This document provides an overview of Hidden Markov Models (HMM). HMMs are statistical models used to model systems where an underlying process produces observable outputs. In HMMs, the observations are modeled as a Markov process with hidden states that are not directly observable, but can only be inferred through the observable outputs. The document describes the key components of HMMs including transition probabilities, emission probabilities, and the initial distribution. Examples of applications like speech recognition and bioinformatics are provided. Finally, common algorithms for HMMs like Forward, Baum-Welch, Backward, and Viterbi are listed for performing inference on the hidden states given observed sequences.

1. Hidden Markov Models
ADVANCED BIOMETRIC
MEVLANA UNIVERSITY, KONYA
Presented
By
Muzammil Abdulrahman
2013

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
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Example: speech signals
Problem: how to construct a model of the
structure or process given only observations

3. Markov Model

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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
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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


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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


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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
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
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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

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Problem is then to build a model to explain
observed sequence of heads and tails

11. HMM Uses

Uses

Speech recognition

Text processing

Gesture classifications

Bioinformatics

12. HMM Algorithms
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They are used to do the inference on Zs given
the sequences of the observed actions
X 1 , X 2 ...... X n
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The ff algorithms are used in HMM
Forward
Estimate the Parameters of HMM (T, E & I) using Baum Welch
Backward
Viterbi

13. Thank You