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- ゆうりゅう けいれい
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4 1 N i p i log p i i p i log p i i N i q i N i p i log q i N i p i { ( log q i ) ( log p i ) } = N i p i log p i q i > 0 KL-divergence 0 < p i < 1, 0 < q i < 1, i p i = i q i = 1 i p i log p i autreasjkloiuxdecollkfdryuhhgreuiolpjhgfreasjoijnbvcxzswert N i p i log p i p i? p i n i i N = i n i p i = n i /N i? 1 () 4
5 i (1) (1) (1) (11) i n i = 1 i n i = 0 0 log 0 = 0 0 log log log 1 = 0 autreasjkloiuxdecollkfdryuhhgreuiolpjhgfreasjoijnbvcxzswert 1 1 (0?) N N 1 5
6 ( ) MDL (Minimum Description Length principle) MDL ( ) i p i log p i autreasjkloiuxdecollkfdryuhhgreuiolpjhgfreasjoijnbvcxzswert 1 autreasjkloiuxdecollkfdryuhhgreuiolpjhgfreasjoijnbvcxzswert O(1) O(N) N i p i q i q i i q i i p i q i y y i < y y i+1 i i q i ( ) q i exp ( (y i θ) 2 ) 2σ 2 i, i = y i+1 y i MDL θ P θ q i = P (i θ) 6
7 i n i log P (i θ) + (θ ) P, θ 4? θ θ θ = θ i π i θ i log π i 4? 399 MDL 10 2 n log n π(n) 1/n π(n) =? n n log n. log n log n + log log n + log log log n... log 5 π(n) n 4 θ π(θ) (prior distribution)
8 MDL π(n) π(n) π(n) = 1 n 2 10 π(n) π(n) 1/n Zipf 1/f n 6 2 π(n) MDL n L log L /L improper prior Lindley paradox 7 MDL 6 7 8
9 5 (b) (a) (b) (c) (d) (c) (b) (c) 8 ( )(c) overfit (d) 8? 9
10 6? ( ) AIC ( ) AIC i p i q i AIC
11 log P ( θ) - (θ ) P θ θ θ σ 2 y = f θ (x) P ({y i } θ) exp ( ) 1 2σ 2 (y i f θ (x i )) 2 1 2σ 2 (y i f θ (x i )) 2 + (θ ) i i 13 2 θ 33 4 i (y i f(x i )) ( ) 2. (cross validation) 3. (MDL,BIC, ) AIC 7? m m 2 m 13 σ 2 11
12 14 m m 2 m m 256 m = χ m m 250 AIC MDL χ 2? n = 1, 2,... n 2 1 m m 1 m m m 15 12
13 ( ) 16 ( ) Complexity is complicated. Complexity is a complicated matter.? 19 III 8 16 Knuth ( ) Knuth n, m 19 13
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18 i p i log p i discommunication ( ) 18
19 11 ( ) 24? [ ] AIC MDL/BIC 26 19
20 (intention)
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