CS221 Section 8
1
CS 221
Section 8: The Filtering Algorithm
1.
The
Maximum a Posteriori
Estimate Chocolate Bar
We return to the example of the Bayesian chocolate company. The bosses have heard of
your success calculating the probability of the lever positions given the observations of
the chocolate colors.
As it is, the company brands their chocolate as the “Completely
Random Chocolate Bar,” since the faulty levers mean they can’t know what the chocolate
contains. But now they want to rebrand it as the “Most Likely
Chocolate Bar.” The
idea is they will label the chocolate as “Most Likely Almond,” “Most Likely Coconut,”
etc., depending on what you calculated was the
maximum a posteriori
(MAP) label of
the chocolate coming off the assembly line at that time. They hope this will attract more
customers versus their old “Completely Random Chocolate.”
You must automate the process of calculating the assignment probabilities. The factory
will be making tens of thousands of chocolate bars a day, so you don’t have time to
manually calculate probabilities for each one of them. You must also be able to calculate a
chocolate bar’s label immediately after you make the observation of whether it is light or
dark chocolate, as the labeling machine comes up next and you must tell it what to print.
But you have forgotten about the filtering algorithm that you learned about in CS221.
You decide to start rederiving probabilities and see if you spot a pattern. You will switch
notation from using the variables
X
and
Y
to using
s
and
o
for your states (lever positions)
and observations (chocolate shades), respectively. To review, you only know the transition
model
P
(
s
t
+1

s
t
) (how the levers will change position from one step to another), the
observation probabilities
P
(
o
t

s
t
) (what shade of chocolate you observe given its color),
and you have a uniform prior over the first state
P
(
s
0
). Computing the MAP label for a
bar of chocolate at time
t
will be equivalent to computing
P
(
s
t

o
1:
t
), where
o
1:
t
represents
all of your observations from the time the factory started making chocolate that day.
(a) Derive the formula for the MAP label for the first chocolate bar off the line, given its
chocolate type. I.e., derive
P
(
s
1

o
1
) where
s
1
is the state of the levers, and
o
1
is the
test outcome saying whether the chocolate is light or dark.
Also show how you would efficiently compute this with matrix and vector operations.
Assume you have a uniform prior over the lever positions once the factory starts up.
Use the matrix,
O
, which contains all of the observation probabilities,
P
(
o
t

s
t
). In
particular, each column corresponds to a different observation, and each row corre
sponds to a different state:
O
=
0
.
1
0
.
9
0
.
3
0
.
7
0
.
8
0
.
2
0
.
9
0
.
1
This follows directly from the original problem statement:
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CS221 Section 8
2
Inside
P(Color=Light (L)

Inside)
P(Color=Dark (D)

Inside)
Plain (N)
0.1
0.9
Almond (A)
0.3
0.7
Coconut (C)
0.8
0.2
Almond + Coconut (B)
0.9
0.1
Answer:
We begin by applying Bayes’ Rule:
P
(
s
1

o
1
) =
P
(
o
1

s
1
)
P
(
s
1
)
P
(
o
1
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 '09
 KOLLER,NG
 Artificial Intelligence, lever, CS221 Section, lever positions

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