* n x 11,, x 1n N(µ 1, σ 2 ) x 21,, x 2n N(µ 2, σ 2 ) H 0 µ 1 = µ 2 (= µ ) H 1 µ 1 µ 2 H 0, H 1 *2 σ 2 σ 2 0, σ 2 1 *1 *2 H 0 H
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1 * n x 11,, x 1n Nµ 1, σ x 1,, x n Nµ, σ H 0 µ 1 = µ = µ H 1 µ 1 µ H 0, H 1 * σ σ 0, σ 1 *1 * H 0 H 0, H 1 H 1 1
2 H 0 µ, σ 0 H 1 µ 1, µ, σ 1 L 0 µ, σ x L 1 µ 1, µ, σ x x H 0 L 0 µ, σ 0 x H 1 L 1 µ 1, µ, σ 1 x L 1 µ 1, µ, σ 1 x L 0 µ, σ 1 0 x 1 *3 1 L 1 µ 1, µ, σ 1 T = log x L 0 µ, σ 0 x χ * 4*5 χ H 0 H 1 H 0 : µ, σ H 1 : µ 1, µ, σ T > χ 1, 0.95 *6 *3 *4 n *5 T T = log L 0 µ, σ 0 x L 1 µ 1, µ, σ 1 x logx = log 1 x *6 χ 1, 0.95 χ t χ F χ
3 . x 11,, x 1n Nµ 1, σ x 1,, x n Nµ, σ H 0 µ 1 = µ = µ H 1 µ 1 µ α i H 0, H 1 *7 *8 ii i iii ii L 1 µ 1, µ, σ 1 T = log x L 0 µ, σ 0 x = l 1 µ 1, µ, σ 1 x l0 µ, σ 0 x *9 iv m= H 1 H 0 χ T > χ m, 1 α T χ H 1 H x 11,, x 15 Nµ 1, σ x 1,, x 5 Nµ, σ * 10 H 0 µ 1 = µ = µ H 1 µ 1 µ x 11 = 10, x 1 = 13, x 13 = 14, x 14 = 8, x 15 = 15 x 1 = 13, x = 1, x 3 = 18, x 4 = 5, x 5 = *7 µ σ *8 *9 l 1 µ 1, µ, σ 1 x, l0 µ, σ 0 x *
4 H 0 µ 1 = µ = µ 10 Nµ, σ * 11 µ = 1 10 σ 0 = 1 10 = 1 10 = 3. L 0 µ, σ x = 5 1 exp x ij µ πσ σ 10 1 = exp πσ 1 σ l 0 µ, σ x = 10 log πσ 1 σ x ij µ x ij µ x ij = = x ij µ { } l 0 µ, σ 0 x = 10 log π σ 0 1 σ 0 x ij µ σ 0 = 1 10 x ij µ x ij µ x ij µ = 10 σ 0 l 0 µ, σ 0 x = 10 log π σ σ σ 0 0 = 5 log 46.4π 5 *11 1 4
5 * 1 L 1 µ 1, µ, σ x = 5 { 1 exp x 1j µ 1 } πσ σ = exp 1 πσ σ x 1j µ 1 + l 1 µ 1, µ, σ x = 10 log πσ 1 σ µ 1 = 1 5 µ = 1 5 { 1 exp x j µ } πσ σ x j µ x 1j µ 1 + x 1j = = 1 5 x j = = 18 5 σ 1 = 1 x 1j µ = 1 10 = 14. x j µ x j µ 3 { } 3 l 1 µ 1, µ, σ 1 x = 10 log π σ 1 1 x 1j µ 1 + σ 1 σ 1 = 1 x 1j µ 1 + x j µ 10 x j µ 4 4 l 1 µ 1, µ, σ 1 x = 10 log π σ σ σ 1 1 = 5 log 8.4π 5 *1 5
6 L 1 µ 1, µ, σ 1 T = log x L 0 µ, σ 0 x = l 1 µ 1, µ, σ 1 x l0 µ, σ 0 x = { 5 log 8.4π 5 5 log 46.4π 5} 46.4π = 10 log 8.4π 46.4 = 10 log * 13 χ H 1 µ 1, µ, σ H 0 µ, σ χ 1, * 14 T 4.9 > 3.84 χ 1, i * 15 ii 4.1 i 1 x 1,, x n x 1,, x n * 16 x 1,, x n x 1,, x n * 17 * 18 * 19 *13 log PC *14 10 *15 *16 *17 *18 *19 6
7 4. ii 4..1 fx n, p = n C x p x 1 p n x 10 3 Lp 10, 3 = 10 C 3 p 3 1 p 7 5 p p = 0.1, 0.3, p p = 0.1 L0.1 10, 3 = 10 C p = 0.3 L0.3 10, 3 = 10 C p = 0.5 L0.5 10, 3 = 10 C p=0.1 p = p=0.3 p = p=0.5 p = p = 0.3 p = 0.1, 0.5 p = * p = 0.1, 0.3, 0.5 p = 0.3 p 0 1 Lp 10, 3 p p Lp 10, 3 p dlp 10, 3 dp = 0 6 *0 7
8 * 1 5 p 10 log * lp 10, 3 = log Lp 10, 3 lp 10, 3 p dlp 10, 3 dp = 0 7 * 3 p p p 5 n x 11,, x 1n Nµ 1, σ x 1,, x n Nµ, σ H 0 µ 1 = µ = µ H 1 µ 1 µ H 0 H 1 H 0 H 0 H 1 µ 1, µ, σ x L 0 µ, σ 0 x L 1 µ 1, µ, σ 1 x *1 y = fx f x = 0 f x = 0 fx = x 3 x = 0 6 * Lp 10, 3 = 10 C 3 p 3 1 p 7 lp 10, 3 = log 10 C log p + 7 log 1 p p dl dp p = 0, dl p = 0 dp *3 x = 0, n 8
9 5.1 i H 0 L 0 µ, σ 0 x ii H 0, H 1 L 1 µ 1, µ, σ 1 x H 0 H 0, H 1 a b a H 0 µ 1 = µ = µ H 0 H 1 µ 1, µ H 0 µ 1 = µ µ 1 µ µ σ 0 = 1 n n x ij µ σ 1 = 1 n x 1j µ 1 + n n x j µ H 0 H 0 H 1 L 1 µ 1, µ, σ 1 x L 0 µ, σ 0 x 1 b H 1 µ 1 µ H 1 µ 1 µ µ 1 µ H 0 µ L 0 µ, σ 0 x H 0 H 1 H 1 µ 1, µ L 1 µ 1, µ, σ 1 x L0 µ, σ 0 x L 1 µ 1, µ, σ 1 x L 0 µ, σ 0 x >> 1 * 4 *4 9
10 6 6.1 log L 1 µ 1, µ, σ 1 T = log x L 0 µ, σ 0 x = l 1 µ 1, µ, σ 1 x l0 µ, σ 0 x * 5 6. * 6 T = l 1 µ 1, µ, σ 1 x l0 µ, σ 0 x χ 1 χ H 1 H 0 H 1 µ 1, µ, σ H 0 µ, σ α T = l 1 µ 1, µ, σ 1 x l0 µ, σ 0 x > χ 1, 1 α H 0 H 0 H 0 H 1 a H 0 H 0 H 1 b H 0 T c T H 0 c H 0 H 0 H 1 a H 0 H 0 H 1 b T H 0 H 1 *5 T 0 *6 10
11 7 7.1 X 1, X X = X X1 X, µ = µ1 µ, Σ = X N µ, Σ σ 1 ρσ 1 σ ρσ 1 σ σ ρ 0 X 1, X = x 1, x Lµ 1, µ, ρ, σ1, σ x 1 1, x = πσ 1 σ exp 1 x1 µ 1 1 ρ 1 ρ σ 1 ρx 1 µ 1 x µ + x µ σ 1 σ σ X 1 Nµ 1, σ1, X Nµ, σ fx 1 µ 1, σ1 fx µ, σ 1 = exp x 1 µ 1 1 exp x µ πσ 1 πσ = σ 1 1 exp x 1 µ 1 πσ σ1 x µ σ Lµ 1, µ, ρ, σ 1, σ x 1, x σ * 7 µ µ µ *7 11
12 * 8 x 1,, x n Nµ, σ σ ML = 1 n σ = 1 n 1 n x i x i=1 n x i x i=1 * 9 σ σ ML n n = 10 x i x = 70 i=1 σ ML = = 7, σ = 70 9 = 30 σ ML * σ ML µ = x µ µ * 31 µ = x µ 9,999 µ 1,, µ 9999 µ 10,000 10,000 µ µ + µ µ 9999 µ µ µ µ µ 1,, µ 100 µ µ *8 µ µ = x µ µ µ *9 *30 *31 UMVU 1
13 10,000 σ ML + σ ML, σ ML,9999 < σ * 3 µ * 33 σ = 1 n x i µ 8 n i=1 σ µ µ = x σ ML = 1 n x i µ 9 n i=1 8 9 µ µ 8 µ σ * 34 9 µ µ µ µ * 36 * H 0, H 1 H 1 H 0 H 1 x 11, x 1, x 13 Nµ 1, σ, x 1, x, x 3 Nµ, σ *3 σ ML *33 *34 *35 *36 µ 13
14 H 0 : µ 1 = µ = µ H 1 : µ 1 µ x 11 = 0, x 1 = 30, x 13 = x 1 = 5, x = 18, x 3 = 9 H 1 µ 1 = = 4, µ = = 4 3 µ 1 µ H 1 µ 1 µ H 0 H 1 µ 1 = µ H 0 µ 1 = µ µ 1 = µ H 0 µ 1 = µ * 37 H 1 H 0 H 1 8 Holmes Sharlock Holmes * 38 * 39 we balance the probabilies and choose the most likely * p = 0.1, 0.3, 0.5 * 41 * =1 L 1 µ 1, µ, σ 1 x L 0 µ, σ 0 x 1 *38 *39 *40 Holmes A B C *41 Holmes Bayes 14
t χ 2 F Q t χ 2 F 1 2 µ, σ 2 N(µ, σ 2 ) f(x µ, σ 2 ) = 1 ( exp (x ) µ)2 2πσ 2 2σ 2 0, N(0, 1) (100 α) z(α) t χ 2 *1 2.1 t (i)x N(µ, σ 2 ) x µ σ N(0, 1
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