Hidden Markov Models

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Hidden Markov Models
Tunghai University Fall 2005
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Simple Model - Markov Chains
Markov Property The state of the system at
time t1 only depends on the state of the system
at time t
X2
X1
X3
X4
X5
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Markov Chains
Stationarity Assumption Probabilities are
independent of t when the process
is stationary So, This means that if system
is in state i, the probability that the system
will transition to state j is pij no matter
what the value of t is
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Simple Example
Weather raining today rain
tomorrow prr 0.4 raining today
no rain tomorrow prn 0.6 no
raining today rain tomorrow
pnr 0.2 no raining today no rain
tomorrow prr 0.8
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Simple Example
Transition Matrix for Example Note that
rows sum to 1 Such a matrix is called a
Stochastic Matrix If the rows of a matrix and
the columns of a matrix all sum to 1, we have a
Doubly Stochastic Matrix
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Gamblers Example
At each play we have the following Gambler
wins 1 with probability p Gambler loses 1
with probability 1-p Game ends when gambler
goes broke, or gains a fortune of 100 Both 0
and 100 are absorbing states
or
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Coke vs. Pepsi
Given that a persons last cola purchase was
Coke, there is a 90 chance that her next cola
purchase will also be Coke. If a persons last
cola purchase was Pepsi, there is an 80 chance
that her next cola purchase will also be Pepsi.
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Coke vs. Pepsi
Given that a person is currently a Pepsi
purchaser, what is the probability that she will
purchase Coke two purchases from now?
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Coke vs. Pepsi
Given that a person is currently a Coke drinker,
what is the probability that she will purchase
Pepsi three purchases from now?
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Coke vs. Pepsi
Assume each person makes one cola purchase per
week. Suppose 60 of all people now drink Coke,
and 40 drink Pepsi. What fraction of people
will be drinking Coke three weeks from now?
Let (Q0,Q1)(0.6,0.4) be the initial
probabilities. We will regard Coke as 0 and Pepsi
as 1 We want to find P(X30)
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Hidden Markov Models - HMM
Hidden variables
Observed data
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Coin-Tossing Example
Start
1/2
1/2
tail
tail
1/2
1/4
0.1
Fair
loaded
0.1
0.9
0.9
3/4
1/2
head
head
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Coin-Tossing Example
L tosses
Query what are the most likely values in the
H-nodes to generate the given data?
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Coin-Tossing Example
Query what are the probabilities for fair/loaded
coins given the set of outcomes x1,,xL?
1. Compute the posteriori belief in Hi (specific
i) given the evidence x1,,xL for each of His
values hi, namely, compute p(hi x1,,xL). 2.
Do the same computation for every Hi but without
repeating the first task L times.
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C-G Islands Example
C-G islands DNA parts which are very rich in C
and G
q/4
P
q/4
G
q
A
P
q
Regular DNA
change
q
P
q
P
q/4
C
T
p/3
q/4
p/6
G
A
(1-P)/4
(1-q)/6
(1-q)/3
p/3
P/6
C-G island
C
T
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C-G Islands Example
G
A
change
C
T
G
A
C
T