Assignment 8

Written and Programming Assignment - Probailites

The assignment should be submitted via Blackboard.

Task 1

24 points

You are a meteorologist that places temperature sensors all of the world, and you set them up so that they automatically e-mail you, each day, the high temperature for that day. Unfortunately, you have forgotten whether you placed a certain sensor S in Maine or in the Sahara desert (but you are sure you placed it in one of those two places) . The probability that you placed sensor S in Maine is 5%. The probability of getting a daily high temperature of 80 degrees or more is 20% in Maine and 90% in Sahara. Assume that probability of a daily high for any day is conditionally independent of the daily high for the previous day, given the location of the sensor.

Part a: If the first e-mail you got from sensor S indicates a daily high under 80 degrees, what is the probability that the sensor is placed in Maine?

Part b: If the first e-mail you got from sensor S indicates a daily high under 80 degrees, what is the probability that the second e-mail also indicates a daily high under 80 degrees?

Part c: What is the probability that the first three e-mails all indicate daily highs under 80 degrees?

Task 2

10 points.

In a certain probability problem, we have 11 variables: A, B₁, B₂, ..., B₁₀.

Variable A has 5 values.
Each of variables B₁, ..., B₁₀ have 7 possible values. Each B_i is conditionally indepedent of all other 9 B_jvariables (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.

Task 3

16 points.

For the following joint probabilty distribution answer the following

	C = T		C = F
	B = T	B = F	B = T	B = F
A = T	0.048	0.192	0.196	0.084
A = F	0.012	0.048	0.294	0.126

a. Calculate P (A | B)
b. Calculate P (A | B, C)
c. Calculate P (A, C | B)
d. Given B, is A conditionally independant of C? Justify.

Task 4

50 points

The task in this part is to implement a system that:

Can determine the posterior probability of different hypotheses, given priors for these hypotheses, and given a sequence of observations.
Can determine the probability that the next observation will be of a specific type, priors for different hypotheses, and given a sequence of observations.

As in the slides that we saw in class, 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.

Command Line arguments:

The program takes a single command line argument, which is a string, for example CLLCCCLLL. This string represents a sequence of observations, i.e., a sequence of candies that we have already picked. Each character is C if we picked a cherry candy, and L if we picked a lime candy. Assuming that characters in the string are numbered starting with 1, the i-th character of the string corresponds to the i-th observation. The program should be invoked from the commandline as follows:

compute_a_posteriori observations

For example:

compute_a_posteriori CLLCCLLLCCL

We also allow the case of not having a command line argument at all, this represents the case where we have made no observations yet.

Output:

Your program should create a text file called "result.txt", that is formatted exactly as shown below. ??? is used where your program should print values that depend on its command line argument. Five decimal points should appear for any floating point number.

Observation sequence Q: ???
Length of Q: ???

After Observation ??? = ???: (This and all remaining lines are repeated for every observation)

P(h1 | Q) = ???
P(h2 | Q) = ???
P(h3 | Q) = ???
P(h4 | Q) = ???
P(h5 | Q) = ???

Probability that the next candy we pick will be C, given Q: ???
Probability that the next candy we pick will be L, given Q: ???

Sample output for the Tasks is given here.

How to submit

Submissions should be made using Blackboard.

Submit a ZIPPED directory called assignment8_<netid>.zip (no other forms of compression accepted, contact the instructor or TA if you do not know how to produce a ZIP file) that contains all the files

Your submission for Task 4 needs to execute in omega. Otherwise no credit will be given
Submit solutions to the remaining Tasks in .pdf format or as scanned .png images.
The submission should also contain a file called readme.txt, which should specify precisely:

Name and UTA ID of the student.
Any additional instructions to compile/run your program.