Visible Markov Models

1
Hidden Markov Models (HMMs)

• First Visible VMMs
• Formal Definition
• Recognition
• Training
• HMMs
• Formal Definition
• Recognition
• Training
• Trellis Algorithms
• Forward-Backward
• Viterbi

2
Visible Markov Models

• Probabilistic Automaton
• N distinct states S s1, , sN
• M-element output alphabet K k1, , kM
• Initial state probabilities ? pi, i ? S
• State transition at t 1, 2,
• State trans. probabilities A aij, i,j ? S
• State sequence X X1, , XT, Xt ? S
• Output seq. O o1, , oT, ot ? K

3
VMM Weather Example
4
Generative HMM

• We choose the state sequence probabilistically
• Lets try this using
• the numbers 1-10
• drawing from a hat
• an ad-hoc assignment scheme

5
2 Questions

• Training Problem
• Given an observation sequence O and a space of
  possible models which spans possible values for
  model parameters w A, ?, how do we find the
  model that best explains the observed data?
• Recognition (decoding) problem
• Given a model wi A, ?, how do we compute how
  likely a certain observation is, i.e. P(O wi) ?

6
Training VMMs

• Given observation sequences Os, we want to find
  model parameters w A, ? which best explain
  the observations
• I.e. we want to find values for w A, ? that
  maximises P(O w)
• A, ? chosen argmax A, ? P(O A, ?)

7
Training VMMs

• Straightforward for VMMs
• frequency in state i at time t 1
• (number of transitions from state i to state j)
• --------------------------------------------------
  --------------------------------------------------
  --------------------------------------------------
  -------------------
• (number of transitions from state i)
• (number of transitions from state i to state j)
• --------------------------------------------------
  --------------------------------------------------
  --------------------------------------------------
  -------------------
• (number of times in state i)

8
Recognition

• We need to calculate P(O wi)
• P(O wi) is handy for calculating P(wiO)
• If we have a set of models L w1,w2,,wV then
  if we can calculate P(wiO) we can choose the
  model which returns the highest probability, i.e.
• wchosen argmax wi ? L P(wiO)

9
Recognition

• Why is P(O wi) of use?
• Lets revisit speech for a moment.
• In speech we are given a sequence of
  observations, e.g. a series of LPC vectors
• E.g. LP coefficients taken from frames of length
  20-40ms, every 10-20 ms
• If we have a set of models L w1,w2,,wV then
  if we can calculate P(wiO) we can choose the
  model which returns the highest probability, i.e.
• wchosen argmax wi ? L P(wiO)

10
wchosen argmax wi ? L P(wiO)

• P(wiO) difficult to calculate as we would have
  to have a model for every possible observation
  sequence O
• Use Bayes rule
• P(x y) P (y x) P(x) / P(y)
• So now we have
• wchosen argmax wi ? L P(O wi) P(wi) / P(O)
• P(wi) can be easily calculated
• P(O) is the same for each calculation and so can
  be ignored
• So P(O wi) is the key!!!