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Hidden Markov Models. Tunghai University Fall 2005. Simple Model - Markov Chains. • Markov Property: The state of the system at time t+1 only depends on the state of the system at time t. X 2. X 4. X 3. X 1. X 5. Markov Chains. Stationarity Assumption

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Hidden Markov Models


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    1. Hidden Markov Models Tunghai University Fall 2005

    2. Simple Model - Markov Chains • Markov Property: The state of the system at time t+1 only depends on the state of the system at time t X2 X4 X3 X1 X5

    3. 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

    4. 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

    5. 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

    6. p p p p 0 1 N-1 N 2 Start (10$) 1-p 1-p 1-p 1-p Gambler’s 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

    7. 0.1 0.9 0.8 coke pepsi 0.2 Coke vs. Pepsi Given that a person’s last cola purchase was Coke, there is a 90% chance that her next cola purchase will also be Coke. If a person’s last cola purchase was Pepsi, there is an 80% chance that her next cola purchase will also be Pepsi.

    8. The transition matrix is: (Corresponding to one purchase ahead) 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?

    9. 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?

    10. P00 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(X3=0)

    11. H1 H2 HL-1 HL Hi X1 X2 XL-1 XL Xi Hidden Markov Models - HMM Hidden variables Observed data

    12. H1 H2 HL-1 HL Hi X1 X2 XL-1 XL Xi Fair/Loaded L tosses Head/Tail 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

    13. Start H1 H2 HL-1 HL Hi 1/2 1/2 tail tail 1/2 1/4 0.1 Fair loaded X1 X2 XL-1 XL Xi 0.1 0.9 0.9 3/4 1/2 head head Fair/Loaded Head/Tail Coin-Tossing Example L tosses Query: what are the most likely values in the H-nodes to generate the given data?

    14. Seeing the set of outcomes {x1,…,xL}, compute p(loaded | x1,…,xL) for each coin toss 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 Hi’s values hi, namely, compute p(hi | x1,…,xL). 2. Do the same computation for every Hibut without repeating the first task L times.

    15. 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 P q 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

    16. H1 H2 HL-1 HL Hi X1 X2 XL-1 XL Xi C-G island? A/C/G/T C-G Islands Example G A change C T G A C T