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Homework answers / question archive / There is some Bayes’ net structure over three variables which can represent any given probability distribution over those variables
There is some Bayes’ net structure over three variables which can represent any given probability distribution over those variables.
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? True
? False
Q1.2
2 Points
For a Markov Chain Monte Carlo Method to work properly, the
posterior distribution being estimated must be Gaussian.
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Q1.3
2 Points
Given any probability distribution over N variables, every Bayes’ net
with N nodes can represent that distribution.
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Q1.4
2 Points
When deploying a multi-agent system, all individual agents must share
identical internal algorithms and models of the environment.
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Q1.5
2 Points
? True
? False
? True
? False
? True
? False
All conditional independence properties of the probability distribution
represented by a Bayes’ net are determined by the graph structure.
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Q1.6
2 Points
Given that a Bayes’ net with N nodes that can be design represent any
probability distribution over N variables, that same network can
represent any marginal distribution over M variables where M ≤ N.
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Q1.7
2 Points
In the worst case, the complexity of inference could be exponential in
the size (number of variables) of a Bayes’ net.
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Q1.8
2 Points
The smoothing algorithm for HMMs returns the single most likely
sequence of hidden states given the observations.
? True
? False
? True
? False
? True
? False
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Q1.9
2 Points
A particle filter with n particles at time t, {pt1 , pt2 . . . ptn } uses each pti
to generate a new particle for time t + 1.
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Q1.10
2 Points
All Markov Chains have a stationary distribution.
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Q2 Multiple Choice
20 Points
Select all that apply. 4 points each, 20 points total.
Q2.1
4 Points
Which of the type of problem would be appropriate for a Kalman filter?
? True
? False
? True
? False
? True
? False
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Q2.2
4 Points
The Viterbi Algorithm would be used for which of these problems:
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Q2.3
4 Points
Markov Chain Monte Carlo
Tracking a bouncing ball
Automating control of a space ship that has to dock with another
Determining your location in a maze using only sonar readings
Calculating likelihood if it raining given that I’ve seen an umbrella Tracking a moving target to find next location
Decoding an audio stream into the most likely text
Transforming analog radio signals into bit sequences
Finding out the most likely explanation for the price of a stock
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Q2.4
4 Points
Which of these are inference tasks in temporal models:
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Q2.5
4 Points
Algorithms such as Gibbs Sampling and Metropolis-Hastings, uses samples which, in the long run, end up converging to the posterior probability is an approximation algorithm enumerates the complete joint probability table iv. is similar to Hill Climbing filtering variable elimination prediction smoothing normalization
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Q3 Bayes' Nets
21 Points
(28 points total, 4 points each) The Whizzo Chocolate Company (as described at https://www.youtube.com/watch?v=7-UssHVuCys) makes the The Whizzo Quality Assortment candy box that has several flavors of candy: 30% are Crunchy Frog (which smell of pond water and contain only the finest of baby frogs) and 70% are Spring Surprise (which smell delightful, but when you pop it in your mouth steel bolts spring out and plunge straight through-both cheeks). All candies start out round and look the same. Someone (who can smell) trims some of the candies so that they are square. Then, a second person who can’t smell wraps each candy in a red or brown wrapper. 80% of Spring
Surprise candies are round and 74% have red wrappers. 90% of
Crunchy Frog candies are square and 82% have brown wrappers. All
candies are sold individually in sealed, identical, black boxes! You have
just bought a box, but haven’t opened it yet.
The following are potential Bayes' Nets for the relationships between
variables in this scenario that you will reference in your answers:
Works if there is a stationary distribution
Can be used to compute the Kalman Filter
Are examples of Markov Chain Monte Carlo algorithms
Use transition probabilities in the Bayes Net to compute next
samples
Q3.1
4 Points
Which network(s) can correctly represent P(Flavor, Wrapper, Shape)?
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Q3.2
4 Points
Which network(s) assert(s) P(Wrapper—Shape) = P(Wrapper)?
Network 1
Network 2
Network 3
Network 4
Network 5
Network 6
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Q3.3
4 Points
Which network(s) assert(s) P(Wrapper—Shape, Flavor) = P(Wrapper—
Shape)?
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Q3.4
4 Points
Network 1
Network 2
Network 3
Network 4
Network 5
Network 6
Network 1
Network 2
Network 3
Network 4
Network 5
Network 6
From the problem description, what independence relationships
should hold for this problem? Which network is the best
representation of this problem?
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Q3.5
4 Points
What is the probability that the candy has a red wrapper?
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Q3.6
1 Point
Network 1
Network 2
Network 3
Network 4
Network 5
Network 6
0.785
0.95
0.635
0.8
In the box is a round candy with a red wrapper. What is the probability
that it is a Spring Surprise candy?
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Q4 HMMs
15 Points
Suppose you are a robot navigating a maze (pictured above), wheresome of the cells are free and some are blocked. At each time step,you are occupying one of the free cells. You are equipped withsensors which give you noisy observations, (wU ,wD,wL,wR) of the fourcells adjacent to your current position (UP, DOWN, LEFT, and RIGHTrespectively). Each wI is either FREE or BLOCKED, and is accurate 80%of the time, independently of the other sensors or your currentposition. Assume that if a cell is off the given grid it is treated asblocked.
YES
NO
Imagine that you have experienced a motor malfunction that causes
you to randomly move to one of the four adjacent cell with probability
1/4 (25%). If you move towards a blocked cell, you hit the wall and stay
where you are.
Q4.1
5 Points
Suppose you start in the central cell in the Figure. One time step
passes and you are now in a new, possibly different state and your
sensors indicate (free,blocked,blocked,blocked). Which states have a
non-zero probability of being your new position?
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Q4.2
5 Points
What is the value of P(S1 = (2,2) — WU, WD, WL, WR)?
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Q4.3
5 Points
? (1,2),(2,2),(3,3)
? (2,1),(2,2),(2,3)
? (1,1),(2,2),(3,3)
? (1,2),(2,2),(3,2)
? 0.65
? 0.0512
? 0.02
? 0.33
Suppose that s0 is your starting state and that s1 and s2 are random
variables indicating your state after the first and second time steps.
Which Bayes Net best illustrates the relationships between each si
and the sensor observations associated with that state.
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Q5 More HMMs
20 Points
The weather example V2.0: The weather in College Park is notoriously
fickle. For simplicity we only consider sun and rain and we assume that
the weather changes once per day. The weather satisfies the following
transition probabilities: (a) When it rains, the probability of sun the
following day is 0.6. (b) When the sun shines, the probability of rain the
following day is 0.3.
?
?
?
5/14/22, 1:30 PM Submit Final Exam | Gradescope
Q5.1
5 Points
Which is the transition matrix for College Park weather?
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Q5.2
5 Points
Given the correct transition matrix from the above, assume that we
observe the weather over a ten day period. In particular, we observe
the following:
The sun shines on the first day.
It rains on day 5.
It rains on day 7.
The sun shines on day 10.
What is the probability of sun on day 6, P(r6)?
?
?
?
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Q5.3
5 Points
Which is the most likely weather sequence on days 8 and 9. (HINT:
Work out each of the four possibilities).
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Q5.4
5 Points
TRUE/FALSE: Is there a stationary point for this distribution?
