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6 g(x) ḧ(ˆx) G(x) Ḧ(ˆx) ḡ(x) g(x)

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17 max E [φ(ϱ(x θ))], θ {1,2} x x G(x) θ {1, 2} ϱ(x θ) φ(ϱ)

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19 g(x), h(x^) x, x^ g( ) h( ) g( ) x h( ) ˆx x supp(g) E(g(x)) := g(x i ) log g(x i ), i=1

20 supp(g) g( ) 1 := log 2 e 1.44 g( ) h( ) log( ) log 2 ( ) ln( ) log e ( )

21 h(ˆx) {0.25, 0.5, 0.25} {00, 1, 01} ˆx g(x) {0.125, 0.125, 0.25, 0.25, 0.125, 0.125} {000, 001, 10, 11, 010, 011} g( ) h( )

22 g( ) h( ) h( ) g( ) g( ) h( )

23 i := i + 1 j := j + 1 codebooks ˆn d i, j := 1 S 0 := 0 x i I A (g; h) {ˆx j } j 1 1 ˆx j j? = ˆn d j := j + 1 {ˆx j }ˆn d 1 j := 1 ˆn d M A (h) j? = ˆn d S j := S j ˆn d ˆx j I A (h; h) ˆx j E h [x] := Sˆnd P

24 κ κ I(g(x); h(ˆx)) = E(g(x)) E(f(x ˆx)) = E(h(ˆx)) E(f(ˆx x)) = = E(g(x)) + E(h(ˆx)) E(f(x, ˆx)) h(ˆx) g(x) f(x, ˆx) x g(x) ˆx h(ˆx) x x ˆx x ˆx

25 ˆx x ˆx ˆx ˆx x x ˆx x ˆx h(ˆx) x ˆx g(x) g(x) h(ˆx) f(x, ˆx) ˆx x x g(x) ˆx h(ˆx) P I min f(x,ˆx) Ef [d(x, ˆx)] = d(x, ˆx) f(x, ˆx) dx dˆx P supp(h) supp(g) I

26 I(g(x); h(ˆx)) κ, λ f(x, ˆx) dˆx = g(x) supp(h) x supp(g), µ(x) f(x, ˆx) 0 x supp(g), ˆx supp(h). ν(x, ˆx) h(ˆx) g(x) κ > 0 d(x, ˆx) P I x G(x) g(x) d(x, ˆx) x ˆx ˆx : d(x, ˆx) g(x) dx <. supp(g) P I

27 Ç 1 f(x ˆx) = exp λ ν(x, ˆx) 1 λ µ(x) 1 å d(x, ˆx), ˆx supp(h). λ x ˆx x ˆx g(x) h(ˆx) f(x, ˆx) := f(x ˆx)h(ˆx) f(x ˆx) ˆx δ(x ˆx) x supp(g), ˆx supp(h) supp(h) := supp(g) ˆx = x, x supp(g) h(x) = g(x), x supp(g) P I E h [ˆx]+ ˇµ = E g [x] ˇµ G G domain(g)

28 x ˆx E h [ˆx] P I 1

29 K t P t q t D t+1 (t + 1) K D t+1 D t g D (D t+1 D t ) t P 0,t q 0,t D 0,t+1 (t + 1) D 0,t+1 t C t t u(c t ) β Ûq K

30 Ûq 0 = 0 P q max {u(c s) + βe g t [u(c t+1)]} = {u(c s ) + β u(c C t,{q 0,t,q t } R K t+1 ) g D (D t+1 D t ) dd t+1 } P q + C t + P 0,t q 0,t + P t q t = q 0,t 1 + (P t + D t ) q t 1, C t+1 = q 0,t + D t+1 q t, C t, {q 0,t, q t } R + R K+1 {q 0,t 1, q t 1 } D t u(c t ) = Ct 1 γ /(1 γ) g D (D t+1 D t ). P q E g t [ φ (x θ) ] := u(w t {P 0,t, P t } θ) + βe g t [u([1 x ]θ)],

31 W t {P 0,t, P t } x := D t+1 θ := {q 0,t, q t } t u( ) φ (θ, x) g(x) P q g( ) E h t [φ(ˆx θ)] := u(w t {P 0,t, P t } θ) + βe h t [u([1 ˆx ]θ)], h(ˆx) P I d(x, ˆx) κ. t h(ˆx) g(x) κ d(x, ˆx) φ (x θ) φ(ˆx θ)

32 P q P I h( ) P qi max {u(c s)+βe h t [u(c t+1 )]} = {u(c s )+β u(c C t,{q 0,t,q t } R K t+1 ) h D ( ˆD t+1 ˆD t ) d ˆD t+1 } P qi + C t + P 0,t q 0,t + P t q t = q 0,t 1 + (P t + ˆD t ) q t 1, C t+1 = q 0,t + ˆD t+1 q t, C t, {q 0,t, q t } R + R K+1 {q 0,t 1, q t 1 } ˆD t P I t P q

33 u(c t ) = Ct 1 γ /(1 γ) h D ( ˆD t+1 ˆD t ) P I d(d t+1, ˆD t+1 ) κ, g D (D t+1 D t ). t D t+1 (t + 1) K D s g D (D s+1 D s ) = g D (D t+1 D t ) s > t D 0,t+1 t P QI v({q 0,t 1, q t 1 }, ˆD t ) = { max u(ct ) + βe h t C t,{q 0,t,q t } î v({q0,t, q t }, ˆD t+1 ) ó } P QI

34 C t + P 0,t q 0,t + P t q t = q 0,t 1 + (P t + ˆD t ) q t 1, C t, {q 0,t, q t } R + R K+1 P I h D ( ˆD t+1 ˆD t ) := f(d t+1, ˆD t+1 D t, ˆD t ) dd t+1, supp(g D ) f D (D t+1, ˆD t+1 D t, ˆD [ t ) := arg min f(, ) Ef d(v ({q 0,t, q t }, D t+1 ), v({q 0,t, q t }, ˆD t+1 )) ] I(g D (D t+1 D t ); h D ( ˆD t+1 ˆD t )) κ, g D (D t+1 D t ). v ({q 0,t 1, q t 1 }, D t ) v({q 0,t 1, q t 1 }, ˆD t ) φ (x θ) := v (θ, x) φ(ˆx θ) := v(θ, ˆx) θ φ(ˆx θ) φ (x θ) v(θ, ˆx) v (θ, x) h D ( ) P I P I d(, ) x g( )

35 P QI W t+1 R 0,t+1 R t+1 y 0,t y t W t+1 := q 0,t + (P t+1 + D t+1 ) q t =: =: P 0,t R 0,t+1 q 0,t + (diag(p t ) R t+1 ) q t =: =: R 0,t+1 y 0,t + R t+1 y t; Ŵt+1 ˆR t+1 g R (R t+1 R t ) := g R (R t+1 ) g R (R t+1 ) log N (µ r, Σ r ) r t+1 := ln R t+1 r 0,t+1 := ln R 0,t+1 g r (r t+1 ) N (µ r, Σ r ) g D (D t+1 D t ) v ({q 0,t, q t }, D t+1 ) = A (W t+1 ) 1 γ, A := (1 β) γ /(1 γ) v({q 0,t, q t }, ˆD t+1 )

36 L 2 d(r t+1, ˆr t+1 ) := v ({q 0,t, q t }, D t+1 ) v({q 0,t, q t }, ˆD t+1 ) 2 2 d(r t+1, ˆr t+1 ) := 1 (1 γ) 2 ln v ({q 0,t, q t }, D t+1 ) ln v({q 0,t, q t }, ˆD t+1 ) 2 2 = = (ln W t+1 ln Ŵt+1) 2. L 2 P QI d(r t+1, ˆr t+1 ) = (ln W t+1 ln Ŵt+1) 2 Ä ω t (r t+1 ˆr t+1 + ˇµ r ) ä 2, ω t := diag(p t )q t / W t K ˆµ r ˆr t+1 ˆµ r := µ r + ˇµ r, v( ) v ( )

37 ˇµ r := 1 2 diag 1 (Σ r ˆΣ r ) 1 2 (Σ r ˆΣ r )ω t, ˆΣ r ˆr t+1 ω t Ä ω t (r t+1 ˆr t+1 + ˇµ r ) ä 2 (rt+1 ˆr t+1 + ˇµ r (Ûω t )) (r t+1 ˆr t+1 + ˇµ r (Ûω t )) =: =: d(r t+1, ˆr t+1 ), ˇµ r (Ûω t ) := 1 2 diag 1 (Σ r ˆΣ r )(1 Ûω t ) = = 1 2 diag(σ2 r,1 ˆσ2 r,1,, σ2 r,k ˆσ2 r,k )1(1 Ûω t) Ûω t Ûω t := 1 ω t. diag 1 ( ) diag( )

38 Σ r ˆΣ r ln Ŵt+1 ln W t+1 ˆµ r ˇµ r ˆr t+1 C t, {q 0,t, q t } ˆµ r C t, {q 0,t, q t } (K + 1) C t, {q 0,t, q t } {q 0,t 1, q t 1 }, ˆD t h D ( ˆD t+1 ˆD t ) h r (ˆr t+1 ) K

39 h r (ˆr t+1 ) ω t = 1 E h t [ ˆR t+1 ] = E g t [R t+1] ˆµ r = µ r E h t [Ŵt+1/W t ] = E h t [ ˆR t+1 ] = exp(ˆµ r + 0.5ˆΣ r ) < exp(µ r + 0.5Σ r ) = E g t [R t+1] = E g t [W t+1/w t ] Σ r ˆΣ r ω t ˇµ r

40 K > 1 d(r t+1, ˆr t+1 ) ω t ω t ω t P QI x := Ξ r t+1, (3.7) ˆx := Ξ ˆr t+1, (3.8) ˇµ(Ûω t ) := Ξ ˇµ r (Ûω t ), (3.9) Ξ Σ r P {Σ, Ξ} := eigendecompose(σ r ); P ˇµ r (Ûω t ) Ûω t ˇµ r (Ûω t ) Ûω t

41 σ Σ =, Σ r = ΞΣΞ 1, Ξ = Ξ 1. 0 σk 2 P QI {P 0 (D t ), P (D t )} : R K + v({q 0,t 1, q t 1 }, D t ) : R K+1 + R K + RK+1 + R F D (D t+1, ˆD t+1 D t, ˆD t ) : R K + RK + [0, 1] C t = Ûq D t, q t = Ûq, q 0,t = Ûq 0 ; f r (r t+1, ˆr t+1 ) P I g r (r t+1 ) κ f D (D t+1, ˆD t+1 D t, ˆD t ) g D (D t+1 D t ) h D ( ˆD t+1 ˆD t ) f r (r t+1, ˆr t+1 ) g r (r t+1 ) h r (ˆr t+1 ) f r (r t+1, ˆr t+1 )

42 g r (r t+1 ) h r (ˆr t+1 ) f(x, ˆx) g(x) h(ˆx) D t+1, ˆD t+1 R K + f D (D t+1, ˆD t+1 D t, ˆD t ) = f r (r t+1, ˆr t+1 ) f r (Ξx, Ξˆx) = f(x, ˆx), g D (D t+1 D t ) = g r (r t+1 ) g r (Ξx) = g(x), h D ( ˆD t+1 ˆD t ) = h r (ˆr t+1 ) h r (Ξˆx) = h(ˆx). q({q 0,t 1, q t 1 }, ˆD t ) Ûq P QI P I Ç 1 f(x ˆx) = exp λ ν(x, ˆx) 1 λ µ(x) 1 å λ (x ˆx + ˇµ(Ûω t)) (x ˆx + ˇµ(Ûω t )), ˆx supp(h). x x ˆx

43 f(x ˆx) x N (µ, Σ) µ Σ P κ λ κ λ f(x ˆx) = (2π) K 2 λ 2 I 1 ( 2 K exp 1 Ç 2 (x ˆx + ˇµ(Ûω t)) λ 1 ) Kå 2 I (x ˆx + ˇµ(Ûω t )), ˆx R K ; x = ˆx ˇµ(Ûω t ) + ϵ, ϵ N (0, Ψ), Ψ = λ 2 I K, ˆx N (ˆµ(Ûω t ), ˆΣ), ˆΣ = Σ Ψ; λ = 2 Ä e 2κ Σ ä 1 K. σk 2 > λ 2, k {1,..., K}.

44 κ λ Ç f(x ˆx) = (2π) K 2 Ψ 2 1 exp 1 å 2 (x ˆx + ˇµ(Ûω t)) Ψ 1 (x ˆx + ˇµ(Ûω t )), ˆx supp(h); x = ˆx ˇµ(Ûω t ) + ϵ, λ / λ/2 0 0 ϵ N (0, Ψ), Ψ = 0 0 σ 2, k σk 2 ˆx N (ˆµ(Ûω t ), ˆΣ), ˆΣ = Σ Ψ; {σ 2 k }K 1 := sortdescending({σ2 k }K 1 ), k := arg min k {1,...,K} {σ2 k σ2 k > λ 2 }, λ = 2 Ä e 2κ σk σ 2 K Σ ä k. (K k ) ˆx {ˆµ k +1(Ûω t ),, ˆµ K (Ûω t )} k {1,..., K} : σk 2 λ 2.

45 variance composition λ 2 σ σ^1 ψ 1 2 σ σ^2 ψ 2 2 σ σ^3 ψ 3 2 σ 4 2 ψ 4 2 σ 5 2 ψ random variables, k K 1 [0, σ2 k ] κ λ Ψ ψ 2 k = 0, k {1,..., K} ˆΣ Σ ˆσ k 2 = σ2 k, k {1,..., K} κ λ ψ 2 k k ˆσ 2 k σ2 k ψ2 k k κ λ ψ 2 k σ 2 k ˆσ2 k ψ2 l l k ψ 2 k ˆσ l 2 σk 2 ˆσ k 2 ψ2 k

46 σk 2 ˆx ˆσ 2 k = 0, k > k. x 1 x 2 ˆx 2 g( ) h( ) g( ) h( )

47 ˆx 1 r t+1 = ˆr t+1 ˇµ r (Ûω t ) + ϵ r,t+1, r t+1 R K, Σ r = ˆΣ r + Ψ r, Σ r K K,

48 ˆr t+1 N (ˆµ r (Ûω t ), ˆΣ r ), ϵ r,t+1 N (0, Ψ r ), ˆµ r (Ûω t ) ˇµ r (Ûω t ) ˆΣ r Ψ r ˆΣ r := Ξ ˆΣΞ 1, Ψ r := ΞΨΞ 1 ˇµ r (Ûω t ) ˆµ r (Ûω t ) ˇµ r ˆµ r ˆr t+1 ˆΣ r ˆr t+1 Σ r r t+1 ˆΣ r Σ r ˆµ r ˆr t+1 Ψ r = Ψ ˇµ r ˇµ r (Ûω t ) ˇµ r = 1 2 diag 1 (Σ r ˆΣ r ) (1 ω t )

49 µ r r t+1 ˆΣ r Σ r ˆr t+1 r t ˆρ r,kl ρ r,kl, k, l {1,..., K}; ˆρ r,kl = ρ r,kl Ä Km=1 ξkm 2 ä 1/2 Ä Km=1 σ2 m ξlm 2 1/2 mä σ2 Ä Km=1 ξkm 2 σ2 m ψ1 2 ä 1/2 Ä Km=1 ξlm 2 σ2 m ψ1ä 2 1/2, k, l {1,..., K}, ψ 2 1 := Ä e 2κ Σ ä 1 K < min m {1,...,K} σ2 m, K ξkm 2 = 1, m=1 k {1,..., K}. ˆr t+1 r t ˆρ r,kl ρ r,kl, k, l {1,..., K}.

50 ˆΣ r Σ r ˆr t+1 r t+1

51 κ h( ) g( ) P QI h( ) g( ) ˆΣ r ˆr t+1 Σ r r t+1

52 R t+1 log N (µ r, Σ r ) r t+1 N (µ r, Σ r ) µ r = [0.10; 0.20] Σ r = [0.10, 0.08; 0.08, 0.16] E g t [R t+1] = [1.16; 1.32] κ = ω t = [0.5; 0.5] ˆR t+1 log N (ˆµ r, ˆΣ r ) ˆr t+1 N (ˆµ r, ˆΣ r ) ˆµ r = [0.11; 0.21] ˆΣ r = [0.06, 0.08; 0.08, 0.12] E h t [ ˆR t+1 ] = [1.15; 1.31]

53 ˇµ r ω t (Σ r ˆΣ r ) ˆµ r ˆr t+1 E h t [ ˆR t+1 ] E g t [R t+1]

54 r t+1 ˆr t+1 ˆΣ ˆx Σ x ˆΣ r Σ Ξ ˆρ r,12 = 0.90 > 0.63 = ρ r,12

55 P0,t P t κ E f t [d(r t+1, ˆr t+1 )] = Ψ r 0 κ g( ) h( )

56 g( ) ħ( ) ħ( ) g( ) E ħ t [ ˆR t+1 ] = E g t [R t+1] E ħ t [ˆr t+1] = ˆµ r µ r = E g t [r t+1] ħ( ) h( )

57 ω t ω t = 0 h( ) := ħ( ) κ (i) i {0,..., m,..., n} ˆΣ (i) r κ Σ r h r (ˆr κ (i) ) g r (r κ ) κ (i) ˆΣ (i) r ˆΣ (m) r ˆΣ r ˆΣ (m) r Σ r m κ κ (m)

58 h r (ˆr κ (0) ) δ(ˆr ˆµ (0) N (ˆµ (1) r ) h r (ˆr κ (1) ) r, ˆΣ (1) r ) h r (ˆr κ (m) ) h r (ˆr κ (n) ) g r (r κ ) κ N (ˆµ (m) (m) r, ˆΣ N (ˆµ (n) (n) r, ˆΣ N (µ r, Σ r ) r ) r ) κ

59 ħ R ( ˆR t+1 ) ħ C (Ĉt+1) ħ D ( ˆD t+1 ) P QI K = 1 ˆΣ r < Σ r E ħ [ ˆR t+1 ] = E g [R t+1 ] h r (ˆr t+1 ) κ γ π( ) E π [R 0 ] := 1 E π [M t+1 ] E π [R R 0 ] := Covπ [R t+1, M t+1 ] E π ; [M t+1 ]

60 E ħ [R 0 ]» V ħ [R 0 ] E ħ [R R 0 ]» V ħ [R] E ħ [ c]» V ħ [ c] E ħ [ d]» V ħ [ d] ħ γ E [R0 ] E [R κ» R0 ] E [ d] E [R0 ] E [R»»» R0 ] E [ d] E [R0 ] E [R»»» R0 ] E [ d] E [R0 ] E [R»»» R0 ] E [ d] E [R0 ] E [R»»» R0 ] E [ d]»» V [R0 ] V [R] V [ d] V [R0 ] V [R] V [ d] V [R0 ] V [R] V [ d] V [R0 ] V [R] V [ d] V [R0 ] V [R] V [ d] ħ g g g g g β = 0.99 ħ g

61 M t+1 := β u (C t+1 ) u (C t ) = β u (D t+1 ) u (D t ) t t + 1 ħ r ( ) g r ( ) ħ ˆR log N (ˆµ r, ˆΣ r ) ˆµ r ˆΣ r β := 0.99 γ E ħ [R R 0 ] R 0 γ ˆΣ r Σ r ˆµ r E ħ [ ˆR t+1 ] Σ r κ κ Σ r ˆµ r

62 ˆµ r ħ r ( ) h r ( ) Σ r µ r E g [R t+1 ] = E ħ [ ˆR t+1 ] g r ( ) ħ g r (r) g ħ g κ κ := 3.4 κ = 0.2 γ = 3 ħ r (ˆr) g r (r) g r (r) r ˆr

63 Density of r^, r Density of r^, r (Kahneman-Tversky) ħ (r^), g(r) (r^), g KT (r) ħ r^, r r^, r ˆr N (ˆµ r, ˆΣ r ) ˆµ r := ˆΣ r := r N (µ r, Σ r ) µ r := Σ r := g κ = 0.2 γ = 3 ˆr r gr KT (µ r, Σ r ) µ r := Σ r g KT γ SR := E [R] R0» V [R]» V [M] E [M] =: HJ,

64 »»»» κ γ E [R0 ] V [R0 ] E [R R0 ] V [R] E [ c] V [ c] E [ d] V [ d] ρ (r, c) ħ g g g g g g g g g g β = 0.99 ħ g

65 ħ β γ κ ˆΣ r E ħ [ ˆR t+1 ] = E g [R t+1 ]

66 κ γ = 3 κ = κ κ K

67 ħ κ κ κ

68 κ

69 weighting function, W(Π) cumulative distribution function, Π W t = 0 C t = 0 W t

70 C t κ Σ r κ W (Π) := Π η (Π η + (1 Π) η ) 1 /η, Π( ) W ( ) η 1 η = 1 η µ r µ r E g [R t+1 ] = E ħ [ ˆR t+1 ] gr KT (r) ħ r (ˆr)

71 g r (r) gr KT (r) g κ = 0.2 γ = 3 gr KT (r) γ = 3 Σ r ˆΣ r g KT γ = 3 κ = ħ r (ˆr) κ gr KT (r) κ κ

72 »»»» κ γ E [R0 ] V [R0 ] E [R R0 ] V [R] E [ c] V [ c] E [ d] V [ d] ρ (r, c) ħ g g KT g KT g KT g KT g KT β = 0.99 ħ g g KT

73 Σ r = ˆΣ r + Ψ r. ˆΣ r Σ r

74 κ ω t

75 κ

76

77 Σ r ˆΣ r

78 ˆΣ r E h [ ˆR t+1 ] = E g [R t+1 ] ˆσ r ˆσ r,1 = ˆσ r,2 := ˆσ r ˆρ r,12 0.5ˆσ r (1 ˆρ r,12 ) ˆσ r ˆρ r,12

79

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81

82 (c t+1 c t )

83 r t+1 t M t+1

84 P Q max {C s,{q 0,s,q s }} t E g t β s t u(c s ) = s=t R K + β s t u(c s ) g D (D s D s 1 ) dd s s=t P Q C s + P 0,s q 0,s + P sq s = q 0,s 1 + (P s + D s ) q s 1, s t, C s, {q 0,s, q s } R + R K+1, s t Ñ é Ñ é T T lim P T 0,s q 0,T diag(p s + D s ) 1 diag(p s ) q T 1 0 s=t s=t g D, lim s Eg t [βs t u (C s )(P 0,s q 0,s + P sq s )] = 0; u(c s ) = Cs 1 γ /(1 γ) g D (D s+1 D s ), s t.

85 {q 0,t 1, q t 1 } D t v ({q 0,t 1, q t 1 }, D t ) v ({q 0,t 1, q t 1 }, D t ) = { max u(ct ) + βe g [ t v ({q 0,t, q t }, D t+1 ) ]} C t,{q 0,t,q t } C t + P 0,t q 0,t + P t q t = q 0,t 1 + (P t + D t ) q t 1, C t, {q 0,t, q t } R + R K+1 g D (D t+1 D t ). P QI max {C s,{q 0,s,q s }} t E h t β s t u(c s ) = s=t R K + β s t u(c s ) h D ( ˆD s ˆD s 1 ) d ˆD s s=t P QI

86 C s + P 0,s q 0,s + P sq s = q 0,s 1 + (P s + ˆD s ) q s 1, s t, C s, {q 0,s, q s } R + R K+1, s t Ñ é Ñ é T T lim P T 0,s q 0,T diag(p s + ˆD s ) 1 diag(p s ) q T 1 0 s=t s=t h D, lim s Eh t [β s t u (C s )(P 0,s q 0,s + P sq s )] = 0; u(c s ) = Cs 1 γ /(1 γ) h D ( ˆD s+1 ˆD s ) P I d(d s, ˆD s ) κ, s t, g D (D s+1 D s ), s t. v({q 0,t 1, q t 1 }, ˆD t ) = { max u(ct ) + βe h t C t,{q 0,t,q t } î v({q0,t, q t }, ˆD t+1 ) ó } C t + P 0,t q 0,t + P t q t = q 0,t 1 + (P t + ˆD t ) q t 1, C t, {q 0,t, q t } R + R K+1

87 h D ( ˆD t+1 ˆD t ) := f(d t+1, ˆD t+1 D t, ˆD t ) dd t+1, supp(g D ) f D (D t+1, ˆD t+1 D t, ˆD [ t ) := arg min f(, ) Ef d(v ({q 0,t, q t }, D t+1 ), v({q 0,t, q t }, ˆD t+1 )) ] I(g D (D t+1 D t ); h D ( ˆD t+1 ˆD t )) κ, g D (D t+1 D t ). h D ( ) P I g D ( )

88 P I d(, ) I( ; )

89 P QI P QI h D ( ˆD t+1 ) P 0,t = E h t P t = E h t [ β u ] [ Ç å (C t+1 ) u = E h Ct+1 γ ] t β, (C t ) C t [ β u (C t+1 ) Ä P u (C t ) t+1 + ˆD ä ] [ Ç å t+1 = E h Ct+1 γ Ä t β P C t+1 + ˆD ä ] t+1. t E h ï t β Ä Ûq ä γ Ä ˆDt+1 P ( ˆDt+1 ) + ˆD ä ò t+1

90 P Q P 0,t q 0,t + (P t ) q t = βw t, C t = (1 β)w t = (q t 1 ) D t, {q0,t, q t } = {0, Ûq}, [ ] P0,t = P 0(D t ) = β(ûq D t ) γ E h 1 t (Ûq ˆDt+1 ) γ, P t = P (D t ) = β [ ] 1 β ( Ûq D t ) γ E h 1 t ˆD (Ûq ˆDt+1 ) γ t+1, v t = v({q 0,t 1, q t 1 }, D t ) = AW 1 γ t, W t = q 0,t 1 + (P t + D t ) q t 1, A = (1 β) γ. 1 γ P QI

91 ˆµ r (Ûω t ) = µ r + ˇµ r (Ûω t ) = µ r diag 1 (Σ r ˆΣ r )1(1 Ûω t ) µ r diag 1 (Σ r ˆΣ r ) 1 2 (Σ r ˆΣ r )ω t = µ r + ˇµ r = = ˆµ r ω t ˆµ r (Ûω t ) < ˆµ r Σ r ˆΣ r ˆµ r ˆµ r (Ûω t ) ˆµ r ι θ ι := {q 0,t,ι, q t,ι } ω t,ι {P 0,t, P t } W t {q 0,t,ι, q t,ι } φ(x θ ι ) ˇµ r,ι ˆµ r,ι ˇµ r,ι ˇµ r,ι ˆµ r,ι θ ι ˆµ r,ι θ ˆµ r

92 ˇµ r (Ûω t ) ˆµ r (Ûω t ) ˇµ r ˆµ r Ûω t R + f(x, ˆx) = f(x ˆx)h(ˆx) = = (2π) K 2 Ψ 1 2 e 1 2 (x ˆx+ˇµ) Ψ 1 (x ˆx+ˇµ) (2π) K 2 ˆΣ 12 e 1 2 (ˆx ˆµ) ˆΣ 1 (ˆx ˆµ) = = (2π) 2K 2 Σ ˆΣ 1 á 2 ˆΣ ˆΣ exp 1 x µ Σ ˆΣ 1 ë x µ 2 ˆx ˆµ ˆΣ ˆΣ, ˆx ˆµ Ξ r t+1 x Ξ ˆr t+1 ˆx Ξ µ r µ Ξ ˆµ r ˆµ Ξ Σ r Ξ Σ Ξ ˆΣr Ξ ˆΣ f r (r t+1, ˆr t+1 ) f(x, ˆx) = (2π) 2K 2 Σ r ˆΣ r ˆΣr ˆΣr 1 2 á exp 1 r t+1 µ r Σ r 2 ˆr t+1 ˆµ r ˆΣ r 1 ë ˆΣr r t+1 µ r = ˆΣr ˆr t+1 ˆµ r =: f r (r t+1, ˆr t+1 ).

93 f(, ) f r (, ) N Θ 1 Θ 2 f(χ, ˆχ Θ 1, Θ 2 ) = f r (χ, ˆχ Θ 1, Θ 2 ) N (Θ 1, Θ 2 ) χ, ˆχ R K. g( ) g r ( ) h( ) h r ( ) P t h r (ˆr t+1 ) = (2π) K 2 ˆΣ r 1 2 e 1 2 (ˆr( ˆD t+1 ˆD t ) ˆµ r ) ˆΣ 1 r (ˆr( ˆD t+1 ˆD t ) ˆµ r ) =: h D ( ˆD t+1 ˆD t ), ˆr( ˆD t+1 ˆD t ) = ln ˆR( ˆD t+1 ˆD t ) = ln ( diag (P (D t )) 1 Ä P ( ˆD t+1 ) + ˆD ä) t+1 := ( Ç å β 1 Ç å) := ln diag 1 β ( Ûq D t ) γ β E 1 β ( Ûq ˆDt+1 ) γ E + ˆD t+1. E := E h t [ ] 1 ˆD (Ûq ˆDt+1 ) γ t+1.

94 g r (r t+1 ) = (2π) K 2 Σ r 1 2 e 1 2 (r(d t+1 D t ) µ r ) Σ 1 r (r(d t+1 D t ) µ r ) =: g D (D t+1 D t ), r(d t+1 D t ) = ln R(D t+1 D t ) = ln ( diag (P (D t )) 1 (P (D t+1 ) + D t+1 ) ) := ( Ç å β 1 Ç å) := ln diag 1 β ( Ûq D t ) γ β E 1 β ( Ûq D t+1 ) γ E + D t+1. E r(d t+1 D t ) ˆr( ˆD t+1 ˆD t ) f r (r t+1, ˆr t+1 ) f D (D t+1, ˆD t+1 D t, ˆD t )

95

96

97

98

99

100 P I L := d(x, ˆx) f(x, ˆx) dx dˆx+ supp(h) supp(g) ñ + λ f(x, ˆx) ln f(x, ˆx) dx dˆx supp(h) supp(g) Ç å Ç f(x, ˆx) dx ln supp(h) supp(g) supp(g) }{{} h(ˆx) ô g(x) ln g(x) dx κ + supp(g) ñ ô + µ(x) f(x, ˆx) dˆx g(x) + supp(h) + ν(x, ˆx) [ f(x, ˆx)]. å f(x, ˆx) dx } {{ } h(ˆx) dˆx f(ˆx x) g(x) f(x, ˆx) 0 =: δl δf(x, ˆx) Ç å = d(x, ˆx) + λ ln f(x, ˆx) + 1 ln f(x, ˆx) dx 1 + supp(g) + µ(x) ν(x, ˆx).

101 f(x ˆx) = e 1 λ ν(x,ˆx) 1 λ µ(x) 1 λ d(x,ˆx). d(r t+1, ˆr t+1 ) = (ln W t+1 ln Ŵt+1) 2 P QI d(r t+1, ˆr t+1 ) Ä ω t (r t+1 ˆr t+1 + ˇµ r ) ä 2, ω t := diag(p t )q t / W t K ˆΣr ˆr t+1 ˆµ r ˆµ r := µ r + ˇµ r, ˇµ r ˇµ r := 1 2 diag 1 (Σ r ˆΣ r ) 1 2 (Σ r ˆΣ r )ω t.

102 t (t + 1) ln W t+1 ω t r t+1 ω t P I min f r (r,ˆr) Ef [d(r t+1, ˆr t+1 )] ï Äω min f r (r,ˆr) Ef t (r t+1 ˆr t+1 + ˇµ r ) ä 2 ò λ µ(x)

103 ν(x, ˆx) ω 1K,t max ω t E f [d(r t+1, ˆr t+1 )] max E f ï Äω ω t t (r t+1 ˆr t+1 + ˇµ r ) ä 2 ò ω t 1 ÛωN t Ûω N t R +, ω k,t 0 k {1,..., K}. ( Ûω A t, ÛωN t R + ) f r (r t+1, ˆr t+1 ) := arg min f r (r,ˆr) Ef k 1 ( Ûω t A K ) 2 ( rk,t+1 ˆr k,t+1 + ˇµ r,k (Ûω t A ) ) 2, ˇµ r,k (Ûω A t ) := 1 2 (σ2 r,k ˆσ2 r,k )(1 ÛωA t ) k {1,..., K}; ω k,t := Ûω N t, k = k, ω k,t := 0, k k, k U({1,..., K}).

104 Ûω A t Ûω N t Ûω A t ω k,t 0 k {1,..., K} Ûω t := 1 ω t = 1 diag(p t )q t / W t, d(r t+1, ˆr t+1 ) := (r t+1 ˆr t+1 + ˇµ r (Ûω t )) (r t+1 ˆr t+1 + ˇµ r (Ûω t )), ˇµ r (Ûω t ) := 1 2 diag 1 (Σ r ˆΣ r )(1 Ûω t ) = = 1 2 diag(σ2 r,1 ˆσ2 r,1,, σ2 r,k ˆσ2 r,k )1(1 Ûω t).

105 P Q P QI dd t := diag(d t )µ D dt + diag(d t )σ D db t, dp 0,t := r 0,t P 0,t dt, dp t := diag(p t )(µ P diag 1 (σ P σ P ))dt + diag(p t)σ P db t, D t µ D K σ D K K B t K P 0,t r 0,t P t P t := P (D t ) µ P = diag(p t ) 1 P D diag(d t)µ D diag(p t) 1 σ D diag(d 2 P t) D D diag(d t)σ D 1 2 diag 1 (σ P σ P ), σ P = diag(p t ) 1 P D diag(d t)σ D. d ln P t = µ P dt + σ P db t, Ç dw t = W t ω t (µ P diag 1 (σ P σ P ) + diag(p t) 1 D t r 0,t 1) + r 0,t C å t dt+ W t + W t ω t σ P db t, Ç d ln W t = ω t (µ P diag 1 (σ P σ P ) + diag(p t) 1 D t r 0,t 1) + r 0,t C t W t 1 å 2 ω t σ P σ P ω t dt + ω t σ P db t,

106 W t := P 0,t q 0,t + P t q t, ω 0,t := P 0,t q 0,t / W t, ω t := 1 W t diag(p t )q t. dt = 1 Σ r := σ P σ P, r t+1 := µ P + σ P (B t+1 B t ) =: µ r + N (0, Σ r ) ln W t+1 ln W t Ç ω t (diag(p t) 1 D t r 0,t 1) + r 0,t C t W t ω t diag 1 (Σ r ) 1 2 ω t Σ rω t å + ω t r t+1, ˆµ r ln Ŵt+1 ln W t Ç ω t (diag(p t) 1 D t r 0,t 1) + r 0,t C t W t ω t diag 1 ( ˆΣ r ) 1 2 ω t ˆΣ r ω t å + ω t ˆr t+1. d(r t+1, ˆr t+1 ) = (ln W t+1 ln Ŵt+1) 2 Ç ω t (r t+1 ˆr t+1 ) ω t diag 1 (Σ r ˆΣ r ) 1 å 2 2 ω t (Σ r ˆΣ r )ω t.

107 ln W t+1 ln Ŵt+1 Σ r ˆΣ r ˆµ r ˆµ r := µ r diag 1 (Σ r ˆΣ r ) 1 2 (Σ r ˆΣ r )ω t =: µ r + ˇµ r. E f [d(r t+1, ˆr t+1 )] E f [Ç ω t (r t+1 ˆr t+1 ) ω t diag 1 (Σ r ˆΣ r ) 1 2 ω t (Σ r ˆΣ r )ω t å 2 ], E f [d(r t+1, ˆr t+1 )] ω t Ψ rω t, ˆµr =µ r +ˇµ r Ψ r := E f [(r t+1 ˆr t+1 + ˇµ r )(r t+1 ˆr t+1 + ˇµ r ) ]. d(r t+1, ˆr t+1 ) Ç ω t (r t+1 ˆr t+1 ) ω t diag 1 (Σ r ˆΣ r ) 1 2 ω t (Σ r ˆΣ r )ω t å 2 = = Ä ω t (r t+1 ˆr t+1 + ˇµ r ) ä 2.

108 f r (r t+1, ˆr t+1 ) E f [d(r t+1, ˆr t+1 )] = E f ï Ä ω t (r t+1 ˆr t+1 + ˇµ r ) ä 2 ò = ω t Ψ rω t, Ψ r := E f [(r t+1 ˆr t+1 + ˇµ r )(r t+1 ˆr t+1 + ˇµ r ) ]. max ω t E f [d(r t+1, ˆr t+1 )] max ω ω t t Ψ rω t ω t 1 ÛωN t Ûω N t R +, ω k,t 0 k {1,..., K}. Ψ r V Ä ω t (r t+1 ˆr t+1 + ˇµ r ) ä = k ω 2 k,t ψ2 r,k + 2 k l ω k,t ω l,t ψ r,kl k ω 2 k,t ψ2 r,k + 2 k l ω k,t ω l,t ψ r,k ψ r,l ( Ûω N t ) 2 max k {1,...,K} ψ2 r,k.

109 ω k,t := Ûω N t, k = k := arg max k {1,...,K} ψ2 r,k ω k,t := 0, k k., Ψ r k U({1,..., K}). ω k,t := Ûω N t, k = k U({1,..., K}), ω k,t := 0, k k. Ûω A t R + Ûω A t Ûω N t 1 min f r (r,ˆr) K Ef [d(r t+1, ˆr t+1 ) ω k,t = Ûω t A ; ω l,t = 0, l k] k min f r (r,ˆr) Ef 1 K d(r t+1, ˆr t+1 ) ω k,t = Ûω t A ; ω l,t = 0, l k k min f r (r,ˆr) Ef min f r (r,ˆr) Ef min f r (r,ˆr) Ef min f r (r,ˆr) Ef k k k k 1 K Ä ωt (r t+1 ˆr t+1 + ˇµ r ) ä 2 ω k,t = Ûω t A ; ω l,t = 0, l k Ç 1 ω K t (r t+1 ˆr t diag 1 (Σ r ˆΣ r ) (Σ r ˆΣ r )ω t )å 1 ( Ûω A ) Ç 2 t r K k,t+1 ˆr k,t (σ2 r,k ˆσ2 r,k ) 1 å 2 2 (σ2 r,k ˆσ2 r,k )ÛωA t 1 ( Ûω t A ) 2 (rk,t+1 ˆr K k,t+1 + ˇµ r,k (Ûω t A )) 2 ω k,t = Ûω t A ; ω l,t = 0, l k

110 Ûω A t, ÛωN t R + f r (r t+1, ˆr t+1 ) := arg min f r (r,ˆr) Ef k 1 ( Ûω t A K ) 2 (rk,t+1 ˆr k,t+1 + ˇµ r,k (Ûω A t )) 2 ; ω k,t := Ûω N t, k = k, ω k,t := 0, k k, k U({1,..., K}). Ûω A t := 1 ω t =: Ûω t k 1 K Ûω2 t (r k,t+1 ˆr k,t+1 +ˇµ r,k (Ûω t )) 2 (r t+1 ˆr t+1 +ˇµ r (Ûω t )) (r t+1 ˆr t+1 +ˇµ r (Ûω t )) =: d(r t+1, ˆr t+1 ),

111 d(r t+1, ˆr t+1 ) I(g r (r t+1 ); h r (ˆr t+1 )) d(r t+1, ˆr t+1 ) = (r t+1 ˆr t+1 + ˇµ r (Ûω t )) (r t+1 ˆr t+1 + ˇµ r (Ûω t )) = = (Ξx Ξˆx + Ξˇµ(Ûω t )) (Ξx Ξˆx + Ξˇµ(Ûω t )) = = (x ˆx + ˇµ(Ûω t )) Ξ Ξ(x ˆx + ˇµ(Ûω t )) = = (x ˆx + ˇµ(Ûω t )) (x ˆx + ˇµ(Ûω t )) =: d(x, ˆx), I(g r (r t+1 ); h r (ˆr t+1 )) = E(g r (r t+1 )) + E(h r (ˆr t+1 )) E(g r (r t+1 ), h r (ˆr t+1 )) = = E(g r (Ξx)) + E(h r (Ξˆx)) E(g r (Ξx), h r (Ξˆx)) = = E(g(x)) + ln(abs Ξ ) + E(h(ˆx)) + ln(abs Ξ ) E(g(x), h(ˆx)) ln(abs( Ξ Ξ )) = = E(g(x)) + E(h(ˆx)) E(g(x), h(ˆx)) =: I(g(x); h(ˆx)). κ λ

112 x R K ˆx R K ν(x, ˆx) = 0 x, ˆx R K. e 1 λ µ(x) = (πλ) K 2 RK 1 = f(x ˆx) dx = e 1 R K λ ν(x,ˆx) λ 1 µ(x) λ 1 (x ˆx+ˇµ(Ûω t)) (x ˆx+ˇµ(Ûω t )) =: =: R K b(x)(2π) K λ 2 2 I K e 2 (x ˆx+ˇµ(Ûω t)) Ä ä λ 2 I K 1(x ˆx+ˇµ(Ûωt )) dx = = b(x)ϕ(ˆx ˇµ(Ûω t) x λ R K 2 I K) dx =: (b ϕ) (ˆx ˇµ(Ûω t )). δ(ξ) = 1 = b(ξ) ϕ(ξ λ 2 I K), b(ξ) = δ(ξ) ϕ(ξ λ 2 I K) = δ(ξ)e2πi(ˆx ˇµ(Ûω t)) ξ+π 2 λξ ξ = δ(ξ) = 1 ξ = 0, 0 ξ 0. b(x) = dξ = R K dξ = 1; R K e 1 λ µ(x) = (2π) K 2 λ 2 I 2 1 K = (πλ) K 2.

113 µ(x) = λ K 2 ln(πλ). ϵ := x ˆx + ˇµ(Ûω t ) N (0, λ 2 I K), x (ˆx ˇµ(Ûω t )) ϵ x = ˆx ˇµ(Ûω t ) + ϵ. (ˆx ˇµ(Ûω t )) N ϵ N x N (ˆx ˇµ(Ûω t )) ˆx ˆx N (ˆµ(Ûω t ), ˆΣ). Ψ := E f [(x ˆx + ˇµ(Ûω t ))(x ˆx + ˇµ(Ûω t )) ] = E ϕ [ϵϵ ] = λ 2 I K, Σ = ˆΣ + Ψ Σ ˆΣ N f(x ˆx) h(ˆx) f(x, ˆx) supp(h) f(x, ˆx) dˆx = g(x) x supp(g) µ(x)

114 κ = 1 2 (ln Σ ln Ψ ) = 1 Ç λ ln Σ ln 2 2 I å K, λ = 2 Ä e 2κ Σ ä 1 K. κ λ σ 2 k > λ 2 k {1,..., K} k {1,..., K} : σ 2 k λ 2 supp(h) f(x, ˆx) dˆx = g(x) x supp(g) µ(x) x R K ˆx R k {ˆµ k +1(Ûω t ),, ˆµ K (Ûω t )} ϵ k +1,, ϵ K µ(x) ν(x, ˆx) = 0 x R K, ˆx R k {ˆµ k +1(Ûω t ),, ˆµ K (Ûω t )}. e 1 λ µ(x) = (2π) K 2 Å( λ2 ) k σ 2 k +1 σ2 K ã 1 2 RK 1 = f(x ˆx) dx = e 1 R K λ ν(x,ˆx) λ 1 µ(x) λ 1 (x ˆx+ˇµ(Ûω t)) (x ˆx+ˇµ(Ûω t )) dx =: =: R K b(x)(2π) K 2 Ψ 1 2 e 1 2 (x ˆx+ˇµ(Ûω t)) Ψ 1 (x ˆx+ˇµ(Ûω t )) dx = = b(x)ϕ(ˆx ˇµ(Ûω t) x Ψ) dx =: (b ϕ) (ˆx ˇµ(Ûω t )). R K

115 δ(ξ) = 1 = b(ξ) ϕ(ξ Ψ), b(ξ) = δ(ξ) ϕ(ξ Ψ) = δ(ξ)e2πi(ˆx ˇµ(Ûω t)) ξ+π 2 (λ k 1 ξ2 k +2 K k +1 σk 2ξ2 k ) = 1 ξ = 0, = δ(ξ) = 0 ξ 0. b(x) = dξ = R K dξ = 1, R K e 1 λ µ(x) = (2π) K 2 Ψ 1 2 = (2π) K 2 Ñ Çλ å k 1 2 σ 2 2 k +1 Ké σ2. Ñ Ñ Çλ K µ(x) = λ 2 ln(2π) + 1 å éé k 2 ln σ 2 2 k +1 σ2 K. ϵ := x ˆx + ˇµ(Ûω t ) N (0, Ψ), x (ˆx ˇµ(Ûω t )) ϵ x = ˆx ˇµ(Ûω t ) + ϵ. (ˆx ˇµ(Ûω t )) N

116 ϵ N x N (ˆx ˇµ(Ûω t )) ˆx ˆx N (ˆµ(Ûω t ), ˆΣ). {e 1,..., e K } R K Ψ := E f [(x ˆx + ˇµ(Ûω t ))(x ˆx + ˇµ(Ûω t )) ] = E ϕ [ϵϵ λ ] = 2 I k 0 0 K k k=1 e k Σ k +1Ke k, Σ = ˆΣ + Ψ Σ ˆΣ N f(x ˆx) h(ˆx) f(x, ˆx) supp(h) f(x, ˆx) dˆx = g(x) x supp(g) µ(x) Ñ κ = 1 2 (ln Σ ln Ψ ) = 1 λ ln Σ ln 2 2 I k ln K k é e k Σ k +1Ke k, k=1 λ = 2 Ä e 2κ σk σ 2 K Σ ä k.

117 Ξ x = ˆx ˇµ(Ûω t ) + ϵ r t+1 = ˆr t+1 ˇµ r (Ûω t ) + ϵ r,t+1, ϵ r,t+1 := Ξϵ. Ξ Ξ 1 Σ = ˆΣ + Ψ Σ r = ˆΣ r + Ψ r, P ˆΣ r := Ξ ˆΣΞ 1 Ψ r := ΞΨΞ 1. ˆx N (ˆµ(Ûω t ), ˆΣ) ˆr t+1 = Ξˆx N (Ξˆµ(Ûω t ), Ξ ˆΣΞ ) ˆr t+1 ˆµ r (Ûω t ) P ˆr t+1 N (ˆµ r (Ûω t ), ˆΣ r ) ϵ N (0, Ψ) ϵ r,t+1 = Ξϵ N (Ξ0, ΞΨΞ ) P ϵ r,t+1 N (0, Ψ r ) θ ι 1 := {q 0,t,ι 1, q t,ι 1 } (ι 1) θ θ ι 1 θ ι

118 ˇµ r,ι 1 ˇµ 2 r 2 = 1 2 diag 1 (Σ r ˆΣ r,ι 1 ) 1 2 (Σ r ˆΣ r,ι 1 )ω t,ι diag 1 (Σ r ˆΣ r ) (Σ r ˆΣ 2 r )ω t = 2 1 = 2 diag 1 (Ψ r,ι 1 ) 1 2 Ψ r,ι 1ω t,ι diag 1 (Ψ r ) Ψ 2 rω t = 2 1 = 2 diag 1 (Ψ r ) 1 2 Ψ rω t,ι diag 1 (Ψ r ) Ψ 2 rω t = 2 = 1 Ä ä Ψ r ωt,ι 1 ω 2 t 2 2 = = 1 Ç 1 2 Ψ r diag(p t )q W t,ι 1 1 å diag(p t )q 2 t = t W t 2 = 1 2 Ψ 1 r diag(p t ) Ä q W t,ι 1 q ä 2 t t Ψ 1 2 r diag(p t ) q W t t,ι 1 q t 2 2, 2 q t,ι 1 ˇµ r,ι 1 ι φ( θ) ˆµ r ˇµ r φ(ξ ˆr t+1 θ ι, ˆµ r (ˇµ r,ι 1 )) φ(ξ ˆr t+1 θ, ˆµ r (ˇµ r )) 2 2 < ε φ ε φ > 0 q t,ι q t 2 2 θ ι θ 2 2 < ε θ

119 ε θ > 0 ˇµ r,ι ˇµ r Ψ 1 2 r diag(p t ) q W t t,ι q t 2 2 < Mε θ =: ε µ 2 0 < M < M q t,ι = q t,ι 1, ˇµ r,ι = 1 2 diag 1 (Σ r ˆΣ r,ι ) 1 2 (Σ r ˆΣ r,ι )ω t,ι = = 1 2 diag 1 (Ψ r ) 1 2 Ψ 1 r diag(p t )q W t,ι = t = 1 2 diag 1 (Ψ r ) 1 2 Ψ 1 r diag(p t )q W t,ι 1 = t = 1 2 diag 1 (Σ r ˆΣ r,ι 1 ) 1 2 (Σ r ˆΣ r,ι 1 )ω t,ι 1 = ˇµ r,ι 1. φ(x θ ι 1 ) φ(x θ ) 2 2 = φ(ξ ˆr t+1 θ ι, ˆµ r (ˇµ r,ι )) φ(ξ ˆr t+1 θ, ˆµ r (ˇµ r )) 2 2 = = φ(ξ ˆr t+1 θ ι, ˆµ r (ˇµ r,ι 1 )) φ(ξ ˆr t+1 θ, ˆµ r (ˇµ r )) 2 2 < ε φ. ˇµ r,ι ˆµ r,ι ˇµ r,ι ˆµ r,ι ω t,ι ˇµ r ˆµ r ω t

120 r t+1 N (µ r, Σ r ) Σ r = ΞΣΞ 1 Σ P ρ r,kl := Km=1 ξ km ξ lm σm 2 Ä Km=1 ξkm 2 ä 1/2 Ä Km=1 σ2 m ξlm 2 1/2. mä σ2 ˆr t+1 N (ˆµ r, ˆΣ r ) ˆΣ r = Ξ ˆΣΞ 1 ˆΣ Km=1 ξ ˆρ r,kl := km ξ lmˆσ m 2 Ä Km=1 ξ kmˆσ2 2 ä 1/2 Ä Km=1 m ξ lmˆσ2 2 1/2 = mä Km=1 ξ = km ξ lm (σm 2 ψm) 2 Ä Km=1 ξkm 2 (σ2 m ψm) 2 ä 1/2 Ä Km=1 ξlm 2 (σ2 m ψm) 2 ä 1/2. ψ 2 m = ψ 2 n, m, n {1,..., K} Km=1 ξ ˆρ r,kl = km ξ lm (σm 2 ψ1 2) Ä Km=1 ξkm 2 (σ2 m ψ1 2)ä 1/2 Ä Km=1 ξlm 2 (σ2 m ψ1 2)ä 1/2 = Km=1 ξ = km ξ lm σm 2 Ä Km=1 ξkm 2 σ2 m ψ1 2 ä 1/2 Ä Km=1 ξlm 2 σ2 m ψ1ä 2 1/2, Ξ K m=1 ξ km ξ lm = δ kl δ kl k = l ˆρ r,kl = ρ r,kl Ä Km=1 ξkm 2 ä 1/2 Ä Km=1 σ2 m ξlm 2 1/2 mä σ2 Ä Km=1 ξkm 2 σ2 m ψ1 2 ä 1/2 Ä Km=1 ξlm 2 σ2 m ψ1ä 2 1/2, k, l {1,..., K}, ˆρ r,kl ρ r,kl ψ1 2 ψm 2 ψn, 2 m, n {1,..., K}

121 {σ 2 k }K 1 k {1,..., K} ψk 2 := σk 2 k = K, σk 2 + ψ2 + k K, σk 2 σ2 K > ψ2 + 0 k {1,..., K 1} Km=1 ξ ˆρ r,kl = km ξ lm (σm 2 σk 2 ) K 1 m=1 ξ kmξ lm ψ+ 2 ( Km=1 ξkm 2 (σ2 m σk 2 ) ) K 1 1/2 ( Km=1 m=1 ξ2 km ψ2 + ξlm 2 (σ2 m σk 2 ) ) K 1 1/2 = m=1 ξ2 lm ψ2 + Km=1 ξ = km ξ lm σm 2 + ξ kk ξ lk ψ+ 2 Ä Km=1 ξkm 2 σ2 m σk 2 ψ2 + + ä 1/2 Ä Km=1 ξ2 kk ψ2 + ξlm 2 σ2 m σk 2 ψ /2, ξ2 lk +ä ψ2 Ξ ˆρ r,kl σ 2 K ˆρ r,kl, K := lim σ 2 K 0+ ˆρ r,kl = K 1 m=1 ξ kmξ lm σm 2 + ξ kk ξ lk ψ+ 2 ( K 1 m=1 ξ2 km σ2 m ψ+ 2 + ) 1/2 ( K 1 ξ2 kk ψ2 + m=1 ξ2 lm σ2 m ψ+ 2 + ) 1/2. ξ2 lk ψ2 + ψ+ 2 := 0 ˆρ r,kl ρ r,kl ξ kk ξ lk K 1 ˆρ r,kl, K = lim ˆρ r,kl ψ 2 σ + =0 K 2 0+ = m=1 ξ kmξ lm σm 2 ( K 1 ) 1/2 ( K 1 ) 1/2 = ψ 2 + =0 m=1 ξ2 km σ2 m m=1 ξ2 lm σ2 m = lim ρ r,kl σk 2 0+ = ρ r,kl, K. ψ 2 + =0 ψ 2 + =0 ρ r,kl, K > 0 ξ kk = ±1/ 2 ξ lk = 1/ 2 ˆρ r,kl, K = K 1 m=1 ξ kmξ lm σm 2 2 1ψ2 + ( K 1 m=1 ξ2 km σ2 m 2 1 ) 1/2 ( K 1 ψ2 + m=1 ξ2 lm σ2 m 2 1 ) 1/2. ψ2 + ˆρ r,kl, K

122 ψ 2 + ψ2 + = 0 Ñ é ˆρ r,kl, K 1 1 ψ+ 2 = ˆρ r,kl, K ψ 2 2 K 1 + =0 m=1 ξ2 km σ2 m ψ2 2 K 1 + m=1 ξ2 lm σ2 m 1 2 ψ2 + ψ 2 + =0 1 ( K 1 m=1 ξ2 km σ2 m 1 ) 1/2 ( K 1 2 ψ2 + m=1 ξ2 lm σ2 m 1 ) 1/2 = 2 ψ2 + ψ 2 + =0 Ñ é 1 1 = ρ r,kl, K 2 K m=1 ξ2 km σ2 m 2 K 1 m=1 ξ2 lm σ2 m 1 ( K 1 ) 1/2 ( K 1 ) 1/2. m=1 ξ2 km σ2 m m=1 ξ2 lm σ2 m K 1 m=1 ξ2 km σ2 m K 1 m=1 ξ2 lm σ2 m 1 ( K 1 ) 1/2 ( K 1 ) 1/2, m=1 ξ2 km σ2 m m=1 ξ2 lm σ2 m K 1 m=1 ξ2 km σ2 m = K 1 m=1 ξ2 lm σ2 m ρ r,kl, K (0, 1] ˆρ r,kl, K / ψ 2 + ψ 2 + =0 > 0 ˆρ r,kl ρ r,kl ρ r,kl, K (0, 1] ˆρ r,kl, K / ψ 2 + ψ 2 + =0 < 0 ˆρ r,kl ρ r,kl ρ r,kl, K = 0 ˆρ r,kl, K / ψ 2 ψ + 2 < 0 + =0 ˆρ r,kl ρ r,kl = 0 ρ r,kl, K < 0

123 x g(x) x g(x), g(x) g(x) = 1 /8 x = 7, 1/8 x = 9, 1 /4 x = 11, 1 /4 x = 13, 1 /8 x = 19, 1 /8 x = g (x) Probability masses of x x g(x) x n q E( g(x)) := g(x i ) log g(x i ), i=1 n q := supp( g).

124 {x i } n q 1 { g(x i )} n q 1 Ç E( g(x)) = log log 1 å = ˆx ḧ(ˆx) x Probability masses of x^ 1 /4 ˆx = 8, ḧ(ˆx) = 1/2 ˆx = 12, 1 /4 ˆx = h (x^) x^ ḧ(ˆx) ḧ( ) g( ) ḧ(ˆx) g(x) ˆn q E(ḧ(ˆx)) = ḧ(ˆx j ) log ḧ(ˆx j) = j=1 Ç = log log 1 å 2 = 1.5, ˆn q := supp(ḧ).

125 G(x) n d n d := 1 δ δ := gcd({ g(x i )} n q 1 ), gcd(a 1,..., a n ) a 1,..., a n R G(x i ) G(x i 1 ) + δ... G (x) Cumulative distribution of x x G(x) x G(x) g(x) 1/8 1/8 2/8 1/8 4 /8 1 /4 6 /8 1 /4 7 /8 1 /8 8 /8 1 /8 E[L(x)] G(x) G(x)

126 x x n q E[L(x)] = g(x i )L(x i ) E( g(x)), i=1 L(x i ) x i E( g(x)) E[L (x)] < E( g(x)) + 1 E[L (x)] = E( g(x)) = 2.5 log 1 / g(x) x A := {000000,..., } x {0,..., 63} Ḧ(ˆx) ˆn d ˆn d := 1ˆδ ˆδ := gcd({ḧ(ˆx j)}ˆn q 1 ), Ḧ(ˆx j ) Ḧ(ˆx j 1) + ˆδ. Ḧ(ˆx)

127 .. H (x^) Cumulative distribution of x^ x^ Ḧ(ˆx) ˆx Ḧ(ˆx) ḧ(ˆx) 1 /4 1 /4 3/4 1/2 4 /4 1 /4 E[L(ˆx)] Ḧ(ˆx) E[L (ˆx)] = E(ḧ(ˆx)) = 1.5 ḧ(ˆx) D overhead Ç å ˆn 1 q D overhead = ˆn d D ḧ(ˆx) = ˆn A d ḧ(ˆx j ) log ḧ(ˆx j) 1 / A j=1 = ˆn d Ä log A E(ḧ(ˆx))ä, = D overhead [0, ˆn d log A ] supp(ḧ(ˆx)) A D ( π 1 π 2 )

128 π 1 π 2 D ( π 1 (χ) π 2 (χ)) := supp( π 1 ) i=1 π 1 (χ i ) log π 1(χ i ) π 2 (χ i ). D overhead D overhead = ˆn d Ä log A E(ḧ(ˆx))ä = 4 (6 1.5) = 18. P overhead P overhead = }{{} }{{} 01 }{{} } {{} = = 3 (6 + 6) = 47, P overhead = supp(ḧ) supp(ḧ) (log A + log A ) + supp(ḧ) supp(ḧ) 2 log A j=1 j=1 L (ˆx j ) + log A log ḧ(ˆx j) + log A supp(ḧ) 2 log A + supp(ḧ) log supp(ḧ) + log A = = ˆn q Ä 2 log A + log ˆn q ä + log A, ḧ(ˆx) P overhead ˆn q L (ˆx j ) ˆx j ḧ(ˆx j) 0

129 D overhead ˆn d overhead := min{d overhead, P overhead} = = min{18, 47} = 18. A D overhead O(ˆn d ) P overhead O(ˆn q log ˆn q ) ˆn q ˆn d P overhead supp(ḧ) A max i L (ˆx i ) P overhead

130 i := i + 1 j := j + 1 codebooks ˆn d i, j := 1 S 0 := 0 x i I A (g; h) {ˆx j } j 1 1 ˆx j j? = ˆn d j := j + 1 {ˆx j }ˆn d 1 j := 1 ˆn d M A (h) j? = ˆn d S j := S j ˆn d ˆx j I A (h; h) ˆx j E h [x] := Sˆnd max I( g(x); ḧ(ˆx)). g(x) x ˆx f(x, ˆx) g(x)ḧ(ˆx) I( g(x); ḧ(ˆx)) := D Ä f(x, ˆx) g(x) ḧ(ˆx) ä n q ˆn q = i=1 j=1 f(x i, ˆx j ) log Ñ f(xi, ˆx j ) g(x i )ḧ(ˆx j) é. I( g(x); ḧ(ˆx)) = E( g(x)) + E(ḧ(ˆx)) E( g(x), ḧ(ˆx)) = = E( g(x)) E( f(x ˆx)) = E(ḧ(ˆx)) E( f(ˆx x)),

131 ḧ(ˆx) g(x) E( f(ˆx x)) = 0 ˆx χ π(χ) Π (χ) Π (χ) 1 E( π(χ)), δ

132 Å δ := gcd { π(χ i )} supp( π) ã 1. χ π(χ) Π (χ)

133 A x ˆx A I A ( g(x); ḧ(ˆx)) I( g(x); ḧ(ˆx)) = E(ḧ(ˆx)) E( f(ˆx x)) = = E(ḧ(ˆx)) 0 = 1.5 ; I A ( g(x); ḧ(ˆx)) = EA(ḧ(ˆx)) E A( f(ˆx x)) = = Çḧ(ˆx) E(ḧ(ˆx)) + D å 1 0 = = 6 A E A Çḧ(ˆx) (ḧ(ˆx)) = E(ḧ(ˆx)) + D å 1. A ˆn d

134 ˆn d ( ) ˆn d ( ) ˆn d I( g(x); ḧ(ˆx)) = = 6 ; ˆn d I A ( g(x); ḧ(ˆx)) = 4 6 = n d E( g(x)) = = 20, ˆn d E(ḧ(ˆx)) = = 6, ˆn d I( g(x); ḧ(ˆx)) + overhead = = = 24, ḧ(ˆx)

135 M(ḧ(ˆx)) = E[L (ˆx)] = E(ḧ(ˆx)) = 1.5 ; M A (ḧ(ˆx)) = L A(ˆx) = E A (ḧ(ˆx)) = 6. ˆn d M(ḧ(ˆx)) = = 6 ; ˆn d M A (ḧ(ˆx)) = 4 6 = 24. ˆx ˆx j 1/ˆn d j = 1, 2,..., ˆn d ˆn d ˆx j

136 I(ḧ(ˆx); ḧ(ˆx)) I A (ḧ(ˆx); ḧ(ˆx)) g(x) ḧ(ˆx) ˆn d I(ḧ(ˆx); ḧ(ˆx)) ˆn d I A (ḧ(ˆx); ḧ(ˆx)) x x ḡ(x) x ḡ(x),

137 Probability density of x Probability masses of x g (x).. g (x) x x ḡ(x) g(x) x E(ḡ(x)) := ḡ(x) log ḡ(x) dx, supp(ḡ) supp(ḡ) n q g(x) (i+1) g(x i ) := ḡ(x) dx = ḡ(x i ) i i = 1, 2,..., n q n q supp(ḡ(x)) n q := m(supp(ḡ)) m σ supp(ḡ) supp( g) := n q,

138 := 2 g(x) ḡ(x) 0 E( g(x)) E(ḡ(x)) + supp( g) := E(ḡ(x)) + n q, 2.5 E(ḡ(x)) + 6.

139 ˆn d I A ( g(x); ḧ(ˆx)) K D ˆn d I A ( g(x); ḧ(ˆx)) n ID Û I AD =: K D, n ID Û I AD ˆn d M A (ḧ(ˆx)) K S ˆn d M A (ḧ(ˆx)) n M ˆM A =: K S, n M ˆM A ˆn d I A (ḧ(ˆx); ḧ(ˆx)) K C ˆn d I A (ḧ(ˆx); ḧ(ˆx)) n IC Û I AC =: K C, n IC Û I AC

140 K K := min{k D, K S, K C }. K = 24 K {ˆx }ˆn d j 1 ˆx j ḧ(ˆx j) j = 1, 2,..., ˆn mc {ˆx j }ˆn nq 1 ˆn mc ˆn mc ˆn d ˆn nq ˆn nq ˆn d ˆn d P E h [x]

141 K 2K K K A E(π(Aχ)) A E( π( k χ k )) φ : R K R K E(π(φ(χ))) φ : R K R K, K > K E( ) I( ; ) K χ R K R K K K φ(χ) φ( θ) θ n ι = 2 n ι = n n θ

142 codebooks ˆn d, φ( ) i, j, ι := 1 S 0 := 0 E := := 0 θ 0 x i I A (g; h) i := i + 1 j := j + 1 {ˆx j } j 1 1 ˆx j j? = ˆn d j := j + 1 {ˆx j }ˆnd 1 j := 1 ˆn d M A (h) j? = ˆn d S j := S j ˆn d φ(ˆx j θ ι ) I A (h; h) ˆx j θ ι E := max( Sˆnd, E ) θ ι := θ ι 1 + (θ ι θ ι 1 ) 1 E ( Sˆnd ) E, ι n ι ι =? j := 1 n ι ι := ι + 1 E h [φ(x θ )] := E θ := θ nι

143 φ( ) P K K := min{k D, K S, K C / n ι }. n ι = 2 K D 24 K S 24 K C 48 K = K C / n ι, K C / n ι = K = = ˆn d I A (ḧ(ˆx); ḧ(ˆx)) = = ˆn d M A (ḧ(ˆx)) = = ˆn d I A ( g(x); ḧ(ˆx)) = = ˆn d I( g(x); ḧ(ˆx)) + overhead.

144 I( g(x); ḧ(ˆx)) = K C overhead =: κ, ˆn d n ι ˆn d κ κ K C K κ K κ κ I( g(x); ḧ(ˆx)) = E( g(x)) + E(ḧ(ˆx)) E( g(x), ḧ(ˆx)) =: E( g(x)) + E(ḧ(ˆx)) E( f(x, ˆx)) = n q ˆn q n q ˆn = g(x i ) log g(x i ) ḧ(ˆx j ) log ḧ(ˆx q j) + i=1 j=1 i=1 j=1 f(x i, ˆx j ) log f(x i, ˆx j ), κ I(ḡ(x); h(ˆx)) = E(ḡ(x)) + E( h(ˆx)) E(ḡ(x), h(ˆx)) =: E(ḡ(x)) + E( h(ˆx)) E( f(x, ˆx)) = = supp(ḡ) ḡ(x) log ḡ(x) dx h(ˆx) log h(ˆx) dˆx+ supp( h) + f(x, ˆx) log f(x, ˆx) dˆx dx. supp(ḡ) supp( h)

145 Π (χ) δ n d := 1/δ Π (χ) {χ i } n d 1 χ i = Π 1 (iδ) E( π(χ)) n i i n d n q min L(χ i ) = {L(χ i )} n i=1 i=1 i=1 n q n i L(χ i ) = n d i=1 n i n d L(χ i ) n q 2 L(χi) 1 i=1 L (χ i ) := log n i n d. n i n i := Π (χ i ) Π (χ i 1 ) δ =: π(χ i), δ n i n d = π(χ i ).

146 n d i=1 n L q (χ i ) = n d i=1 n i n d log n i n q = n n d π(χ) log π(χ) =: n d E( π(χ)), d i=1 χ i Π (χ i ) χ i < χ i χ i n d i=1 L(χ i) = n q i=1 n il(χ i ) E[L(χ)] = n q i=1 π(χ i)l(χ i ) L (χ i ) = log π(χ i ), n E[L q (χ i )] = π(χ i ) log π(χ i ) =: E( π(χ)). i=1 L(χ i ) := log π(χ i ) 1 (E( π(χ)) + 1). δ

147 1 /8 1 /4 1 /8 7 1 / / / /8 = / / /4 = 13

148 7 1 / / /4 = 12 max {u(c t) + βe h t [u(c t+1 )]} = {u(c t ) + β u(c t+1 ) h( ˆD t+1 ) d ˆD t+1 } {C t,q t } R + C t + P t q t = (P t + D t )q t 1 =: W t, C t+1 = ˆD t+1 q t =: Ŵ t+1 q t 1 D t h( ˆD t+1 ) := f(d t+1, ˆD t+1 ) dd t+1, supp(g) f(d t+1, ˆD t+1 ) := arg min î f(, ) Ef u(c(d t+1 )) u(c( ˆD t+1 )) 2ó I(g(D t+1 ); h( ˆD t+1 )) κ, g(d t+1 ) ; ˆD t+1 D t+1

149 1 /2 max f(, ) Ef t [u(c t+1) + βu(c t+2 )] = = [u(c t+1 ) + βu(c t+2 )] f({c t+1, q t+1 }, D t+1 ) d{c t+1, q t+1 } dd t+1 R 3 + C t+1 + P t+1 q t+1 = (P t+1 + D t+1 )q t =: W t+1, C t+2 = D t+2 q t+1 =: W t+2 q t D t+2 I(g(D t+1 ); h({c t+1, q t+1 })) κ, h({c t+1, q t+1 }) := f({c t+1, q t+1 }, D t+1 ) dd t+1, supp(g) g(d t+1 ) ;

150 D t+1 ˆD t+1

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159 [ max θ Eg t φ (x θ) ] = supp(g) φ (x θ) g(x) dx, P θ g(x), max θ Eh [φ(ˆx θ)] = φ(ˆx θ) h(ˆx) dˆx, P supp(h) θi h(ˆx) := f(x, ˆx) dx, supp(g) { [ f(x, ˆx) := arg min f(x,ˆx) Ef d(φ (x θ), φ(ˆx θ)) ] } I(g(x); h(ˆx)) κ, g(x). θ φ ( ) φ( ) L p p d(φ (x θ), φ(ˆx θ)) = φ (x θ) φ(ˆx θ) p p = φ (x θ) φ(ˆx θ) p =: d(x, ˆx). supp(g(x)) supp(h(ˆx)) φ (x θ) = φ(ˆx θ)

160 p = 2 d(x, ˆx) = φ (x θ) φ(ˆx θ) 2 = = φ (x θ) + φ (x θ) x (ˆx x) + o( x ˆx ) φ 2 (x θ) = ( φ )2 (x θ) = x (ˆx x) + o( x ˆx ). φ (x θ)/ x ˆx x f(x, ˆx) E κ,p [ φ (x θ) ] := E h [φ(ˆx θ)] h(ˆx) := f(x, ˆx) dx, supp(g) [ f(x, ˆx) := arg min f(x,ˆx) Ef d(φ (x θ), φ(ˆx θ)) ] I(g(x); h(ˆx)) κ, g(x). [ max θ Eg φ (x θ) ], [ max θ Eκ,p φ (x θ) ]. g(x) E(g(x)) < κ < κ κ L p p P θi P θ max θ Eκ,p [ φ (x θ) ] [ max θ Eg φ (x θ) ].

161

162 P QI r t+1 N (µ r, Σ r ) f(ω t r t+1 ω t ˆr t+1) = (2π) 1 λ 1 Ñ Ä 2 ω 2 2 exp t (r t+1 ˆr t+1 + ˇµ r ) ä 2 é, ˆr 2( λ /2) t+1 R K ; ω t r t+1 = ω t ˆr t+1 ω t ˇµ r + ϵ r,t+1, ω t Σ rω t = ω t ˆΣ r ω t + Ψ r, ϵ r,t+1 N (0, Ψ r ), Ψ r = λ 2, ω t ˆr t+1 N (ω t ˆµ r, ω t ˆΣ r ω t ), ω t ˆΣ r ω t = ω t Σ rω t Ψ r ; λ = 2e 2κ ω t Σ rω t. +

163 f(r t+1 ˆr t+1 ) = (2π) K 2 2 Ψ r 1 Ç + exp 1 å 2 (r t+1 ˆr t+1 + ˇµ r ) Ψ + r (r t+1 ˆr t+1 + ˇµ r ), ˆr t+1 R K ; r t+1 = ˆr t+1 ˇµ r + ϵ r,t+1, Σ r = ˆΣ r + Ψ r, ϵ r,t+1 N (0, Ψ r ), Ψ r = λ 2 (ω t ω t) 2 ω t ω t, ˆr t+1 N (ˆµ r, ˆΣ r ), ˆΣr = Σ r Ψ r ; λ = 2e 2κ ω t ω t Σ r. ω t Σ rω t ω t ω t Σ r λ κ Ψ r ϵ r,t+1 ϵ r,t+1 Ψ r ω t

164 ˆr t+1 ˆρ r,kl k, l {1,..., K} λ ˆΣ r

165 Σ r ˆΣ r Ä (1 α) ˆΣ ä r + αηi K α [0, 1] η ηi K Ä ˆΣr + α 1 ä Ψ r Σr Ä ä ˆΣr + Υ r Υ r Υ r Ψ r Σ r ˆΣ Σ

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