QUIZ!!
T/F
: Rejection Sampling without weighting is not consistent.
FALSE
T/F:
Rejection Sampling (often) converges faster than Forward Sampling.
FALSE
T/F:
Likelihood weighting (
often
) converges faster than Rejection Sampling.
TRUE
T
/F:
The Markov Blanket
of X contains other children
of
parents of X.
FALSE
T/F:
The Markov Blanket
of X contains other parents of children of X.
TRUE
T/F:
GIBBS sampling requires you to weight samples by their likelihood.
FALSE
T
/F:
In GIBBS sampling, it is a good idea to reject the first M<N samples.
TRUE
Decision Networks:
T/F:
Utility nodes never have parents.
FALSE
T/F:
Value of Perfect Information (VPI) is always non

negative.
TRUE
1
CSE
511
a:
Artificial Intelligence
Spring
2013
Lecture 19: Hidden Markov Models
04/10/2013
Robert
Pless
Via
Kilian
Q.
Weinberger, slides
adapted from Dan Klein
–
UC
Berkeley
Recap: Decision Diagrams
Weather
Forecast
Umbrella
U
A
W
U
leave
sun
100
leave
rain
0
take
sun
20
take
rain
70
W
P(W)
sun
0.7
rain
0.3
F
P(Frain)
good
0.1
bad
0.9
F
P(Fsun)
good
0.8
bad
0.2
Example:
MEU decisions
4
Weather
Forecast
=bad
Umbrella
U
A
W
U(A,W)
leave
sun
100
leave
rain
0
take
sun
20
take
rain
70
W
P(WF=bad)
sun
0.34
rain
0.66
Umbrella = leave
Umbrella = take
Optimal decision = take
Value of Information
Assume we have evidence E=e. Value if we act now:
Assume we see that E
’
= e
’
. Value if we act then:
BUT
E
’
is a random variable whose value
is
unknown
, so we don
’
t know
what e
’
will
be.
Expected
value if E
’
is revealed and then we act:
Value of information: how much MEU goes up
by revealing E
’
first:
VPI == “Value of perfect information”
VPI Example: Weather
6
Weather
Forecast
Umbrella
U
A
W
U
leave
sun
100
leave
rain
0
take
sun
20
take
rain
70
MEU with no evidence
MEU if forecast is bad
MEU if forecast is good
F
P(F)
good
0.59
bad
0.41
Forecast distribution
VPI Properties
Nonnegative
Nonadditive

consider, e.g., obtaining
E
j
twice
Order

independent
7
Now for something completely different
8
“Our youth now love luxury. They
have bad manners, contempt for
authority; they show disrespect for
their elders and love chatter in place
of exercise; they no longer rise when
elders enter the room; they contradict
their parents, chatter before
company; gobble up their food and
tyrannize their teachers.”
9
“Our youth now love luxury. They
have bad manners, contempt for
authority; they show disrespect for
their elders and love chatter in place
of exercise; they no longer rise when
elders enter the room; they contradict
their parents, chatter before
company; gobble up their food and
tyrannize their teachers.”
–
Socrates
469
–
399
BC
10
Adding time!
11
Reasoning over Time
Often, we want to
reason about a sequence
of
observations
Speech recognition
Robot localization
User attention
Medical monitoring
Need to introduce time into our models
Basic approach: hidden Markov models (HMMs)
More general: dynamic Bayes
’
nets
12
Markov Model
13
Markov Models
A
Markov model
is a chain

structured BN
Each node is identically distributed (
stationarity
)
Value of X at a given time is called the
state
As a BN:
….
P(X
t
X
t

1
)…..
Parameters
: called
transition probabilities
or
dynamics, specify how the state evolves over time
(also, initial
probs
)
X
2
X
1
X
3
X
4
Conditional Independence
Basic conditional independence:
Past and future independent of the present
Each time step only depends on the previous
This is called the (first order) Markov property
Note that the chain is just a (growing) BN
We can always use generic BN reasoning on it if we
truncate the chain at a fixed length
X
2
X
1
X
3
X
4
15
Example: Markov Chain
Weather:
States: X = {rain, sun}
Transitions:
Initial distribution:
1.0
sun
What
’
s the probability distribution after one step?
rain
sun
0.9
0.9
0.1
0.1
This is a
CPT, not a
BN!
16
Mini

Forward Algorithm
Question: What
’
s P(X) on some day t?
An instance of variable elimination!
sun
rain
sun
rain
sun
rain
sun
rain
Forward simulation
18
Example
From initial observation of sun
From initial observation of rain
P(
X
1
)
P(
X
2
)
P(
X
3
)
P(
X
)
P(
X
1
)
P(
X
2
)
P(
X
3
)
P(
X
)
19
Stationary Distributions
If we simulate the chain long enough:
What happens?
Uncertainty accumulates
Eventually, we have no idea what the state is!
Stationary distributions:
For most chains, the distribution we end up in is
independent of the initial distribution
Called the
stationary distribution
of the chain
Usually, can only predict a short time out
Hidden Markov Model
22
Hidden Markov Models
Markov chains not so useful for most agents
Eventually you don
’
t know anything anymore
Need observations to update your beliefs
Hidden Markov models (HMMs)
Underlying Markov chain over states S
You observe outputs (effects) at each time step
As a Bayes
’
net:
X
5
X
2
E
1
X
1
X
3
X
4
E
2
E
3
E
4
E
5
Example
An HMM is defined by:
Initial distribution:
Transitions:
Emissions:
Ghostbusters HMM
P(X
1
) = uniform
P(XX
’
) = usually move clockwise, but
sometimes move in a random direction or
stay in place
P(R
ij
X) = same sensor model as before:
red means close, green means far away.
1/9
1/9
1/9
1/9
1/9
1
/
9
1
/
9
1
/
9
1/9
P(X
1
)
P(XX
’
=<1,2>)
1/6
1/6
0
1
/
6
1
/
2
0
0
0
0
X
5
X
2
R
i,j
X
1
X
3
X
4
R
i,j
R
i,j
R
i,j
E
5
Conditional Independence
HMMs have two important independence properties:
Markov hidden process, future depends on past via the present
Current observation independent of all else given current state
Quiz: does this mean that observations are independent
given no evidence?
[No, correlated by the hidden state]
X
5
X
2
E
1
X
1
X
3
X
4
E
2
E
3
E
4
E
5
Real HMM Examples
Speech recognition HMMs:
Observations are acoustic signals
(continuous valued)
States are specific positions in specific
words (so, tens of thousands)
Machine translation HMMs:
Observations are words (tens of thousands)
States are translation options
Robot tracking:
Observations are range readings
(continuous)
States are positions on a map (continuous)
Filtering / Monitoring
Filtering, or monitoring, is the task of tracking the
distribution B(X) (the belief state) over time
We start with B(X) in an initial setting, usually uniform
As time passes, or we get observations, we update B(X)
The Kalman filter was invented in the
60
’
s and first
implemented as a method of trajectory estimation for the
Apollo program
Example: Robot Localization
t=
0
Sensor model: never more than
1
mistake
Motion model: may not execute action with small prob.
1
0
Prob
Example from
Michael Pfeiffer
Example: Robot Localization
t=1
1
0
Prob
Example: Robot Localization
t=2
1
0
Prob
Example: Robot Localization
t=
3
1
0
Prob
Example: Robot Localization
t=4
1
0
Prob
Example: Robot Localization
t=5
1
0
Prob
Inference Recap: Simple Cases
E
1
X
1
X
2
X
1
Passage of Time
Assume we have current belief P(X  evidence to date)
Then, after one time step passes:
Or, compactly:
Basic idea: beliefs get
“
pushed
”
through the transitions
With the
“
B
”
notation, we have to be careful about what time
step t the belief is about, and what evidence it includes
X
2
X
1
Example: Passage of Time
As time passes, uncertainty
“
accumulates
”
T = 1
T = 2
T = 5
Transition model: ghosts usually go clockwise
Observation
Assume we have current belief P(X  previous evidence):
Then:
Or:
Basic idea: beliefs reweighted by likelihood of evidence
Unlike passage of time, we have to renormalize
E
1
X
1
Example: Observation
As we get observations, beliefs get
reweighted, uncertainty
“
decreases
”
Before observation
After observation
Example HMM
The Forward Algorithm
We are given evidence at each time and want to know
We can derive the following updates
We can normalize
as we go if we
want to have
P(xe) at each
time step, or just
once at the end…
Online Belief Updates
Every time step, we start with current P(X  evidence)
We update for time:
We update for evidence:
The forward algorithm does both at once (and doesn
’
t normalize)
Problem: space is X and time is X
2
per time step
X
2
X
1
X
2
E
2
Next Lecture:
Sampling! (
Particle Filtering)
44
Enter the password to open this PDF file:
File name:

File size:

Title:

Author:

Subject:

Keywords:

Creation Date:

Modification Date:

Creator:

PDF Producer:

PDF Version:

Page Count:

Preparing document for printing…
0%
Comments 0
Log in to post a comment