Project 5: Ghostbusters

Abstract

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!

Introduction

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.

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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:

bustersAgents.py Agents for playing the Ghostbusters variant of Pacman.

inference.py Code for tracking ghosts over time using their sounds.

Files you will NOT edit:

busters.py The main entry to Ghostbusters (replacing Pacman.py)

bustersGhostAgents.py New ghost agents for Ghostbusters

distanceCalculator.py Computes maze distances

game.py Inner workings and helper classes for Pacman

ghostAgents.py Agents to control ghosts

graphicsDisplay.py Graphics for Pacman

graphicsUtils.py Support for Pacman graphics

keyboardAgents.py Keyboard interfaces to control Pacman

layout.py Code for reading layout files and storing their contents

util.py Utility functions

Files to Edit and Submit: You will fill in portions of bustersAgents.py

and inference.py during the assignment. Please do not change the other

files in this distribution or submit any of our original files other than these

files.

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.

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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.

python busters.py

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 inference.py to model belief distributions and weight distributions. This class is an extension of the built-in Python dictionary class, where

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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 inference.py

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 inference.py. 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,

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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 inference.py 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

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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

self.getPositionDistribution.

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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

movements.

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

Test

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 bustersAgents.py. 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.

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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 inference.py. 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

Observation

Next, we will implement the observeUpdate method in the ParticleFilter

class in inference.py. 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

self.getJailPosition().

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.

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Question 7 (3 points): Approximate Inference

with Time Elapse

Implement the elapseTime function in the ParticleFilter class in inference.py. 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

inference.py. Your initialization should be consistent with a uniform prior.

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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 inference.py. A correct implementation will weight

and resample the entire list of particles based on the observation of all ghost

distances.

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.

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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 inference.py

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. bustersAgents.py and

inference) to online judge.

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