6.01:
Introduction to EECS 1
Week 10
November 10, 2009
1
6.01: Introduction to EECS I
Bayesian estimation, etc.
Week 10
November 10, 2009
What about the bet?
•
Total number of legos in the bag
= 4
•
Random variable
R
: number of red legos in the bag.
•
Domain
D
R
=?
•
Assume
uniform prior
on
R
(all values equally likely):
•
Random variable
L
0
: color of ﬁrst lego we draw out of the bag
•
Observation model:
Pr(
L
0
= red

R
=
n
) =?
Pr(
L
0
= white

R
=
n
) = 1
−
Pr(
L
0
= red

R
=
n
)
•
We want to know:
Pr(
R
=
n

L
0
=
whatever color we observed
)
Bayes!!
What do we know after ﬁrst Lego draw?
0
1
Number of Red Legos
normalize
4
3
2
divide by sum
Pr(
R
=
r
)
Pr(
L
0
=
l
0

R
=
r
)
Pr(
L
0
=
l
0
,R
=
r
)
Pr(
R
=
r

L
0
=
l
0
)
What do we know after second Lego draw?
0
1
Number of Red Legos
normalize
4
3
2
divide by sum
Pr(
L
1
=
l
1

R
=
r, L
0
=
l
0
)
= Pr(
L
1
=
l
1

R
=
r
)
Pr(
L
1
=
l
1
,R
=
r

L
0
=
l
0
)
Pr(
R
=
r

L
0
=
l
0
,L
1
=
l
1
)
Pr(
R
=
r

L
0
=
l
0
)
Hidden Markov Models
System with a state that changes over time, probabilistically.
•
Discrete time steps
0
,
1
,...,t
•
Random variables for states at each time:
S
0
,S
1
,S
2
,...
•
Random variables for observations:
O
0
,O
1
,O
2
,...
State at time
t
determines the probability distribution:
•
over the observation at time
t
•
over the state at time
t
+ 1
Hidden Markov Models
System with a state that changes over time, probabilistically.
•
Discrete time steps
0
,
1
,...,t
•
Random variables for states at each time:
S
0
,S
1
,S
2
,...
•
Random variables for observations:
O
0
,O
1
,O
2
,...
•
Initial state distribution:
Pr(
S
0
=
s
)
•
State transition model:
Pr(
S
t
+1
=
s

S
t
=
r
)
•
Observation model:
Pr(
O
t
=
o

S
t
=
s
)
Inference problem: given actual sequence of observations
o
0
,...,o
t
,
compute
Pr(
S
t
+1
=
s

O
0
=
o
0
,...,O
t
=
o
t
)