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. Hidden Markov Models - HMM Tutorial #5 © Ydo Wexler & Dan Geiger
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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 X1X1 X2X2 X3X3 X4X4 X5X5
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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 p ij no matter what the value of t is
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4 Weather: –raining today rain tomorrow p rr = 0.4 –raining today no rain tomorrow p rn = 0.6 –not raining today rain tomorrow p nr = 0.2 –not raining today no rain tomorrow p rr = 0.8 Simple Example
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5 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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6 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 $100 are absorbing states 01 2 N-1 N p p p p 1-p Start (10$) or
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7 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. coke pepsi 0.1 0.9 0.8 0.2
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8 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? The transition matrix is: (Corresponding to one purchase ahead)
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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?
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10 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 (Q 0,Q 1 )=(0.6,0.4) be the initial probabilities. We will regard Coke as 0 and Pepsi as 1 We want to find P(X 3 =0) P 00
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11 Hidden Markov Models - HMM H1H1 H2H2 H L-1 HLHL X1X1 X2X2 X L-1 XLXL HiHi XiXi Hidden variables Observed data
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