2. HMM Motivation
Real-world has structures and processes which
have (or produce) observable outputs
Usually sequential (process unfolds over time)
Cannot see the event producing the output
Example: speech signals
Problem: how to construct a model of the
structure or process given only observations
3. Markov Model
The future is independent on the past given the
present
Let X 1 , X 2 ...... X n be discrete random
variables of states, then the Markov chain can be
represented as
X1
X2
X3
…
Xn
P( X t | X 1 , X 2 ,K , X t −1 ) = P ( X t | X t −1 )
4. Markov Model Example
State transition matrix
Weather
Once each day weather is observed
Cloudy
Sunny
Rainy
0.4
0.3
0.3
Cloudy
0.2
0.6
0.2
Sunny
0.1
0.1
0.8
What is the probability the weather for
the next 7 days will be:
State 1: rain
State 2: cloudy
State 3: sunny
Rainy
sun, sun, rain, rain, sun, cloudy, sun
Each state corresponds to a physical
observable event
5. MM
Then What is the Problem of MM ?
Some informations will be missing
Why ??
We can’t expect to perfectly observe the
complete states of the systems
6. HMM
The solution to the missing informations will be solve by HMM
It’s a Sequential Model
Let Z1 , Z 2 ......Z n be a hidden random variables and
X 1 , X 2 ...... X n be the observed states
Z1
Z2
Z3
Zn
X1
X2
X3
Xn
The Trellis Diagram of HMM
7. HMM
The joined prob. of the above variables is
n
p( X t ... X n , Z1,K , Z n ) = p( Z1) p( X1 / Z1)π k = 2 p( Z k / Z k −1) p( X k / Z k )
8. HMM Parameters
Transition probs.
Emission probs.
Initial Distribution
Xs are the hidden states
Ys are the observed states
a’s are the Transition probs.
b’s are the emission probs.
The initial prob is sometimes assumed to be 1
9. HMM Examples
• Typed word recognition, assume all characters are separated.
• Character recognizer outputs probability of the image being particular character,
P(image|character).
a
0.5
b
0.03
c
0.005
0.31
z
Hidden state
Observation
10. HMM Examples
Coin toss:
Heads, tails sequence with 2 coins
You are in a room, with a wall
Person behind wall flips coin, tells result
Coin selection and toss is hidden
Cannot observe events, only output (heads, tails) from
events
Problem is then to build a model to explain
observed sequence of heads and tails
12. HMM Algorithms
They are used to do the inference on Zs given
the sequences of the observed actions
X 1 , X 2 ...... X n
The ff algorithms are used in HMM
Forward
Estimate the Parameters of HMM (T, E & I) using Baum Welch
Backward
Viterbi