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Announcements

Announcements. Midterm scores (without challenge problem): Median 85.5, mean 79, std 16. Roughly, 85-100 ~A, 70-85 ~B, <70 ~C. Hidden Markov Models. Generative, rather than descriptive model. Objects produced by random process. Dependencies in process, some random events influence others.

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Announcements

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  1. Announcements • Midterm scores (without challenge problem): • Median 85.5, mean 79, std 16. • Roughly, 85-100 ~A, 70-85 ~B, <70 ~C.

  2. Hidden Markov Models • Generative, rather than descriptive model. • Objects produced by random process. • Dependencies in process, some random events influence others. • Time is most natural metaphor here. • Simplest, most tractable model of dependencies is Markov. • Lecture based on: Rabiner, “A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition.”

  3. Markov Chain • States: S1, … SN • Discrete time steps, 1, 2, … • State at time t is q(t). • Initial state, q(1). pi(i) = P(q(1) = Si). • P(q(t) = Sj | q(t-1)= Si, q(t-2)=Sk, … ) = P(q(t) = Sj | q(t-1) = Si). This is what makes it Markov. • Time independence: a(ij) = P(q(t) = Sj | q(t-1) = Si).

  4. Examples • 1D random walk in finite space. • 1D curve generated by random walk in orientation.

  5. States of Markov Chain • Represent state at time t as vector: w(t) = (P(q(t)=S1), P(q(t)=S2), … P(q(t) = SN)) • Put transitions, a(ij) into matrix A. • A is Stochastic, meaning columns sum to 1. • Then w(t) = A*w(t-1).

  6. Asymptotic behavior of Markov Chain • w(n) = A(A(…(A(v(1))))) = An(w(1)). • w(n) will be leading eigenvector of A. • This means asymptotic behavior independent of initial conditions • Some special conditions required: • Reach every state from every state (ergodic). • Markov chain may not converge (periodic)

  7. Hidden Markov Model • Observations, v(1), v(2) …, v(M). • We never know the state, but at each time step a state produces an observation. • Observation distribution: b(j,k) = P(v(k) at t| q(t) = Sj). Note this is also taken to be time independent. • Example, HMM that generates contours varying from smooth to rough.

  8. Three problems • Probability of observations given model. • Use to select model given observations (eg, speech recognition). • To refine estimate of HMM. • Given model and observations, what were likely states? • States may have semantics (rough/smooth contour). • May provide intuitions about model. • Find model to optimize probability of observations. • Learning the model.

  9. Probability of Observations • Solved with dynamic programming. • Whiteboard (see Rabiner for notes).

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