Written Assignment - Probabilites and Bayesian Networks
Task 1
15 points.
In
a certain probability problem, we have 11 variables: A, B1,
B2,
..., B10.
- Variable A has 8 values.
- Each of variables B1, ..., B10 have 5
possible values. Each Bi is
conditionally indepedent of all other 9 Bjvariables
(with j != i) given A.
Based
on these facts:
Part
a: How
many numbers do you need to store in the joint distribution table of
these 11 variables?
Part
b: What
is the most space-efficient way (in terms of how many numbers you need
to store) representation for the joint probability distribution of
these 11 variables? How many numbers do you need to store in your
solution? Your answer should work with any variables satisfying the
assumptions stated above.
Part c: Does this scenario follow the Naive-Bayes model?
Task 2
30 points
As
in the slides that we saw in class (Prior and Posterior probabilties), there are five types of bags of
candies. Each bag has an infinite amount of candies. We have one of
those bags, and we are picking candies out of it. We don't know what
type of bag we have, so we want to figure out the probability of each
type based on the candies that we have picked.
The
five possible hypotheses for our bag are:
- h1 (prior:
10%): This type of bag contains 100% cherry candies.
- h2 (prior:
20%): This type of bag contains 75% cherry candies and 25% lime candies.
- h3 (prior:
40%): This type of bag contains 50% cherry candies and 50% lime candies.
- h4 (prior:
20%): This type of bag contains 25% cherry candies and 75% lime candies.
- h5 (prior:
10%): This type of bag contains 100% lime candies.
Given the following sequences of observations show how the Posterior probabilties change.
a. CCCCCL
b. CLCLCL
c. CCCLLL
Task 3
10 points
George
doesn't watch much TV in the evening, unless there is a baseball game
on. When there is baseball on TV, George is very likely to watch.
George has a cat that he feeds most evenings, although he forgets every
now and then. He's much more likely to forget when he's watching TV.
He's also very unlikely to feed the cat if he has run out of cat food
(although sometimes he gives the cat some of his own food). Design a
Bayesian network for modeling the relations between these four events:
- baseball_game_on_TV
- George_watches_TV
- out_of_cat_food
- George_feeds_cat
Your
task is to connect these nodes with arrows pointing from causes to
effects. No programming is needed for this part, just include an
electronic document (PDF, Word file, or OpenOffice document) showing
your Bayesian network design.
Task 4
10 points
For
the Bayesian network of Task 3, the text file at this link contains
training data from every evening of an entire year. Every line in this
text file corresponds to an evening, and contains four numbers. Each
number is a 0 or a 1. In more detail:
Based
on the data in this file, determine the probability table for each node
in the Bayesian network you have designed for Task 3. You need to
include these four tables in the drawing that you produce for question
3. You also need to submit the code/script that computes these
probabilities.
Task 5
8 points
Given the network in Task 4, calculate P (not(George Feeds Cat) / Baseball Game on TV) using Inference by Enumeration
Task 6
12
points.
Figure 1: Yet another Bayesian Network.
Part a: On the
network shown in Figure 1, what is the Markovian blanket of node L?
Part
b: On
the network shown in Figure 1, what is P(C, H)? How is it derived?
Part c: On
the network shown in Figure 1, what is P(O | not(J), E)? How is it
derived?
