Homework 4 (due 10/30 before 1:30 PM via BlackBoard)
1. Bayes' Rule
a) Write down Bayes’ rule (1 point)
Suppose the following: Someone has figured out, miraculously, that there's a link between the Eagles playing a football game and the weather.
It turns out that:
If the temperature is below 50 degrees Fahrenheit, the Eagles win 40% of the time.
If the temperature is between 50 and 70 degrees, the Eagles win 85% of the time.
If the temperature is greater than 70 degrees, the Eagles win 95% of the time.
We go out and measure the temperature over ten days, and figure out that the temperature is hotter than 70 degrees on 3 of the 10 days, and between 50 and 70 degrees on a different 3 of the 10 days.
b) What percentage of the time is the temperature under 50 degrees? (1 point)
c) Let's use the Eagles as a probability thermometer: given that they've just won a game, what's the temperature? Show all work.
i) What probabilities are you trying to solve for? (3 points)
ii) Using Bayes' rule, what would the equations to solve for (i) look like? (2 points)
iii) What is the probability of a win by the Eagles? (3 points) (Based on the information in the problem, NOT based on real life please)
iv) Given an Eagles' win, which temperature is it most likely to be? (2 points)
v) What is the probability that it actually is that temperature, given an Eagles' win? (3 points)
2) According to the logic of double dissociation, we show that two mechanisms are distinct and relatively independent if we can, essentially, shut one off without really affecting the other, and then shut the other off without really affecting the one. In our class discussion, one important division of memory was the "extrinsic vs. intrinsic" distinction: the idea that the memory itself and the source of the memory (where we learned it, its familiarity, etc.) are dissociable.
a) Consider the two studies we discussed in class: the mugshots study (where subjects watched a staged crime, then later either viewed or did not view a set of mugshots completely unrelated to the crime, and were finally asked to pick out the criminals from a lineup) OR the sentencerating study (where subjects were either asked to rate a set of ridiculous sentences for "interestingness" or not, and then later asked which sentences were true). How does either study show a dissociation between the memory itself and the source of the memory? (3 points)
b) What kind of evidence would you need to show a double dissociation? That is, part (a) gives us half the evidence we need to show a double dissociationwhat would the other half need to be? (2 points)
3) Reinforcement Learning
Suppose we're looking at the following state chain:
Class > Eat > Study > EXIT


v
Party > EXIT
The reward structure is the set of numbers in parentheses:
Class (0) > Eat (0) > Study (+2) > EXIT


v
Party (+1) > EXIT
Rate of learning "c" is always .5, the value at all states starts at 0, and v(EXIT) always equals 0.
For all the following questions, please write your answers in the following order:
v(Class), v(Eat), v(Study), v(Party)
Putting all your answers into a table is acceptable.
a) It's Monday: You go to Class, Eat, Study, and then end your day (EXIT). What do the values look like after one run through this chain? (2 points)
b) It's Tuesday, so you do the same thing: You go to Class, Eat, Study, end your day. What do the values look like after two runs? (2 points)
c) It's Wednesday, so of course you do the same thing: Go to Class, Eat, Study, end your day. What do the values look like after three runs? (2 points)
d) It's Thursdayscrew studying! You go to Class, Eat, Party, and then end your day. What do the values look like now? (2 points)
e) It's Fridayclearly a time for studyiI mean partying. You go to Class, Eat, Party, and then end your day. What do the values look like now? (2 points)
4) Naive Bayes
Let's suppose you're browsing through newspaper headlines to find information about the upcoming election. Let's further suppose that you're interested in whether an article has to do with either the ECONOMY (E) or POLLS (P).
You've estimated, based on articles in previous years, that the probability of an article being about the ECONOMY is p(E)=.45, and the probability of an article being about POLLS is p(P)=.25.
It turns out that the likelihood of seeing various words depends on what kind of article you're considering:
p("points"E) = .01
p("six"E) = .04
p("reasons"E) = .02
p("deficit"E) = .05
p("points"P) = .04
p("six"P) = .01
p("reasons"P) = .04
p("deficit"P) = .04
Common words such as "to", "for", and "of" are equally likely in all article types.
a) Suppose you see the following sentence fragments:
"...points to six reasons for deficit..."
"...reasons for deficit of six points..."
Which genre would a Naive Bayes classifier assign to each? Show all your work. (3 points)
b) Regardless of what the calculations to (a) tell you, it seems likely that the first sentence "...points to six reasons for deficit..." has to do with the Economy, and the second sentence "...reasons for deficit of six points..." has to do with Polls. What information does the Naive Bayes classifier not model (but you do use in reading) which allows you to make this judgment? (3 points)
5) You're asked to judge whether or not Joe Schmoe committed a murder. You're discussing this with your best friend, and show him a picture of Joe Schmoe.
Your best friend notices that Joe Schmoe has beady eyes, and says "You know, now that I think about it, I had this neighbor with beady eyes. They finally arrested him yesterday for multiple burglaries. So this Joe Schmoe probably did commit the murder!"
Name two heuristics/errors in decision making found in this statement, and why they're wrong. (4 points)