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わんどたつざわ
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3 1/2 3

4 {X n } P ({0}) =P ({1}) =1/2 P n k=1 X k lim n n =1/2 1 4

5 5

6 FAIR-COIN GAME Players: Skeptic, Reality Protocol: K 0 := 1. FOR n =1, 2,...: Skeptic announces M n 2 R. Reality announces x n 2 {0, 1}. K n := K n 1 + M n (x n 1 2 ). Collateral Duties: Skeptic must keep K n non-negative. Reality must keep K n from tending to infinity. 6

7 Skeptic P n k=1 x k lim n n =

8 8

9 9

10 10

11 11

13 Pascal Fermat (1654) 13

14 Pascal Pascal ( ) 14

15 Pascal (1/4) 100 Peter Paul

16 Pascal (2/4) Peter 0? Paul Peter 0 Paul

17 Pascal (3/4) Peter 0 a Paul 2a Peter b (b+c)/2 Paul c 17

18 Pascal (4/4) 25 Peter Paul 0 Peter 0 50 Paul

19 von Mises, 19

20 Fermat Fermat (1607or ) 20

21 Fermat 21

22 Arnauld ( ) The Port-Royal Logic Pascal 22

23 The Port-Royal Logic G. Shafer (1978) Non-additive Probabilities in the Work of Bernoulli and Lambert I. Hacking (1975) The Emergence of Probability 23

24 => => 24

25 25

26 sup. 26

27 K 0 := 1. n =1, 2,... : M n 2 R. x n 2 { 1, 1} K n := K n 1 + M n x n. K n sup n K n = 1 27

28 2 : Skeptic, Reality : K 0 := 1. n =1, 2,... : Skeptic M n 2 R. Reality x n 2 { 1, 1} K n := K n 1 + M n x n. Skeptic K n sup n K n = 1 Skeptic Skeptic 28

29 29

30 Skeptic K n Reality sup n K n < 1 S n /n! 0 Skeptic Skeptic. 30

31 Skeptic =) E sup n K n = 1 =) sup n K n < 1 E Reality =)sup n K n < 1 E {x n } =) E quasi-borel 31

32 Skeptic S E lim n K n = 1 S E Skeptic S n /n! 0 E S E 32

33 1/2 1/2+e 1/2+e 1/2-e e 33

34 > 0 Skeptic lim sup n!1 1 n nx i=1 x i apple.. 34

35 1 n ny (1 + x i ). i=1 D nx ln(1 + x i ) nx x i 2 n X x 2 i nx x i n 2, i=1 i=1 i=1 i=1 P n i=1 x i n apple D n +. 35

36 Skeptic E E Skeptic E 1,E 2, T 1 k=1 E k 36

37 37

38 => => => => 38

39 : Forecaster, Skeptic, Reality : K 0 := 1. n =1, 2,... : Forecaster m n 2 R v n Skeptic M n 2 R V n Reality x n 2 R 0 0. K n := K n 1 + M n (x n m n )+V n ((x n m n ) 2 v n ). Skeptic K n lim n K n = 1 Skeptic. 39

40 : (, F) : Forecaster, Skeptic, Reality : K 0 := 1. n =1, 2,... : Forecaster (, F) p n Skeptic p n f n R f ndp n f n Reality x n 2 R K n := K n 1 + f n (x n ) f ndp n. 40

41 P(E) := inf{ > 0 : (9S)supK S n(w 0 w n ) 1/ for all w 1 w 2 2 E}, P(E) :=1 P(E c ).. 41

42 Kolmogorov 0-1 law {X n } E tail event {X n } E tail event P (E) =1 P (E) =0 42

43 (Takemura-Vovk-Shafer 2008) E tail event 1. P(E) = 1 certain. 2. P(E) = 0 impossible. 3. P(E) =1,P(E) = 0, fully uncertain. 43

44 fully uncertain tail event : Skeptic, Reality : K 0 := 1. n =1, 2,... : Skeptic M n 2 R. Reality x n 2 R K n := K n 1 + M n x n. E: n x n = 0, tail event P(E) =1, P(E) =0. 44

45 45

46 : Player, Casino : K 0 := 1. n =1, 2,... : Player M n 2 R. Casino x n 2 { 1, 1} K n := K n 1 + M n x n. Player K n lim n K n = 1 Player. 46

47 Casino Player 47

48 Reality Reality Skeptic 48

49 : Forecaster, Skeptic, REality : K 0 := 1. n =1, 2,... : Forecaster m n 2 R v n Skeptic M n 2 R V n Reality x n 2 R 0 0. K n := K n 1 + M n (x n m n )+V n ((x n m n ) 2 v n ). Skeptic K n lim n K n = 1 Skeptic. 49

50 Skeptic 1X n=1 v n n 2 < 1) lim n!1 1 n nx (x i m i )=0 i=1 Reality 1X n=1 v n n 2 = 1) 1 n nx (x i m i ) 0 i=1. 50

51 (Shafer-Vovk 2001) Randomized Strategy + Martin (Vovk 2013) Deterministic Strategy (M.-Takemura 2012, M. 20xx) Randomized Strategy Deterministic Strategy 51

52 coherent Skeptic Reality coherent Reality K n+1 apple K n 52

53 (M.-Takemura 2012) Skeptic E Reality E E P Skeptic S (S + P )/2 Reality P P E Reality E S 53

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2012 2012 9 7 1 2 3 von Mises, (,) algorithmic probability, Solomonoff, Hutter, MDL Vitányi 4 1900 von Mises Kolmogorov algorithmic probability, 5 6 7 Aristotle B.C. 384-322 Tyche automaton ( ) 8 Augustine