CSCI 3202
Artificial Intelligence

In this assignment you will experiment with discrete probability distributions, and use these distributions to make predictions about the environment.

Part 1

When the Titanic struck and iceberg and sank, there were 2201 people on board.  Some survived, some died.  How does survival relate to other attributes of the individuals?  We explore this question with a probabilistic approach.  We consider a sample space of individuals who are characterized by four random variables:

• Class: What status did the individual have on the ship?  (1st, 2nd, or 3rd class passenger; crew member)
• Age:  what age was the individual?  (child, adult)
• Gender:  what gender was the individual? (male, female)
• Survival:  did the individual survive the shipwreck? (yes, no)

For example,  the sample point (Class=1st, Age=adult, Gender=male, Survival=yes) characterizes a subset of individuals.  The data set of 2201 individuals is available from http://www.cs.colorado.edu/~mozer/courses/3202/titanic.dat.

(a) Using the data set, compute the full joint distribution, i.e., P(Class, Age, Gender, Survival).  This distribution has 4x2x2x2 = 32 probabilities.  Display the distribution as follows:

P(Class, Age, Gender, Survival)	Survival=yes				Survival=no
	Gender=male		Gender=female		Gender=male		Gender=female
	Age=child	Age=adult	Age=child	Age=adult	Age=child	Age=adult	Age=child	Age=adult
Class=1st
Class=2nd
Class=3rd
Class=crew

(b) Using the data set and the joint distribution, compute P(Survival=yes | Class, Age, Gender). Warning:  Be alert to the possibility of a cell whose value is undefined.  Display the distribution as follows:

P(Survival=yes | Class, Age, Gender)	Gender=male		Gender=female
	Age=child	Age=adult	Age=child	Age=adult
Class=1st
Class=2nd
Class=3rd
Class=crew

 
(c) Construct the unconditional distribution P(Survival).

(d) Construct the conditional distributions P(Gender|Survival), P(Adult|Survival), and P(Class|Survival).

(e) Using the distributions you computed in parts (c) and (d), estimate P(Survival=yes | Class,Age,Gender) under the Naive Bayes assumption.  See the text and class notes for a description of Naive Bayes.  It boils down to this equation: 

(f) How well does the Naive Bayes assumption do in matching the probabilities you obtained in (b)?  Are there any advantages of estimating the conditional probability using the Naive Bayes assumption?

Part 2

In this portion of the assignment, you are to do an analysis of pit probabilities in the Wumpus World, analogous to the analysis that was done in the text and in class.  The particular situation you should consider is as follows:

	OK breeze		OK breeze
	OK		OK
OK	OK	OK	OK

The rooms labeled "OK" have been visited and contain no wumpus or pit.  In the two rooms labeled "breeze", the agent sensed a breeze.  Estimate the probability of a pit in each of the remaining rooms.  Show the logic of your work.