Project 5: Ghostbusters


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Project 5: Ghostbusters

Probabilistic inference in a Hidden Markov Model tracks the movement of hidden ghosts in the Pacman world. This project will help
you to implement exact inference using the forward algorithm and
approximate inference via particle filters.
Figure 1: I can hear you, ghost.
Running won’t save you from my Particle filter!
Pacman spends his life running from ghosts, but things were not always so.
Legend has it that many years ago, Pacman’s great grandfather Grandpac
learned to hunt ghosts for sport. However, he was blinded by his power and
could only track ghosts by their banging and clanging.
In this project, you will design Pacman agents that use sensors to locate and
eat invisible ghosts. You’ll advance from locating single, stationary ghosts
to hunting packs of multiple moving ghosts with ruthless efficiency.
The code for this project contains the following files, available as a zip archive
in the folder VE492 Projects on Canvas.
Files you’ll edit: Agents for playing the Ghostbusters variant of Pacman. Code for tracking ghosts over time using their sounds.
Files you will NOT edit: The main entry to Ghostbusters (replacing New ghost agents for Ghostbusters Computes maze distances Inner workings and helper classes for Pacman Agents to control ghosts Graphics for Pacman Support for Pacman graphics Keyboard interfaces to control Pacman Code for reading layout files and storing their contents Utility functions
Files to Edit and Submit: You will fill in portions of
and during the assignment. Please do not change the other
files in this distribution or submit any of our original files other than these
Evaluation: Your code will be autograded for technical correctness. Please
do not change the names of any provided functions or classes within the code,
or you will wreak havoc on the autograder. However, the correctness of your
implementation will be the final judge of your score. If necessary, we will
review and grade assignments individually to ensure that you receive due
credit for your work.
Academic Dishonesty: We will be checking your code against other submissions in the class for logical redundancy. If you copy someone else’s code
and submit it with minor changes, we will know. These cheat detectors are
quite hard to fool, so please don’t try. We trust you all to submit your own
work only; please don’t let us down. If you do, we will pursue the strongest
consequences available to us.
Getting Help: You are not alone! If you find yourself stuck on something,
contact the course staff for help. Office hours and the discussion forum on
Piazza are there for your support; please use them. If you can’t make our
office hours, let us know and we will schedule more. We want these projects
to be rewarding and instructional, not frustrating and demoralizing. But, we
don’t know when or how to help unless you ask.
Discussion: Please be careful not to post spoilers.
Ghostbusters and BNs
In this version of Ghostbusters, the goal is to hunt down scared but invisible
ghosts. Pacman, ever resourceful, is equipped with sonar (ears) that provides
noisy readings of the Manhattan distance to each ghost. The game ends when
Pacman has eaten all the ghosts. To start, try playing a game yourself using
the keyboard.
The blocks of color indicate where the each ghost could possibly be, given
the noisy distance readings provided to Pacman. The noisy distances at the
bottom of the display are always non-negative, and always within 7 of the
true distance. The probability of a distance reading decreases exponentially
with its difference from the true distance.
Your primary task in this project is to implement inference to track the
ghosts. For the keyboard based game above, a crude form of inference was
implemented for you by default: all squares in which a ghost could possibly be
are shaded by the color of the ghost. Naturally, we want a better estimate
of the ghost’s position. Fortunately, Bayes Nets provide us with powerful
tools for making the most of the information we have. Throughout the rest
of this project, you will implement algorithms for performing both exact and
approximate inference using Bayes Nets. The project is challenging, so we
do encouarge you to start early and seek help when necessary.
Question 0 (0 points): DiscreteDistribution Class
Throughout this project, we will be using the DiscreteDistribution class
defined in to model belief distributions and weight distributions. This class is an extension of the built-in Python dictionary class, where
the keys are the different discrete elements of our distribution, and the corresponding values are proportional to the belief or weight that the distribution
assigns that element. This question asks you to fill in the missing parts of
this class, which will be crucial for later questions (even though this question
itself is worth no points).
First, fill in the normalize method, which normalizes the values in the distribution to sum to one, but keeps the proportions of the values the same.
Use the total method to find the sum of the values in the distribution. For
an empty distribution or a distribution where all of the values are zero, do
nothing. Note that this method modifies the distribution directly, rather
than returning a new distribution.
Second, fill in the sample method, which draws a sample from the distribution, where the probability that a key is sampled is proportional to its
corresponding value. Assume that the distribution is not empty, and not all
of the values are zero. Note that the distribution does not necessarily have to
be normalized prior to calling this method. You may find Python’s built-in
random.random() function useful for this question.
The correctness of your implementation can be easily checked. We have
provided Python doctests as a starting point, and you can feel free to add
more and implement other tests of your own. You can run the doctests using:
python -m doctest -v
Note that, depending on the implementation details of the sample method,
some correct implementations may not pass the doctests that are provided.
To thoroughly check the correctness of your sample method, you should
instead draw many samples and see if the frequency of each key converges to
be proportional of its corresponding value.
Question 1 (2 points): Observation Probability
In this question, you will implement the getObservationProb method in
the InferenceModule base class in This method takes in an
observation (which is a noisy reading of the distance to the ghost), Pacman’s
position, the ghost’s position, and the position of the ghost’s jail, and returns
the probability of the noisy distance reading given Pacman’s position and
the ghost’s position. In other words, we want to return P(noisyDistance |
pacmanPosition, ghostPosition) .
The distance sensor has a probability distribution over distance readings
given the true distance from Pacman to the ghost. This distribution is modeled by the function busters.getObservationProbability(noisyDistance,
trueDistance), which returns P(noisyDistance | trueDistance) and is
provided for you. You should use this function to help you solve the problem, and use the provided manhattanDistance function to find the distance
between Pacman’s location and the ghost’s location.
However, there is the special case of jail that we have to handle as well.
Specifically, when we capture a ghost and send it to the jail location, our
distance sensor deterministically returns None, and nothing else. So, if the
ghost’s position is the jail position, then the observation is None with probability 1, and everything else with probability 0. Conversely, if the distance
reading is not None, then the ghost is in jail with probability 0. If the distance reading is None then the ghost is in jail with probability 1. Make sure
you handle this special case in your implementation.
Please submit on the online judge for testing. As a general note, it is possible
for some of the autograder tests to take a long time to run for this project,
and you will have to exercise patience. As long as the autograder doesn’t
time out, you should be fine (provided that you actually pass the tests).
Question 2 (3 points): Exact Inference Observation
In this question, you will implement the observeUpdate method in ExactInference class of to correctly update the agent’s belief distribution over ghost positions given an observation from Pacman’s sensors. You
are implementing the online belief update for observing new evidence. The
observeUpdate method should, for this problem, update the belief at every
position on the map after receiving a sensor reading. You should iterate your
updates over the variable self.allPositions which includes all legal positions plus the special jail position. Beliefs represent the probability that the
ghost is at a particular location, and are stored as a DiscreteDistribution
object in a field called self.beliefs, which you should update.
Before typing any code, write down the equation of the inference problem you
are trying to solve. You should use the function self.getObservationProb
that you wrote in the last question, which returns the probability of an observation given Pacman’s position, a potential ghost position, and the jail position. You can obtain Pacman’s position using gameState.getPacmanPosition(), and the jail position using self.getJailPosition().
In the Pacman display, high posterior beliefs are represented by bright colors,
while low beliefs are represented by dim colors. You should start with a large
cloud of belief that shrinks over time as more evidence accumulates. As you
watch the test cases, be sure that you understand how the squares converge
to their final coloring.
Note: your busters agents have a separate inference module for each ghost
they are tracking. That’s why if you print an observation inside the observeUpdate function, you’ll only see a single number even though there may be
multiple ghosts on the board.
Please submit on the online judge for testing.
Question 3 (3 points): Exact Inference with
Time Elapse
In the previous question you implemented belief updates for Pacman based on
his observations. Fortunately, Pacman’s observations are not his only source
of knowledge about where a ghost may be. Pacman also has knowledge
about the ways that a ghost may move; namely that the ghost can not move
through a wall or more than one space in one time step.
To understand why this is useful to Pacman, consider the following scenario
in which there is Pacman and one Ghost. Pacman receives many observations
which indicate the ghost is very near, but then one which indicates the ghost
is very far. The reading indicating the ghost is very far is likely to be the
result of a buggy sensor. Pacman’s prior knowledge of how the ghost may
move will decrease the impact of this reading since Pacman knows the ghost
could not move so far in only one move.
In this question, you will implement the elapseTime method in ExactInference. The elapseTime step should, for this problem, update the belief at every position on the map after one time step elapsing. Your agent has access to
the action distribution for the ghost through self.getPositionDistribution. In order to obtain the distribution over new positions for the ghost, given
its previous position, use this line of code:
newPosDist = self.getPositionDistribution(gameState, oldPos)
Where oldPos refers to the previous ghost position. newPosDist is a DiscreteDistribution object, where for each position p in self.allPositions,
newPosDist[p] is the probability that the ghost is at position p at time
t + 1, given that the ghost is at position oldPos at time t. Note that
this call can be fairly expensive, so if your code is timing out, one thing
to think about is whether or not you can reduce the number of calls to
Before typing any code, write down the equation of the inference problem you
are trying to solve. In order to test your predict implementation separately
from your update implementation in the previous question, this question will
not make use of your update implementation.
Since Pacman is not observing the ghost, this means the ghost’s actions will
not impact Pacman’s beliefs. Over time, Pacman’s beliefs will come to reflect
places on the board where he believes ghosts are most likely to be given the
geometry of the board and what Pacman already knows about their valid
For the tests in this question we will sometimes use a ghost with random
movements and other times we will use the GoSouthGhost. This ghost tends
to move south so over time, and without any observations, Pacman’s belief
distribution should begin to focus around the bottom of the board.
Please submit on the online judge for testing.
Question 4 (2 points): Exact Inference Full
Now that Pacman knows how to use both his prior knowledge and his observations when figuring out where a ghost is, he is ready to hunt down ghosts
on his own. This question will use your observeUpdate and elapseTime implementations together, along with a simple greedy hunting strategy which
you will implement for this question. In the simple greedy strategy, Pacman
assumes that each ghost is in its most likely position according to his beliefs,
then moves toward the closest ghost. Up to this point, Pacman has moved
by randomly selecting a valid action.
Implement the chooseAction method in GreedyBustersAgent in Your agent should first find the most likely position of each remaining uncaptured ghost, then choose an action that minimizes the maze
distance to the closest ghost.
To find the maze distance between any two positions pos1 and pos2, use
self.distancer.getDistance(pos1, pos2). To find the successor position
of a position after an action:
successorPosition = Actions.getSuccessor(position, action)
You are provided with livingGhostPositionDistributions, a list of DiscreteDistribution objects representing the position belief distributions for
each of the ghosts that are still uncaptured.
Please submit on the online judge for testing.
Question 5 (2 points): Approximate Inference
Initialization and Beliefs
Approximate inference is very trendy among ghost hunters this season. For
the next few questions, you will implement a particle filtering algorithm for
tracking a single ghost.
First, implement the functions initializeUniformly and getBeliefDistribution in the ParticleFilter class in A particle (sample)
is a ghost position in this inference problem. Note that, for initialization,
particles should be evenly (not randomly) distributed across legal positions
in order to ensure a uniform prior.
Note that the variable you store your particles in must be a list.
A list is simply a collection of unweighted variables (positions in this case).
Storing your particles as any other data type, such as a dictionary, is incorrect
and will produce errors. The getBeliefDistribution method then takes
the list of particles and converts it into a DiscreteDistribution object.
Please submit on the online judge for testing.
Question 6 (3 points): Approximate Inference
Next, we will implement the observeUpdate method in the ParticleFilter
class in This method constructs a weight distribution over
self.particles where the weight of a particle is the probability of the
observation given Pacman’s position and that particle location. Then, we
resample from this weighted distribution to construct our new list of particles.
You should again use the function self.getObservationProb to find the
probability of an observation given Pacman’s position, a potential ghost position, and the jail position. The sample method of the DiscreteDistribution
class will also be useful. As a reminder, you can obtain Pacman’s position using gameState.getPacmanPosition(), and the jail position using
There is one special case that a correct implementation must handle. When all particles receive zero weight, the list of particles should
be reinitialized by calling initializeUniformly. The total method of the
DiscreteDistribution may be useful.
Please submit on the online judge for testing.
Question 7 (3 points): Approximate Inference
with Time Elapse
Implement the elapseTime function in the ParticleFilter class in This function should construct a new list of particles that corresponds to each existing particle in self.particles advancing a time step,
and then assign this new list back to self.particles. When complete, you
should be able to track ghosts nearly as effectively as with exact inference.
Note that in this question, we will test both the elapseTime function in
isolation, as well as the full implementation of the particle filter combining
elapseTime and observe.
As in the elapseTime method of the ExactInference class, you should use:
newPosDist = self.getPositionDistribution(gameState, oldPos)
This line of code obtains the distribution over new positions for the ghost,
given its previous position (oldPos). The sample method of the DiscreteDistribution class will also be useful.
Please submit on the online judge for testing.
Question 8 (1 points): Joint Particle Filter Observation
So far, we have tracked each ghost independently, which works fine for the
default RandomGhost or more advanced DirectionalGhost. However, the
prized DispersingGhost chooses actions that avoid other ghosts. Since
the ghosts’ transition models are no longer independent, all ghosts must
be tracked jointly in a dynamic Bayes net!
The Bayes net has the following structure, where the hidden variables G
represent ghost positions and the emission variables E are the noisy distances
to each ghost. This structure can be extended to more ghosts, but only two
(a and b) are shown below.
You will now implement a particle filter that tracks multiple ghosts simultaneously. Each particle will represent a tuple of ghost positions that is a
sample of where all the ghosts are at the present time. The code is already
set up to extract marginal distributions about each ghost from the joint inference algorithm you will create, so that belief clouds about individual ghosts
can be displayed.
Complete the initializeUniformly method in JointParticleFilter in Your initialization should be consistent with a uniform prior.
You may find the Python itertools package helpful. Specifically, look at
itertools.product to get an implementation of the Cartesian product.
However, note that, if you use this, the permutations are not returned in
a random order. Therefore, you must then shuffle the list of permutations in
order to ensure even placement of particles across the board.
As before, use self.legalPositions to obtain a list of positions a ghost
may occupy. Also as before, the variable you store your particles in
must be a list.
Please submit on the online judge for testing.
Question 9 (3 points): Joint Particle Filter Observation
In this question, you will complete the observeUpdate method in the JointParticleFilter class of A correct implementation will weight
and resample the entire list of particles based on the observation of all ghost
To loop over all the ghosts, use:
for i in range(self.numGhosts):

You can still obtain Pacman’s position using gameState.getPacmanPosition(),
but to get the jail position for a ghost, use self.getJailPosition(i), since now
there are multiple ghosts each with their own jail positions.
Your implementation should also again handle the special case when all particles receive zero weight. In this case, self.particles should be recreated from
the prior distribution by calling initializeUniformly.
As in the update method for the ParticleFilter class, you should again use
the function self.getObservationProb to find the probability of an observation
given Pacman’s position, a potential ghost position, and the jail position.
The sample method of the DiscreteDistribution class will also be useful.
Please submit on the online judge for testing.
Question 10 (3 points): Joint Particle Filter
Time Elapse and Full Test
Complete the elapseTime method in JointParticleFilter in
to resample each particle correctly for the Bayes net. In particular, each ghost
should draw a new position conditioned on the positions of all the ghosts at
the previous time step.
As in the last question, you can loop over the ghosts using:
for i in range(self.numGhosts):

Then, assuming that i refers to the index of the ghost, to obtain the distributions over new positions for that single ghost, given the list (prevGhostPositions) of previous positions of all of the ghosts, use:
newPosDist = self.getPositionDistribution(gameState,\
prevGhostPositions, i, self.ghostAgents[i])
Note that completing this question involves grading both question 9 and
question 10. Since these questions involve joint distributions, they require
more computational power (and time) to grade, so please be patient!
Please submit on the online judge for testing.
Submission and Due Date
Please submit the files you are asked to edit (i.e. and
inference) to online judge.

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