- #1
SamTaylor
- 20
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Hi,
I try to teach myself Hidden Markov Models. I am using this text
"www.cs.sjsu.edu/~stamp/RUA/HMM.pdf" [Broken] as Material. The Introduction with
the example was reasonable but now I have trouble in unterstanding
some of the derivation.
I can follow the math and use the formulas to get results but
I also want to understand the meaning behind it.
One question that arise at hidden markov models is to determine
the likelihood of the oberserved sequence O with the given Model
as written below:
b = probability of the observation
a = transiting from one state to another
[itex]\pi[/itex] = starting probability
O = observation
X = state sequence
[itex]\lambda[/itex] = hidden markov model
( Section 4, Page 6 in the Text )
[itex]P(O | X, \lambda) = b_{x0}()* b_ {x1}() *\cdots* b_{xT-1}(O_{T-1}) [/itex]
[itex]P(X|\lambda) = \pi_{x_0}a_{x_0,x_1}a_{x_1,x_2}*\cdots* a_{x_{T-1},x_{T-2}}[/itex]
[itex]P(O, X|\lambda) = \frac{P(O \cap X \cap \lambda)}{P( \lambda)}[/itex]
[itex]P(O | X, \lambda) *P(X|\lambda) = \frac{P(O \cap X \cap \lambda)}{P(X \cap \lambda)} \frac{P(X \cap \lambda)}{P(\lambda)}=\frac{P(O \cap X \cap \lambda)}{P(\lambda)}[/itex]
[itex]P(O, X|\lambda) = P(O | X, \lambda) * P(X|\lambda) [/itex]
[itex]P(O | \lambda) = \sum\limits_{X} P(O,X|\lambda)[/itex]
[itex]P(O | \lambda) = \sum\limits_{X} P(O | X, \lambda) * P(X|\lambda)[/itex]
My question is: Why do I get the likelihood of the oberserved sequence
by summing up over all all possible state sequences. Can someone please
explain it differently?
I try to teach myself Hidden Markov Models. I am using this text
"www.cs.sjsu.edu/~stamp/RUA/HMM.pdf" [Broken] as Material. The Introduction with
the example was reasonable but now I have trouble in unterstanding
some of the derivation.
I can follow the math and use the formulas to get results but
I also want to understand the meaning behind it.
One question that arise at hidden markov models is to determine
the likelihood of the oberserved sequence O with the given Model
as written below:
b = probability of the observation
a = transiting from one state to another
[itex]\pi[/itex] = starting probability
O = observation
X = state sequence
[itex]\lambda[/itex] = hidden markov model
( Section 4, Page 6 in the Text )
[itex]P(O | X, \lambda) = b_{x0}()* b_ {x1}() *\cdots* b_{xT-1}(O_{T-1}) [/itex]
[itex]P(X|\lambda) = \pi_{x_0}a_{x_0,x_1}a_{x_1,x_2}*\cdots* a_{x_{T-1},x_{T-2}}[/itex]
[itex]P(O, X|\lambda) = \frac{P(O \cap X \cap \lambda)}{P( \lambda)}[/itex]
[itex]P(O | X, \lambda) *P(X|\lambda) = \frac{P(O \cap X \cap \lambda)}{P(X \cap \lambda)} \frac{P(X \cap \lambda)}{P(\lambda)}=\frac{P(O \cap X \cap \lambda)}{P(\lambda)}[/itex]
[itex]P(O, X|\lambda) = P(O | X, \lambda) * P(X|\lambda) [/itex]
[itex]P(O | \lambda) = \sum\limits_{X} P(O,X|\lambda)[/itex]
[itex]P(O | \lambda) = \sum\limits_{X} P(O | X, \lambda) * P(X|\lambda)[/itex]
My question is: Why do I get the likelihood of the oberserved sequence
by summing up over all all possible state sequences. Can someone please
explain it differently?
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