Machine Learning

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Machine Learning

• Hidden Markov Models

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The Markov Property

• A stochastic process has the Markov property if
  the conditional probability of future states of
  the process, depends only upon the present state.
• i.e. what Im likely to do next
• depends only on where I am
• now, NOT on how I got here.
• P(qt qt-1,,q1) P(qt qt-1)
• Which processes have the Markov property?

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Markov model for Dow Jones
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The Dishonest Casino

• A casino has two dice
• Fair die
• P(1) P(2) P(5) P(6) 1/6
• Loaded die
• P(1) P(2) P(5) 1/10 P(6) ½
• I think the casino switches back and forth
  between fair and loaded die once every 20 turns,
  on average

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My dishonest casino model
This is a hidden Markov model (HMM)
0.05
0.95
0.95
FAIR
LOADED
P(1F) 1/6 P(2F) 1/6 P(3F) 1/6 P(4F)
1/6 P(5F) 1/6 P(6F) 1/6
P(1L) 1/10 P(2L) 1/10 P(3L) 1/10 P(4L)
1/10 P(5L) 1/10 P(6L) 1/2
0.05
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Elements of a Hidden Markov Model

• A finite set of states Q q1, ..., qK
• A set of transition probabilities between states,
  A
• each aij, in A is the prob. of going from state
  i to state j
• The probability of starting in each state P
  p1, , pK each pK in P is the probability of
  starting in state k
• A set of emission probabilities, B
• where each bi(oj) in B is the probability of
  observing output oj when in state i

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My dishonest casino model
This is a HIDDEN Markov model because the states
are not directly observable. If the fair die
were red and the unfair die were blue, then the
Markov model would NOT be hidden.
0.05
0.95
0.95
FAIR
LOADED
0.05
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HMMs are good for

• Speech Recognition
• Gene Sequence Matching
• Text Processing
• Part of speech tagging
• Information extraction
• Handwriting recognition

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The Three Basic Problems for HMMs

• Given observation sequence O(o1o2oT), of
  events from the alphabet ?, and HMM model ?
  (A,B,?)
• Problem 1 (Evaluation)
• What is P(O ?), the probability of the
  observation sequence, given the model
• Problem 2 (Decoding)
• What sequence of states Q(q1q2qT) best explains
  the observations
• Problem 3 (Learning)
• How do we adjust the model parameters ? (A,B,?)
  to maximize P(O ? )?

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The Evaluation Problem

• Given observation sequence O and HMM ?, compute
  P(O ?)
• Helps us pick which model is the best one

O 1,6,6,2,6,3,6,6
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Computing P(O?)

• Naïve Try every path through the model
• Sum the probabilities of all possible paths
• This can be intractable. O(NT)
• What we do instead
• The Forward Algorithm. O(N2T)

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The Forward Algorithm
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The inductive step,

• Computation of ?t(j) by summing all previous
  values ?t-1(i) for all i

A hidden state at time t-1
transition probability
?t-1(i)
?t(j)
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Forward Algorithm Example
Model
P(1F) 1/6 P(2F) 1/6 P(3F) 1/6 P(4F)
1/6 P(5F) 1/6 P(6F) 1/6
P(1L) 1/10 P(2L) 1/10 P(3L) 1/10 P(4L)
1/10 P(5L) 1/10 P(6L) 1/2
Start prob
P (fair) .7 P (loaded) .3
Observation sequence 1,6,6,2
?2(i)
?1(i)
?3(i)
?4(i)
?1(1)0.051/6 ?1(2)0.051/6
?2(1)0.051/6 ?2(2)0.051/6
?3(1)0.051/6 ?3(2)0.051/6
0.71/6
State 1 (fair)
?3(1)0.951/10 ?3(2)0.951/10
?2(1)0.951/2 ?2(2)0.951/2
?1(1)0.951/2 ?1(2)0.951/2
State 2 (loaded)
0.31/10
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Markov model for Dow Jones
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Forward trellis for Dow Jones
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The Decoding Problem

• What sequence of states Q(q1q2qT) best explains
  the observation sequence O(o1o2oT)?
• Helps us find the path through a model.

ART
N
V
ADV
The dog sat quietly
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The Decoding Problem

• What sequence of states Q(q1q2qT) best explains
  the observation sequence O(o1o2oT)?
• Viterbi Decoding
• slight modification of the forward algorithm
• the major difference is the maximization over
  previous states
• Note Most likely state sequence is not the same
  as the sequence of most likely states

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The Viterbi Algorithm
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The Forward inductive step

• Computation of at(j)

ot-1
ot
at-1(j)
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The Viterbi inductive step

• Computation of vt(j)

Keep track of who the predecessor was at each
step.
ot-1
ot
vt-1(i)
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Viterbi for Dow Jones
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The Learning Problem

• Given O, how do we adjust the model parameters ?
  (A,B,?) to maximize P(O ? )?
• In other words How do we make a hidden Markov
  Model that best models the what we observe?

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Baum-Welch Local Maximization

• 1st step You determine
• The number of hidden states, N
• The emission (observation alphabet)
• 2nd step randomly assign values to
• A - the transition probabilities
• B - the observation (emission) probabilities
• - the starting state probabilities
• 3rd step Let the machine re-estimate
• A, B, p

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Estimation Formulae
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Learning transitions
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Math
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Estimation of starting probs.
This is number of transitions from i at time t
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Estimation Formulae
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Estimation Formulae
k
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What are we maximizing again?
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The game is

• EITHER the current model is at a local maximum
  and
• reestimate current model
• OR our reestimate will be slightly better and
• reestimate ! current model
• SO we feed in the reestimate as the current
  model, over and over until we cant improve any
  more.

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Caveats

• This is a kind of hill-climbing technique
• Often has serious problems with local maxima
• You dont know when youre done

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Sohow else could we do this?

• Standard gradient descent techniques?
• Hill climb?
• Beam search?
• Genetic Algorithm?

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Back to the fundamental question

• Which processes have the Markov property?
• What if a hidden state variable is included?(an
  in an HMM)