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3 3
4 von Mises, (,) algorithmic probability, Solomonoff, Hutter, MDL Vitányi 4
5 1900 von Mises Kolmogorov algorithmic probability, 5
6 6
7 7
8 Aristotle B.C Tyche automaton ( ) 8
9 Augustine ( ), (random) (chance) 17 ( ) 9
10 Thomas Aquinas ( ) (Argumenta probabilia) 10
11 Pascal Fermat (1654) 11
12 Pascal Pascal ( ) 12
13 Pascal (1/4) 100 Peter Paul
14 Pascal (2/4) Peter 0? Paul Peter 0 Paul
15 Pascal (3/4) Peter 0 a Paul 2a Peter b (b+c)/2 Paul c 15
16 Pascal (4/4) 25 Peter Paul 0 Peter 0 50 Paul
17 von Mises, 17
18 Fermat Fermat (1607or ) 18
19 Fermat 19
20 Arnauld ( ) The Port-Royal Logic Pascal 20
21 The Port-Royal Logic G. Shafer (1978) Non-additive Probabilities in the Work of Bernoulli and Lambert I. Hacking (1975) The Emergence of Probability 21
22 Probability Chance external evidence and internal evidence (Arnauld) epistemic probability and aleatory probability subjective probability and objective probability? Bernoulli 22
23 Laplace Laplace ( ) cf. 23
24 24
25
26 von Mises Kolmogorov 26
27 Hilbert Hilbert ( )
28
29 Kolmogorov Kolmogorov ( ) 29
30 (1933) 30
31 A (n ) A m m n m/n P(A) B P(A) 1 A 31
32 Popper
33 Cournot Antoine Cournot
34 de Finetti Keynes Dempster-Shafer imprecise probability 34
35 von Mises Kolmogorov
36 von Mises ( ) von Mises ( )
37 37
38 38
39 2 39
40 collective 40
41 von Mises Collective 41
42 von Mises 42
43 Neyman, Pearson Fisher Bayesian 43
44 44
45 Solomonoff Solomonoff ( ) algorithmic probability, algorithmic information theory artificial intelligence based on machine learning 45
46 Epicurus (B.C.341?-) If more than one theory is consistent with the data, keep them all. William of Ockham ( ) Keep the simplest theory consistent with the observations. 2 46
47 (Solomono 1964) M M( ) = X p: U(p) 2 p. U. 47
48 M : 2 <!! R + M( ) apple 1 M( ) M( 0) + M( 1) M M (optimal) N C C M( ) N( ) 48
49 c.e. M c.e. {µ n } c.e. M M = X n 2 n µ n M c.e. 49
50 M d( ) = 2 M( ) d d M M( ) = 2 d( ) 50
51 (Levy 1973) µ 2! M c.e. x 1:1 µ-martin-löf c n M(x 1:n ) apple c µ(x 1:n ) 51
52 x 2 2! M(x n x <n ) = M(x 1:n) M(x <n ) µ(x n x <n ) = µ(x 1:n) µ(x <n ). 52
53 Solomonoff (Solomono 1978; Hutter) µ M(x n x <n ) µ(x n x <n )! 1 (n! 1) µ 1 53
54 Hellinger h := i {0,1} M( i ) µ( i ) 2. h 2 2 i {0,1} M( i )µ( i ) N := i {0,1} M( i )µ( i ) N 1 h 2 exp h 2. 54
55 d :2 R + d( )= M( ) µ( ) exp 1 i=1 h 1:i. M M( )/µ( ) >c>0 55
56 i {0,1} µ( i)d( i) = i {0,1} M( i)µ( i) exp i=1 h i = i {0,1} M( i )µ( i ) exp (h ) M( )µ( ) exp 1 i=1 h 1:i M( )µ( ) exp =µ( )d( ). i=1 h 1:i d µ- 56
57 µ lim sup n d(x 1:n ) <. i=1 h x1:n <. M(x 1:n x <n ) µ(x 1:n x <n ) 0. 57
58 ML (Hutter and Muchnik 2007) µ M µ- Martin-Löf M(x n x <n ) µ(x n x <n ) 1. µ 58
59 Vovk (1987) Li and Vitányi (2nd ed. 1997), Vitányi and Li (2000) Hutter (2003) Hutter and Muchnik (2004) 59
60 For quite some time I felt that the dependence of ALP (Algorithmic Probability) on the reference machine was a serious flaw in the concept, and I tried to find some objective universal device, free from the arbitrariness of choosing a particular universal machine. When I though I finally found a device of this sort, I realized that I really didn t want it - that I had no use for it at all! Solomonoff (2009) 60
61 <=> <=> 61
62 x x x ( 1 ) y x I(x; y) = K(x) K(x y ) 62
63 1. I(x : y) I(x : x) = K(x).. 63
64 K(x, y) = K(x) + K(y x ) = K(x) + K(y hx, K(x)i). I(x : y) = K(y) K(y x ) = K(x)+K(y) K(x, y) = I(y : x). 64
65 prefix-free M Q M ( )=µ([[{ : M( )= }]) K( )= log Q( ). 65
66 { s } U s, s U( s )= s, s W = {s : s = } n s W s n, s request KC request 1 KC prefix-free M n, 66
67 prefix-free V U( )=, V ( )= Q V ( )= Q(, ). Q(, ) Q( )+O(1). c 2 K( ) c Q(, ) 1. 67
68 s W 2 s +c =2 K( ) c Q(, ) 1. M K( ) c, request prefix-free M U( )= M c, M M K( ) c,, = s s W K(, ) K( )+c, request K( ) K M ( ) K(, ) K( )+O(1). 68
69 I(x : y) x y. 69
70 (Vitányi 2006) x, y NCD(x, y) = C(yx) min{c(x), C(y)} max{c(x), C(y)}. 70
71 71
72 72
73 Downey & Hirshfeldt Algorithmic Randomness and Complexity
74 Nies Computability and Randomness
75 Li & Vitányi. An Introduction to Kolmogorov Complexity and Its Applications Calude. Information and Randomness Hutter. Universal Artificial Intelligence Shafer & Vovk. Probability and Finance; It s Only a Game! 75
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