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Foundations of Statistical NLP Chapter 9. Markov Models 한 기 덕한 기 덕

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2 Contents Introduction Markov Models Hidden Markov Models –Why use HMMs –General form of an HMM –The Three Fundamental Questions for HMMs Fundamental Questions For HMMs Implementation, Properties, and Variants

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3 Introduction Markov Model –Markov processes/chains/models were first developed by Andrei A. Markov –First use linguistic purpose : modeling the letter sequences in Russian literature(1913) –Current use general statistical tool VMM (Visible Markov Model) –Words in sentences is depend on their syntax. HMM (Hidden Markov Model) –operate high level abstraction by postulating additional “hidden” structures.

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4 Markov Models Markov assumption –Future elements of the sequence independent of past elements, given the present element. Limited Horizon –X t = sequence of random variables –S k = state space Time invariant (stationary)

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5 Markov Models(Cont’) Notation –stochastic transition –probability of different initial state Application : Linear sequence of events –modeling valid phone sequences in speech recognition –sequences of speech acts in dialog systems

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6 Markov Chain circle : state and state name arrows connecting states : possible transition arc label : probability of each transition

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7 Visible Markov Model We know what states the machine is passing through. m th order Markov model –n 3, n-gram violate Limited Horizen condition –reformulate any n-gram model as a visible Markov model by simply encoding (n-1)-gram

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8 Hidden Markov Model We don’t know the state sequence that the model passes through, only some probabilistic function of it Example 1 : The crazy soft drink machine –two state : cola preferring(CP), iced tea preferring(IP) –VMM : machine always put out a cola in CP –HMM : emission probability –Output probability given From state

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9 Crazy soft drink machine Problem –What is the probability of seeing the output sequence {lem, ice-t} if the machine always start off in the cola preferring state?

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10 Crazy soft drink machine(Cont’)

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11 Why use HMMs? underlying events probabilistically generate surface events –the words in a text parts of speech Linear interpolation of n-gram Hidden state –the choice of whether to use the unigram, bigram, or trigram probabilities. Two Keys –This is conversion works by adding epsilon transitions. –Separate parameters iab don’t adjust them separately.

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13 Notation A B AAA BB SSS KKK S K S K

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14 General form of an HMM Arc-emission HMM –the symbol emits at time t depends on both the state at time t and at time(t+1). State-emission HMM : ex) crazy drink machine –the symbol emits at time t depends just on the state at time t Figure 9.4 A program for a Markov process.

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15 The Three Fundamental Questions for HMMS

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16 Finding the probability of an observation

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17 The forward procedure Cheap algorithm required only 2N 2 T multiplication

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18 The backward procedure The total probability of seeing the rest of the observation sequence. Use of a combination of forward and backward probabilities is vital for solving the third problem of parameter reestimation. Backward variables Combining forward & backward

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19 Finding the best state sequence State sequence that explains the observations is more than one way. Find X t that maximizes P(X t |O, ) This may yield a quite unlikely state sequence. Viterbi algorithm is more efficient.

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20 Viterbi algorithem The most likely complete path This is sufficient to maximize for a fixed O Definition

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21 Variable calculations for O = (lem, ice_t, cola)

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22 Parameter estimation Given a certain observation sequence Find the values of the model parameter = (A, B, ) Using Maximum Likelihood Estimation Locally maximize by an iterative hill-climbing algorithm usually effective for HMM

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23 Parameter estimation (Cont’)

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24 Parameter estimation (Cont’)

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25 Implementation, Properties, Variants Implementation –Obvious issue : keeping on multiplying very small numbers Use Log function Variants –It is not impossible to estimate many number parameter. Multiple input observations Initialization of parameter values –Try to approach near global maximum

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