, BC = B C ( 6= ;) (18) Kyoto University, p.20/81

21 The Lowering-down law (L-Law) THEOREM 3(LOWERING-DOWN LW). (i) For any (R; L) 2 D Ω D, we have R = RL L (R 6= ;) (19) L (ii) If ; B; C are mutually exclusive then we have = BC B B, BC = B C ( 6= ;) (20) C Kyoto University, p.21/81

22 Valley Change Lemma LEMM 1(VLLEY CHNGE). Let ; B; C; D be mutually disjoint subsets of D, then BC = BD [D ] () C B = D B [D ] (21) where [D ] is an arbitrary coin. Kyoto University, p.22/81

23 1 B = = = B B B B B 1 B 3 = 2 = B 1 B 2 B 2 B 2 B 1 B 1 Expressions Expression: For any 2 there exists ( i ; B i ) 2 D Ω D; i = 1; ; r so that r r (22) B Nonuniqueness of expression: suppose that D = 1 t 2 = B 1 t B 2 t B 3. Then we can write D in many different ways such as D D Kyoto University, p.23/81

24 D ; Mutually Prime Coins DEFINITION 5(MUTULLY PRIME COINS). Two raising coins and with 6= ;; B 6= ; are said to be mutually prime,if B 6= B. EXMPLE 3. D Let Then = t B t C and none of ; B; C is empty. are mutually prime coins. ; B ; C ; C Kyoto University, p.24/81

25 Prime Coin and the Null Model THEOREM 4. Let 1 6= 2 be an arbitrary coin. Then there exists nonzero integers n 1 ; ; n r and mutually prime coins i ; i = 1; ; r such that the following holds = ( 1 ) n 1 ( r ) n r (23) DEFINITION 6(PRIME COIN). raising coin prime coin if there does not exist an expression is called a = ( 1 ) n 1 ( r ) n r so that each ; i = 1; ; r i is a proper subset of. DEFINITION 7(NULL MODEL). coin = (D ) group is called a null model if every raising coin is a prime coin. Kyoto University, p.25/81

26 Canonical Expression THEOREM 5. Every coin 2 has a unique expression = ( 1 ) n 1 ( r ) n r (24) where n 1 ; ; n r are nonzero integers and (i) 1 ; r are prime coins; and ; (ii) 1 r ; are mutually prime. ; DEFINITION 8(CNONICL EXPRESSION). The unique expression of a coin given above is called a canonical expression of. nd we call (i) the order of, j j = and write r; r and (ii) [ [ r 1 the index set, or simply the index,of, = ) = and I( write. Kyoto University, p.26/81

27 Property of Index THEOREM 6. The index I( ) has the following properties. (i) I( ) = I( 1 ); (ii) sub-additivity: I( ^) ρ I( ) [ I(^). Kyoto University, p.27/81

28 3 2 1 = Example Let D = 1 t 2 t t w, where i 6= ;; i = 1; ; w. (i) By applying the R-Law sequentially to the right hand side of w 1 w 1 we can see that = D.Soj j = 1 and I( ) = D. (ii) = 1 Similarly, if 2 = 1 1 then 2 j = 1 and,soj 1 t 2. = ) I( Kyoto University, p.28/81

29 = = 1 2 ( 2 ) = 2 ) j j = Example, ctd. (iii) The = 1 mixed coin 2, by the C-xiom, can be expressed as where 1 2 and 2 are mutually prime. So j j = 2 and I( ) = 1 t 2. = have (iv) Let, then by the Bayes theorem we (v) In general, while raising and lowering coins have order 1, mixed coins have order 2. Kyoto University, p.29/81

30 Coin Integration {, B} d Kyoto University, p.30/81

31 Notation NOTTION 1. Let be a subset of D. [] (i) denotes the raising coin in with raising index. (ii) denotes an arbitrary coin restricted to, the coin group with respect to. (iii) fg denotes an arbitrary coin in with index I( fg) equal to. Note that I( ) = I( fg) = and I( []) ρ. Kyoto University, p.31/81

32 1 ; ( 1 ) 2 ; 2; 2 = f1g 1 ; 12 2 Example Let D = f1; 2; 3g ; = f1; 2g. (i) = 12, where 12 = f1;2g. (ii) is the collection of []. Examples of [] include where 1 = f1g ; 1; proper subset of appears in 1; 1 f2g, etc. Note that only a 1 ; 2 Kyoto University, p.32/81

33 1 2 ; 12 2 Example, ctd. (iii) The following coins ; 12 ; 1 are examples of fg. ll these coins have as their index set. For example, if we = let 1 2 1, then 2 = 12 ( 2 ) 1 12 ( 1 ) 1 = ( 12 ) 2 ( 1 ) 1 ( 2 ) 1 so j j = 3, and I( ) = f1; 2g =. Kyoto University, p.33/81

34 Integrand DEFINITION 9(INTEGRND). Let D be the power set of D. Let 2 D. We denote by () the set of all coins fbg of such that is a subset of B, that is = f fbg j ρ B 2 Dg () We shall call fbg 2 () any an integrand with respect to, or simply an -integrand. The set of () coins will be referred to as the -integrand set. Kyoto University, p.34/81

35 Definition of Integration D Let () and be the -integrand set. We define the 2 -integration, or simply integration, as a function, denoted by, from () into, R Z : ()! so that for any -integrand fbg 2 (), there is a unique coin fb n g 2 such that Z ( fbg) = fb n g (25) The following properties hold for the integration. Kyoto University, p.35/81

36 Definition of Integration, ctd. (i) If B is an -integrand then B Z = Bn (26) (ii) Let = 1 t 2. Let fbg = fb 1 g fb 2 g be an -integrand, fb 1 g be an 1 -integrand, fb 2 g be an 2 -integrand, and 1 B 2 = 2 B 1 = ;. Z Z ( fbg) = 1 t 2 ( fb 1 g fb 2 g) = 1 ( fb 1 g) Z 2 ( fb 2 g) (27) Z (iii) Finally, for any coin 2 : R ; ( ) = Kyoto University, p.36/81

37 d; = fb 1 gd 1 Z fb 2 gd 2 Notation Coin integration: Z Z d ( ) = Properties of the coin integration: Z B d = Bn Z Z ( fb 1 g fb 2 g) d 1 t 2 = Z Kyoto University, p.37/81

38 Property of Integration THEOREM 7. fbg Let be an -integrand. Suppose that C = ;. The we have Z fbg d (28) Z [C] fbg d = [C] Proof. Since C = B ; = ;, so Z Z [C] fbg d = [C] fbg d t ; Z Z = [C] d; fbg d Z = [C] fbg d Kyoto University, p.38/81

39 R L Property of Integration THEOREM 8. For any R ρ D we have Z R dr = 1 (29) THEOREM 9. For any (R; L) 2 D Ω D with R 6= ; we have Z dr = 1 (R 6= ;) (30) Kyoto University, p.39/81

40 Property of Integration THEOREM 10. If ; B; C; D are exclusive subsets of D, then we have Z B db = [] (31) [] Z BC dc = [] B (32) [] BC dc = [D] B (33) Z [D] B db = C ( 6= ;) (34) C Z Kyoto University, p.40/81

41 B = [ μ B] [ μ ] ) B = B : Law of Normalization (N-Law) THEOREM 11. (i) Let μ B = D n B and [ μ B] 2 μb, then = [ μ B] ) B = ( 6= ;) (35) B (ii) Let 6= ;, and ; B; C be mutually exclusive, then = [ μ C] ) BC = B ( 6= ;) (36) BC (iii) If ; B; C are mutually exclusive then BC [ μ B] [ μ ] ) B C = C = B (37) C In particular, Kyoto University, p.41/81

42 B = C C B ) ab C = a C C b C j C Law of Marginalization (M-Law) THEOREM 12. If ; B; C are exclusive subsets of D. Then we have In particular, we have ; 8a ρ ; 8b ρ B (38) B = B ) ab = a b ; 8a ρ ; 8b ρ B (39) B = C C B ) ij C = i C C ; 8i 2 ; 8j 2 B (40) THEOREM 13. ; B; C If are mutually exclusive, [ and C] denote a coin independent of C, then μ BC = [ μ C] BC ) B = [ μ C] B (41) Kyoto University, p.42/81

43 Independence B / B B B / Kyoto University, p.43/81

44 Marginal Coin Group R DEFINITION 10 (MRGINL COINS). L We call a marginal atom coin R ρ ; L ρ of if. coin is said a marginal coin of if is the product of some finite sequence of marginal atom coins of. The set of all marginal coins of is denoted by. THEOREM 14 (MRGINL COIN GROUP). ρ D Let. Then is a subgroup of. We shall refer to as the marginal group of. Kyoto University, p.44/81

45 ^, ^ 1 2 ο ^, = []^ ο Equivalent Coins DEFINITION 11 (EQUIVLENT COINS). If coin group of, then is the marginal, ^ f []^ j [] 2 The set of all coins equivalent to with respect to is called, the coset of with respect to, which is given by g. The coset is also called the orbit of caused by group. Kyoto University, p.45/81

46 Independenc DEFINITION 12 (INDEPENDENCE). Let B = ; and 6= ;; B 6= ;. We say that is independent of B if and only if B ο B,or B B ο. That is, using Dawid s notation B, B ο B (42), B B ο (43) Kyoto University, p.46/81

47 Geometry of Independence B / B B B / Figure 1: Independence as coin equivalence Kyoto University, p.47/81

48 Independence DEFINITION 13 (INDEPENDENCE). Let B = ; and 6= B ;; 6= B B = ;. We say that is independent of if and only if, that is B, B = B (44) Kyoto University, p.48/81

49 Properties of independence THEOREM 15. Let R and L be exclusive nonempty subsets of D. The following equations are equivalent to one another. RL = R L (45) R (46) R = L L (47) L = R Kyoto University, p.49/81

50 R L, R L Properties of independence THEOREM 16. Let R and L be exclusive nonempty subsets of D. Then = [R] (48) = [L] (49) where [R] 2 R and [L] 2 L denote some coins depending only on R and L respectively., L R Kyoto University, p.50/81

51 Marginalization (M-Law) THEOREM 17 (MRGINLIZTION). If B = ;, then B ) 1 B 1 (8 1 ρ ; 8B 1 ρ B) (50) In particular, B ) a b (8a 2 ; 8b 2 B) (51) Kyoto University, p.51/81

52 Conditional Independence B / B B B / Kyoto University, p.52/81

53 B j C, B C B C Conditional independence ; B; C D ; B C C DEFINITION 14 (CONDITIONL INDEPENDENCE). Let be mutually disjoint subsets of. Let be nonempty. Then is said conditionally independent of given if and only if is equivalent to with respect to C, that is B C ο B C (52) THEOREM 18. Let ; B; C be mutually disjoint nonempty subsets of D. Let ; B be nonempty. Then B j C, ο (53) BC C Kyoto University, p.53/81

54 B, B j Independence as CI ; B D B B DEFINITION 15 (INDEPENDENCE). Let be mutually disjoint nonempty subsets of. Then is said to be independent of, written as, if and only if is conditionally independent of given 1. That is, ; (54) Kyoto University, p.54/81

55 Properties THEOREM 19. Let ; B; C be disjoint subsets of D. Then the following coin equations are equivalent to one another. B = C C B (55) C BC C B C = (56) BC BC C = (57) = C (58) BC B = B C (59) C Kyoto University, p.55/81

56 12 = = = = = Example If D = f1; 2; 3g, then 1 2 j 3 holds if and only if Kyoto University, p.56/81

57 Example, ctd. Using usual notations, f (! 1 ;! 2 j! 3 ) = f (! 1 j! 3 )f (! 2 j! 3 ) f (! 1 ;! 2 ;! 3 ) = f (! 1 ;! 3 )f (! 2 j! 3 ) f (! 1 ;! 2 ;! 3 ) = f (! 2 ;! 3 )f (! 1 j! 3 ) f (! 1 j! 2 ;! 3 ) = f (! 1 j! 3 ) f (! 2 j! 1 ;! 3 ) = f (! 2 j! 3 ) Kyoto University, p.57/81

58 B j Factorization Theorem THEOREM 20. Let 6= ;; B 6= ;; C be disjoint subsets of D, then B j C, B = [ μ B] [ μ ] (60) C C, BC = [ μ B] [ μ ] (61) where [ μ B] 2 μ B ; [ μ ] 2 μ. Kyoto University, p.58/81

59 Marginalization THEOREM 21. Let 6= ;; B 6= ;; C be disjoint subset of D, then C B j C ) 1 B 1 j (8 1 ρ ; 8B 1 ρ B) (62) In particular, C B j C ) a b j (8a 2 ; 8b 2 B) (63) Kyoto University, p.59/81

60 B j C 6() B j Simpson s Paradox If ; B; C; D are disjoint, then CD (64) or, in terms of random vectors!! B j! C 6()!! B j(! C ;! D ) (65) Note that as a special case we have C where 6, means 6( and 6). B 6, B j or!! B 6,!! B j! C Kyoto University, p.60/81

61 B j C j,! (! B ;! C )j! D Intersection Theorem THEOREM 22. If ; B; C; D are exclusive, then we have ) CD D (66), BC j BD or, in terms of random variables, )! B j (! C ;! D )!! C j (! B ;! D )! Kyoto University, p.61/81

62 C j,! (! B ;! C ) Intersection Theorem, ctd. COROLLRY 1. If ; B; C are mutually exclusive and nonempty, then we have ) B j C, BC (67) B or, in terms of random vectors, )! B j! C!! C j! B! Kyoto University, p.62/81

63 j B j C Contraction Theorem THEOREM 23 (CONTRCTION). If ; B; C; D are mutually exclusive and nonempty, then we have ) CD D (68), BC j D In terms of random vectors, the contraction theorem can be expressed as )! B j (! C ;! D )!! C j! D! ()! (! B ;! C ) j! D (69) Kyoto University, p.63/81

64 Contraction Theorem, ctd. COROLLRY 2. If ; B; C are mutually exclusive and nonempty, then we have ) B j C, BC (70) C or, in terms of random vectors, )! B j! C!! C! ()! (! B ;! C ) (71) Kyoto University, p.64/81

65 BC j BCD = BC j C j D ) B j D, BCD = D CD BCD, B j Weak Union Theorem THEOREM 24 (WEK UNION). If ; B; C; D are mutually exclusive, then CD (72) Proof. The M-Law implies that D, CD = D (73) which, when putted into BCD (74) gives CD Kyoto University, p.65/81

66 C j C j Graphoid xioms Pearl and Paz (1987), Geiger, et al. (1990), Pear (2000, p.11) Symmetry: j C B B j =) Decomposition: j D BC B j =) Weak Union: BC j D =) B j C D CD Contraction: B j CD D ) =) BC j D Intersection: B j CD D ), BC j BD Kyoto University, p.66/81

67 D j Mixing Rule The following property, due to Dawid (1979), is known as the mixing rule. THEOREM 25 (MIXING). If ; B; C; D are mutually exclusive, then ( BD j C C (75) B =) D B j C Kyoto University, p.67/81

68 ( BD j D j D j Strong Mixing THEOREM 26 (STRONG-MIXING). If ; B; C; D are mutually exclusive, then C D B j C (76) C ( C () B Kyoto University, p.68/81

69 ( B j D j Chaining Rule The following property is known as the chaining rule (Lauritzen, 1982). THEOREM 27 (CHINING RULE). If ; B; C; D are mutually exclusive, then C C (77) C B =) D j Kyoto University, p.69/81

70 C j j C j D 9 >= >; Seperation Theorem THEOREM 28. If ; B; C; D; S are mutually exclusive, then BDS D j BCS S (78), B CD j B DS CS B B S C D Kyoto University, p.70/81

71 C j j C j D 9 >= >; 9 >= Seperation Theorem, ctd. COROLLRY 3. If ; B; C; D are exclusive, then we have BD D j BC, B CD (79) B D B C >; Using the symbols of random variables we can write! C j (! B ;! D )!! D j (! B ;! C )!, (! ;! B ) (! C ;! D ) (80) B! C j (! ;! D )! B! D j (! ;! C )! Kyoto University, p.71/81

72 C j S ; D j C j j C j D F j S ; B C j S ; B D j Properties COROLLRY 4. If ; B; C; D; S are mutually exclusive and BDS D j BCS (81) B DS B CS Then for any subsets E ρ [ B; F ρ C [ D, we have S (82) E In particular, we have S Kyoto University, p.72/81

73 1 n = 1 B C, C j j C j B Mutual Independence DEFINITION 16 (MUTUL INDEPENDENCE). ; ; Let 1 n be mutually exclusive nonempty D subsets of. Then coins 1 ; n are said mutually independent of each other, ; written as 1 n,if n (83) THEOREM 29. If ; B; C are mutually exclusive nonempty subsetd of D, then B B (84) C Kyoto University, p.73/81

74 12 d = 1 2 Independent Model DEFINITION 17 (INDEPENDENT MODEL). D = f1; 2; ; Let dg. The coin (D ) group is called an independent model if d (85) The following theorem gives an important characterization of the independent model. THEOREM 30 (CHRCTERIZTION OF INDEPENDENT MODEL). coin group is an independent model if and only if there exists no prime coin. Kyoto University, p.74/81

75 Saparoid The coin algebra satisfies the axioms of a strong saparoid of Dawid (2001) Kyoto University, p.75/81

76 Meet-semilattice DEFINITION 18. The pair (L; ^), where L is a set and ^ is a binary operation on L, is called a meet-semilattice if the following identities hold for all x; y; z 2 L: associativity: commutativity: idempotency: x ^ (y ^ z) = (x ^ y) ^ z x ^ y = y ^ x x ^ x = x Kyoto University, p.76/81

77 Homomorphisms of Semilattices DEFINITION 19. Let (L 1 ; _) and (L 2 ; _ 0 ) be two join-semilattices. homomorphism between (L 1 ; _) and (L 2 ; _ 0 ) is a function f : L 1! L 2 so that (x _ y) = f (x) _ 0 f (y) f hold for x; y 2 L all 1. Homomorphisms between join-semilattices can be defined similarly. Kyoto University, p.77/81

78 Lattice DEFINITION 20. n algebraic structure (L; _; ^), where L is a (possibly infinite) set and _ and ^ are two binary operations, is a lattice if the following identities hold for all elements x; y and z in L: Commutativity (L1): x _ y = y _ x ; x ^ y = y ^ x ssociativity (L2): x _ (y _ z) = (x _ y) _ z ; x ^ (y ^ z) = (x ^ y) bsorption (L3): x _ (x ^ y) = x ; x ^ (x _ y) = x Kyoto University, p.78/81

79 The Separoids DEFINITION 21. (S;») Let be a join-semilattice. Let ternary relation on S. (S;»; ) Then is a separoid if j be a P1: x y j x P2: x y j z =) y x j z P3: x y j z & w» y =) x w j z P4: x y j z & w» y =) x y j (z _ w) P5: x y j z & x w j (y _ z) =) x (y _ w) j z Kyoto University, p.79/81

80 Strong Separoid DEFINITION 22. (S;»; ) separoid is said to be a strong separoid (S;») if is a lattice and the following additional property holds P6: If z» y & w» y then x y j z & x y j w =) x y j (z ^ w) Kyoto University, p.80/81

81 Coin lgebra Implies Separoid THEOREM 31. (D;») Let be the Boolean lattice. Define the (partial) ternary x yjz relation D in if the coin equation xy = x z z separoid, that is, P1-P6 hold. y holds, where x ^ y = ;. Then (D;»; z ) is a strong Kyoto University, p.81/81