THE HONG KONG POLYTECHNIC UNIVERSITY, DEPARTMENT OF EIE
Lecture Notes of Machine Learning
Lecture 14 Mixtures of Gaussian and the EM algorithm
April 11, 2015
1 MIXTURES OF GAUSSIAN
ù˜ù§·‚?Ø|^EM (Expectation-Maximization)‰VÇ—ÝO"b·‚
k˜|Ôöx
(1)
, x
(2)
,...x
(m)
,Ï•´u nsupervisedÆS¯K§¤±·‚vk?Ûy
&E"
·‚F"|^˜‡éÜ©Ùp(x
(i)
, z
(i)
) = p(x
(i)
|z
(i)
)p(z
(i)
)5[Üù êâ, Ù¥z
(i)
∼
Multinomial(φ) (φ
j
Ê 0,
P
k
j =1
φ
j
= 1,ë êφ
j
‰ Ñ V Çp(z
(i)
= j ))§ ¿ …x
(i)
|z
(i)
= j ∼
N(µ
j
,Σ
j
)§·‚4kL«z
(i)
ŒUŠ‡ê§Ïd3ù‡.¥§z˜‡Ôöx
(i)
´d
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)¤§¤±x
(i)
´lk‡pd©Ù¥˜‡(dz
(i)
•«)J
Ñ5" ù‡¡•pd·Ü.§·‚•‡5¿z
(i)
´Û¹‘ÅCþ§pd·Ü
.9ëê´φ,µ,Σ§•OùCþ§·‚Œ±ïáXeLˆªµ
l(φ,µ,Σ) =
m
X
i=1
logp(x
(i)
;φ,µ,Σ)
=
m
X
i=1
log
k
X
z
(i)
=1
p(x
(i)
|z
(i)
;µ,Σ) p(z
(i)
,φ)
·‚uy§ ÏL¦ ê¦4Š•{§Ã{ùëê)§lþ¡Lˆª
Œ±wѧ‘ÅCþz
(i)
wŠ·‚x
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´lk‡pd©Ù¥Ù¥˜‡)¤§XJ·‚
1