1.Consider the following scenario for object recognition with two types of objects, A and B where the ultimate goal is to build a system that can recognize which of the two object types is present in front of a camera. To recognize the objects we make observations of 5 different properties of the objects, shape, color, size, texture, and labeling. Here we measure shape as either round or square, color as red or blue, size as small, medium, or large, texture as uniform or striped, and labeling as labeled or unlabeled. Making observations of the two object types, we find the following:

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7 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

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1.

Consider the following scenario for object recognition with two types of objects,
A

and
B

where the ultimate
goal is to build a system that can recognize which of the two object types is present in front of a camera. To
recognize the objects we make observations of 5 different properties of the objects,
shape
,
color
,
size
,
texture
, and
labelin
g
. Here we measure
shape

as either
round
or

square
,

color
as

red
or

blue
,

size
as

small
,

medium
, or

large
,

texture
as

uniform
or
striped
, and
labeling

as
labeled

or
unlabeled
.



Making observations of the two object types, we find the following:



Of all

the objects,
70%

are of
type A

and
30%

of
type B
.
80%

of objects of
type A

are
large
,
10%
are

medium
, and

10%
are

small
. Also,

40%
are

red
while

60%
are blue. Of objects of

type B
,

30%
are

large
,

50%
are

medium
, and

20%
are

small
. Similarly,

80%
are

red
while

20%
are

blue
. For both object types there
is no dependence between their size and their color.

Of all the the
small

objects,
75%

are
round

while of the
medium

objects only
35%

are round and only
10%

of the
large

objects. The texture depends on the
co
lor

and
the
size

where the probability of a
uniform

texture is
30%

for
small

and
red

objects,
10%

for
small

and
blue

objects,
75%

for
medium

and
red

objects,
40%

for
medium

and
blue

objects,
90%

for
large

and
red

objects,
and
60%

for
large

and
blue

objects
. The likelihood of labeling on the objects is related to the texture and the
shape. In particular,
80%

of all
square

and
uniform

objects,
40%
of the

round
and

uniform
objects,

30%
of all

square
and

striped
objects, and

10%
of

all
round

and
striped

objects have labeling.


a)

Construct a Bayesian Network for the object recognition scenario that encodes the


information we have about the objects (you can assume that only the direct dependencies listed above
are present). Provide the structure of the network as well as the conditional probability tables.


b)

To perform object recognition with the network we woul
d set the observations that we made and then
infer the conditional probability of the different object types. Derive the probability of object type
A

given that we observe that the object is
unlabeled
,
small
, and
red
. Similarly, derive the probability of
o
bject type

A
given that we can observe that the

object is
striped
, and
large
.


c)

Using the network we can also infer properties of the objects that we could not observe before. Infer
the probability of an object being
labeled

given that it is
blue

and
round
. Also compute the prior
probability of an object being
labeled
.


d)

Bayesian Networks are not unique and can be rearranged into a different structure that nevertheless
represents the same joint probability distribution. For the network in a), build an equivalent network in
which the node for object
type

is a leaf node. In
particular, build a network with the node order
shape
,
color
,
size
,
texture
,
labeling
,
type
, where the order indicates that only nodes earlier in the list can be
parents to nodes later in the list. You should derive both the new structure and the new condi
tional
probability tables.


2.

Monte
-
Carlo simulations can be used to solve expected value problems.


a)

Implement random sampling from the exponential distribution and use Monte
-
Carlo simulation to
compute the
mean
,
variance

and
skewness

of the exponential distribution with
λ

= 0
.
5
. Show your
estimates after 10, 50, and 100 samples.



b)

Use Monte
-
Carlo simulation to visualize the central limit theorem by simulating the distribution of the
means from the exponential and uniform distribution. In particular, for each of these original
distributions sample averages over 5, 15, and 30 samples
by drawing the corresponding number of
samples and computing their average. Repeat this 100 times and plot the histograms of the resulting
distributions of the means.