Prml Mixtures Of Gaussians

2.3.9

a

PRML :Mixtures of Gaussians

V%&f< M-1 !W$S 7

n4|

1/8

Limitation of single Gaussian 100 90 80 70 60 50 40

1

2

3

4

5

6

Figure A.5 Old Faithful dataset

2/8

Limitation of single Gaussian 100

100

80

80

60

60

40

40

1

2

3

Figure 2.21 1

D5,,[ 4

5

6

1

2

3

Figure 2.21 2

D5,,[ 4

5

6

2/8

Mixture of Univariate Gaussians p(x)

x

VkJ~OF5,,[ (gW 3 D)$V$~O 3 DN5,, [r~AkgG@?,['5,.g,[%

Figure 2.22

3/8

Mixture of Univariate Gaussians p(x)

x

VkJ~OF5,,[ (gW 3 D)$V$~O 3 DN5,, [r~AkgG@?,['5,.g,[%

Figure 2.22

=,JtN5,,[rQ$F$kg9k~N8tHF5,,[N? Q$&,6r409lP$[\4FN"3,[,awG-k% 3/8

Mixture of Gaussians

p(x) =

K X

πk N (x|µk , Σk )

(2.188)

k=1

1

1

(a)

0.5

0.2

(b)

0.5

0.3

0.5 0

0 0

Figure 2.23 3

0.5

1

0

0.5

1

D5,,[ (2 !5) r~AkgG@?5,.g,[ 4/8

3NaGO5,.g,[r7CF$k% lL*K>N,[r~Akg9lP$.g,[KJk% ?H(P 9.3.3 aGO%6,[G"k Bernoulli ,[r~AkgG@? .g,[KD$FD@9k% 5,.g,[N8t π OJ
K X

πk = 1

(2.189)

0 6 πk 6 1

(2.190)

k=1

5/8

sum,product

k
K X

p(k)p(x|k)

(2.191)

k=1

D^j0 (2.188) K*$F$π = p(k) r k V\N.,r*VlgNv 0N($N (x|µ , Σ ) = p(x|k) r k NlgN x NroU-N(,[K _J93H,G-k% k

k

k

p(x) =

K X

πk N (x|µk , Σk )

(2.188)

k=1

6/8

responsibilities

ve,[ p(k|x) OsoKEWG"k%responsibilities HbFV% Bayes j}rQ$F$ve,[,J
0 (2.192) N\YJraO 9 OK*$FT&%

(2.192)

7/8

`Y ln p(X|π, µ, Σ) =

N X

ln

n=1

X K k=1

 πk N (xn |µk , Σk )

(2.193)

0 (2.193) K*$F$X = {x , . . . , x }$π ≡ {π , . . . , π }$ µ P≡ {µ , . . . , µ }$Σ ≡ {Σ , . . . , Σ } HjA9k% O ln NfG"k+i$µ , Σ rdj9klg$q7/Jk% 1 DN}!O+jV7tMW;Q$F$`YrGg=K9k µ , Σ r dj9k% b& 1 Dj!O 9 OKRp9k expectation maximization(EM) "k4 j:`G"k% N

1

K k=1

1

K

1

K

K

1

k

k

k

k

8/8