# Machine Learning - PowerPoint PPT Presentation

PPT – Machine Learning PowerPoint presentation | free to download - id: 6e4e9b-M2VlM The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
Title:

## Machine Learning

Description:

### Machine Learning Hidden Markov Models Doug Downey, adapted from Bryan Pardo,Northwestern University – PowerPoint PPT presentation

Number of Views:12
Avg rating:3.0/5.0
Slides: 36
Provided by: BryanP157
Category:
Tags:
Transcript and Presenter's Notes

Title: Machine Learning

1
Machine Learning
• Hidden Markov Models

2
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?

3
Markov model for Dow Jones
4
The Dishonest Casino
• A casino has two dice
• Fair die
• P(1) P(2) P(5) P(6) 1/6
• 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

5
My dishonest casino model
This is a hidden Markov model (HMM)
0.05
0.95
0.95
FAIR
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
6
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

7
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
0.05
8
HMMs are good for
• Speech Recognition
• Gene Sequence Matching
• Text Processing
• Part of speech tagging
• Information extraction
• Handwriting recognition

9
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 ? )?

10
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
11
Computing P(O?)
• Naïve Try every path through the model
• Sum the probabilities of all possible paths
• This can be intractable. O(NT)
• The Forward Algorithm. O(N2T)

12
The Forward Algorithm
13
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)
14
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
0.31/10
15
Markov model for Dow Jones
16
Forward trellis for Dow Jones
17
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
The dog sat quietly
18
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

19
The Viterbi Algorithm
20
The Forward inductive step
• Computation of at(j)

ot-1
ot
at-1(j)
21
The Viterbi inductive step
• Computation of vt(j)

Keep track of who the predecessor was at each
step.
ot-1
ot
vt-1(i)
22
Viterbi for Dow Jones
23
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?

24
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

25
Estimation Formulae
26
Learning transitions
27
Math
28
Estimation of starting probs.
This is number of transitions from i at time t
29
Estimation Formulae
30
Estimation Formulae
k
31
What are we maximizing again?
32
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.

33
Caveats
• This is a kind of hill-climbing technique
• Often has serious problems with local maxima
• You dont know when youre done

34
Sohow else could we do this?