CMPSCI 291b: Reasoning About Uncertainty
David Mix Barrington
Homework Assignment #4
Posted Wednesday 30 April 2008
Due on paper to me or to CMPSCI main office by 4:00 p.m.,
Tuesday 13 May 2008
There were originally five questions for 70 total
points -- the last three-part question for 30 more points was added on
5 May 2008.
Students are responsible for understanding and following
the academic honesty
policies indicated on this page.
Question X.4 corrected 5 May.
- Problem X.4 (10) Suppose my prior hypothesis is that black, yellow,
and chocolate Labrador retrievers are equally numerous.
This is not enough information to solve the problem -- I need to have a prior
that indicates the range of hypotheses I am considering. Let's say that H is
the hypothesis that the three types are equally likely. The hypotheses
HB, HY, and HC say that the distribution of black, yellow, and chocolate is
60-20-20, 20-60-20, and 20-20-60 respectively. My prior will then be that
H is true with probability 70%, and that HB, HY, and HC have probability
10% each. I then observe three
black labs and a chocolate lab. What should be my posterior estimate of the
probability of each of the four hypotheses?
- Problem X.5 (10) In State M, 63% of voters are Democrats and 37%
Republicans, while in State U 73% are Republicans and 27% Democrats. In State
M, 90% prefer the Red Sox to the Yankees and in State U 60% prefer the Red
Sox to the Yankees. Suppose I now meet a person who is either a random State M
resident or a random State U resident with equal probability. What should
be my estimated probability that she is from State M given that she is:
- a Democratic Red Sox fan
- a Democratic Yankee fan
- a Republican Red Sox fan
- a Republican Yankee fan
- Problem X.6 (15)
We place a marker on one of the cells of a three-by-three array (a tic-tac-toe
board). On each turn the marker moves one cell north, east, south, or west,
with each possible move equally likely -- for example, from the northwest corner
it moves south or east with probability 1/2 each. For each of three possible
initial conditions (corner, side, center) determine the probability of each
of the nine positions after one, two, and three turns. Determine the long-term
behavior of the system and the steady-state probability if any.
- Problem X.7 (15)
Repeat Problem X.6 where the marker has an equal probability of remaining where
it is or moving one square -- so from the northwest corner it stays, moves
south, or moves east with probability 1/3 each, and from the center it stays
or moves in one of four directions with probability 1/5 each.
- Problem X.8 (20)
Now consider a Markov decision process where we can decide at each turn
whether to use Strategy A, which keeps the marker stationary with 1/2
probability and moves it in one of the available directions otherwise, and
Plan B, which moves the marker to any of the nine squares with probability
1/9 each. The reward on any turn is 3 for the center square, 2 for a side
square, and 1 for a corner square. What is the policy that maximizes
the reward on the next turn? What is the policy that maximizes discounted
future return with a discount factor of γ = 1/2?
- Problem X.9 (30) This problem views a single
pitch of a baseball game as a two-person zero-sum game with simultaneous
moves. On every pitch the state consists of a number of balls from 0 through
3 and a number of strikes from 0 through 2. On the pitch, the pitcher chooses
whether to throw a good or a bad pitch, and the batter decides to swing or not
swing (take). If the batter swings at a good pitch, he hits the ball in play
which has a 35% chance of being a hit, so the batter gets 35 points. If the
batter takes a good pitch or swings at a bad pitch the number of strikes goes
up by one. If he takes a bad pitch the number of balls goes up by one. If the
number of strikes reaches three the batter is out and gets no points. If the
number of balls reaches four there is a walk and the batter gets a number of
points to be determined below.
- (a,5) Determine the optimal mixed strategies for each player if the
count is 3-2 (three balls and two points) and a walk is worth 100 points.
What is the expected return to the batter from this game?
- (b,10) Compute the expected return to the batter for the other eleven
count situations. You must do this in a sensible order. For example, if the
count is 2-1 a ball makes it 3-1 and a strike makes it 2-2, so to compute the
value of 2-1 you must already know the values of 3-1 and 2-2.
- (c,15) Repeat parts (a) and (b) with the walk valued at 60 points
rather than 100.
Last modified 5 May 2008