homework2 - CS
440:
Introduction
to
AI
...

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Unformatted text preview: CS
440:
Introduction
to
AI
 Homework
2
 Due:
Thursday
September
24th
 
 Your
 answers
 must
 be
 concise
 and
 clear.
 Explain
 sufficiently
 that
 we
 can
 easily
 determine
 what
 you
 understand.
 We
 will
 give
 more
 points
 for
 a
 brief
 interesting
 discussion
 with
 no
 answer
than
for
a
bluffing
answer.
 
 Solutions
 will
 be
 posted
 no
 sooner
 than
 two
 days
 after
 the
 due
 date.
 Homework
 will
 be
 accepted
until
that
point
with
a
penalty
of
10%
per
day
that
it
is
late.
 
 No
assignments
will
be
 accepted
after
the
solutions
have
been
posted.

 Late
homework
will
only
be
accepted
in
class,
 during
office
hours,
or
electronically
by
email
to
the
TA.
 You
 are
 expected
 to
 do
 each
 homework
 on
 your
 own.
 You
 may
 discuss
 concepts
 with
 your
 classmates,
but
there
must
be
no
interactions
about
solutions.
You
may
consult
the
web
but
 the
work
handed
in
must
be
done
on
your
own.
 The
 penalty
 for
 cheating
 on
 any
 assignment
 is
 straightforward.
 On
 the
 first
 occurrence,
 you
 will
receive
a
zero
for
the
assignment
and
your
course
grade
will
be
reduced
by
one
full
letter
 grade.
A
second
occurrence
will
result
in
course
failure.
 
 1)
After
your
yearly
checkup,
the
doctor
has
bad
news
and
good
news.
The
bad
news
is
that
you
 tested
 positive
 for
 a
 serious
 disease
 and
 that
 the
 test
 is
 99%
 accurate
 (i.e.,
 the
 probability
 of
 testing
positive
when
you
have
the
disease
is
0.99
and
the
probability
of
testing
negative
when
 you
don’t
have
the
disease
is
0.99).
The
good
news
is
that
this
is
a
rare
disease,
striking
only
1
in
 10,000
individuals.
Why
is
it
good
news
that
the
disease
is
rare?
What
are
the
chances
that
you
 actually
have
the
disease?
 
 2)
 Our
research
lab
is
having
problems
with
his
autonomous
car
and
wants
to
model
the
system
 using
a
Bayesian
network.
The
objective
is
to
avoid
obstacles
by
reducing
the
speed
of
the
car
 and
turning
to
drive
around
the
obstacle.
The
variable
ObstaclePresent
denotes
the
presence
of
 an
 obstacle
 in
 the
 route.
 If
 an
 obstacle
 is
 present
 it
 is
 very
 likely
 that
 the
 obstacle
 detection
 system
 will
 detect
 this
 obstacle.
 ObstacleDetected
 determines
 whether
 and
 obstacle
 was
 detected
or
not.
Besides
obstacles,
the
autonomous
driver
has
to
pay
attention
in
cars
coming
 from
 behind.
 The
 variable
 CarBehind
 indicates
 has
 value
 true
 if
 there
 is
 a
 car
 behind
 our
 autonomous
car.
The
variable
CarDetected
indicates
that
a
car
was
detected
behind
our
car,
and
 it
is
likely
to
be
true
if
there
is
actually
a
car
behind.

 If
 we
 detect
 an
 obstacle
 the
 driver
 is
 likely
 to
 decide
 to
 turn
 around,
 as
 well
 as
 it
 is
 likely
 to
 decide
to
reduce
the
speed.
However,
the
driver
is
also
likely
to
decide
not
to
reduce
the
speed
 if
 it
 detects
 a
 car
 behind.
 DecideToTurn
 indicates
 that
 the
 driver
 decided
 to
 turn
 and
 DecideToReduce
indicates
that
the
driver
decided
to
reduce
the
speed.

 Currently,
one
of
the
big
problems
of
our
autonomous
driver
is
the
communication
between
the
 decision
components
and
the
actuators.
Even
if
the
driver
decides
to
reduce
the
speed
there
is
 chance
that
the
car
will
not
reduce
the
speed
and
even
if
the
driver
decides
to
turn
there
is
a
 chance
that
the
car
will
not
turn.
There
is
also
a
chance
that
the
car
will
reduce
speed
or
turn
 without
 a
 decision
 from
 the
 driver.
 ReducedSpeed
 indicates
 that
 the
 car
 actually
 reduced
 its
 speed
 while
 Turned
 indicates
 whether
 the
 car
 actually
 turned
 or
 not.
 It
 is
 also
 important
 to
 remember
that
the
car
is
likely
to
fail
if
it
attempts
to
turn
while
driving
in
high
speeds.

 a) Use
 the
 ordering
 ObstacleDetected,
 ObstaclePresent,
 Turned,
 DecideToTurn,
 CarBehind,
 ReducedSpeed,
 CarDetected
 and
 DecideToReduce
 to
 build
 a
 Bayesian
 network
to
represent
this
problem.
How
many
parameters
are
necessary
represent
the
 conditional
probabilities
in
this
network?

 b) 
What
 is
 the
 best
 ordering
 of
 variables
 that
 you
 can
 find
 for
 this
 problem?
 Give
 the
 resulting
 graphical
 representation
 of
 the
 Bayesian
 network
 and
 say
 how
 many
 parameters
would
we
need
at
each
node.


 
 
 
 
 
 
 FellOffCliff The robot fell off a cliff. (a) Use the ordering ChangedDirection, DirtDetected, WallDetected, CliffDetected, DirtPresent, WallPresent, CliffPresent, DirtRemaining, BumpedIntoWall, FellOffCliff to build a Bayesian network. (b) How many parameters are needed to represent the conditional probabilities in 1a? (c) The network in 1a may not be optimal. Give a Bayesian network so that the minimal number of parameters are needed to represent the conditional probabilities. (d) How many parameters are needed to represent the conditional probabilities in 1c? 
 3)
Given
the
following
Bayesian
network
of
Boolean
variables:
 2. Given the following Bayesian network of Boolean variables: P (A) A (a) d d d d d © d ‚ A P (B |A) A P (C |A) B C T (b) T (d) d F (c) F (e) d d d d c d ‚© B P (D|B ) B C P (E |B, C ) D E T (f) TT (h) F (g) TF (i) FT (j) FF (k) 
 and
the
joint
probability
table

 and the joint probabilities: B D ¬D E ¬E E ¬E C 0.00084 0.00336 0.00196 0.00784 ¬C 0.00378 0.03402 0.00882 0.07938 A C 0.00378 0.00162 0.00042 0.00018 ¬B B C 0.0096 0.0384 0.0224 0.0896 ¬C 0.03888 0.00972 0.00432 0.00108 1 ¬A ¬C 0.0048 0.0432 0.0112 0.1008 ¬B C ¬C 0.1512 0.1728 0.0648 0.0432 0.0168 0.0192 0.0072 0.0048 Compute the conditional probabilities (a)–(k). Show your work. 3. Given the following Bayesian network of Boolean variables: 
 a)
Compute
the
probabilities
a
to
k
showing
your
work.

 b)

 A P (A) 0.4 A T F 
 B T F ii)
Use
the
results
from
i)
and
a)
to
obtain
a
value
for
the
probability.
 © P (B |A) P (C ) B C 0.3 0.7 iii)
Use
the
values
from
the
joint
to
check
that
i)
and
ii)
are
correct.


 d 0.5 d d d d c d ‚© P (D|B ) B C P (E |B, C ) D E 0.8 TT 0.5 0.7 TF 0.2 FT 0.3 FF 0.6 i)
Calculate
P(E=T|A=T)
(or
P(e|a))
in
terms
of
the
probabilities
 a
to
 k.
(Use
the
letters
 a
 to
k
and
not
the
number
that
they
represent).
 compute the following probabilities and show your work. Round your results to three digits after the decimal point if necessary. 4)
Given
the
Bayesian
network
 
 
 a) Which
 nodes
 constitute
 the
 Markov
 blanket
 of
 node
 D?
 To
 which
 nodes
 D
 is
 conditionally
independent
given
D’s
Markov
blanket?
 b) Which
 minimal
 set
 of
 nodes
 should
 be
 known
 or
 unknown
 in
 order
 for
 D
 and
 E
 to
 be
 conditionally
independent?
Same
thing
for
F
and
G.

 c) Are
the
nodes
B
and
C
independent?
 d) To
which
nodes
E
is
conditionally
independent
given
C?
D
or
G?

 ...
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