CS621: Artificial Intelligence

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CS621 Artificial Intelligence

• Pushpak BhattacharyyaCSE Dept., IIT Bombay
• Lecture 38-39 Baum Welch Algorithm HMM training

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Baum Welch algorithm

• Training Hidden Markov Model (not structure
  learning, i.e., the structure of the HMM is
  pre-given). This involves
• Learning probability values ONLY
• Correspondence with PCFG
• Not learning production rule but probabilities
  associated with them
• Training algorithm for PCFG is called
  Inside-Outside algorithm

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Key Intuition

• Given Training sequence
• Initialization Probability values
• Compute Pr (state seq training seq)
• get expected count of transition
• compute rule probabilities
• Approach Initialize the probabilities and
  recompute them EM like approach

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Building blocks Probabilities to be used

W1
W2 Wn-1
Wn
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Probabilities to be used, contd

• Exercise 1- Prove the following

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Start of baum-welch algorithm
b
b
r
q
a
a

• String aab aaa aab aaa
• Sequence of states with respect to input symbols

o/p seq
State seq
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• Calculating probabilities from table
• Table of counts
• Tstates
• Aalphabet symbols
• Now if we have a non-deterministic transitions
  then multiple state seq possible for the given
  o/p seq (ref. to previous slides feature). Our
  aim is to find expected count through this.

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Interplay Between Two Equations

wk No. of times the transitions si?sj occurs in
the string
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Learning probabilities
a0.67
b0.17
q
r
a0.16
b1.0
Actual (Desired) HMM
a0.4
b0.48
q
r
a0.48
b1.0
Initial guess
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One run of Baum-Welch algorithm string ababa
State sequences
is considered as starting and ending
symbol of the input sequence string
This way through multiple iterations the
probability values will converge.
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Appling Naïve Bayes
Hence multiplying the transition probabilities is
valid
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Discussions

• Symmetry breaking
• Example Symmetry breaking leads to no change in
  initial values
• Struck in Local maxima
• Label bias problem
• Probabilities have to sum to 1.
• Values can rise at the cost of fall of values
  for others.

a0.5
b0.25
a0.25
b1.0
a0.5
a0.5
b0.5
a1.0
b0.5
a0.25
b0.5
b0.5
Desired
Initialized
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Computational part
Exercise 2 What is the complexity of calculating
the above expression? Hint To find this first
solve Exercise 1 i.e. understand how probability
of given string can be represented as