Written questions [100 points]
Your answers to the following questions should be typed
and submitted as a pdf (e.g. you can type your answers using Word and
print to pdf, or use LaTeX, or write your answers by hand
and then scan your paper).
- [10 points] AIMA 13.8
- [10 points] AIMA 13.15
- [10 points] AIMA 14.4a. "Numerical semantics" refers to the equation for independence. "Topological semantics" refers to the d-separation algorithm for showing independence
- Suppose that the joint probability distribution of
four variables {X, Y, Z, W} can be factorized as: p(x,y,z,w) =
p(x)p(y|x)p(z|x)p(w|y,z). Draw the Bayesian network that corresponds
to this factoring of the joint distribution. Use
d-separation to determine whether or
not each of the following independencies holds:
- [5 points] X is independent of W given Y
- [5 points] X is independent of W given Z
- [5 points] X is independent of W given {Y, Z}
- [5 points] Y is independent of Z given {X, W}
- [5 points] Assume X, Y, Z, and W are binary random variables. How many probabilities must be specified
for the conditional distributions of the Bayesian network?
===== Question 5 has been postponed! Please do not do this question!=====
- Consider the WetGrass Bayesian network shown in
Figure 14.12a in AIMA. Assume we observe
WetGrass=false.
- [10 points] Compute the exact posterior probabilities
p(Cloudy|WetGrass=false), p(Sprinkler|WetGrass=false), and
p(Rain|WetGrass=false).
- [5 points] Show how to generate a sample using rejection
sampling.
- [15 points] Generate 10 samples using rejection sampling. How many
samples do you get where WetGrass=false and how many do
you have to reject? Estimate posterior probabilities for
Cloudy, Sprinkler, and Rain based on those 10
samples.
- [5 points] Show how to generate a sample
using likelihood weighting
- [15 points] Generate 10 samples using likelihood
weighting. Show the weights for each sample. Estimate
posterior probabilities for Cloudy, Sprinkler, and Rain
based on those 10 samples.
Note: To generate a random double between [0,1] in Python use the commands: "import random" and "random.uniform(0,1)"
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Ungraded optional problems (Solutions to these problems
are already posted)
- A discrete probability distribution over some set of random
variables (X1,...,XN) assigns a real-valued number to each
possible assignment of the random variables. These real-valued
numbers are called "probabilities". What are the constraints
to have a valid probability distribution?
- How many probabilities must you specify for the joint
distribution of n binary random variables? For the joint
distribution of n random variables with domain size d?
- Mathematically define: marginalization,
conditional distribution, product rule, chain rule, Bayes'
rule.
- Derive Bayes' rule using the definition of the conditional
distribution and the product rule
- Explain (in a mathematically precise way) what is meant
by the statement: "The conditional
distribution and the joint distribution are proportional"
- AIMA 13.12
- AIMA 13.14
- AIMA 13.17 (Note there is a typo: B should be Y)
- AIMA 13.22 - Naive Bayes model
- Define "Bayesian network".
- Why are Bayesian networks useful?
- Type of question I might ask on a midterm: I
might describe a story and then ask you to
(1) construct a Bayesian network that captures the causual
structure of the story (2) compute the number of
probabilities that must be specified for the BN (3) compute the
number of probabilities that must be specified for the joint
(without the BN) (4) list the independencies using d-separation (3) compute some query (or describe given a
piece of evidence which exact and which approximate inference
algorithm you might use to compute the query)
- AIMA 14.5
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