Description
Introduction
In the past few weeks, you have learned about parameter estimation in probabilistic graphical
models, as well as structure learning. In this programming assignment, you will explore structure
learning in probabilistic graphical models from a synthetic dataset. In particular, we will provide
you synthetic human and alien body pose data. We use a directed model to represent their body
poses. You will tackle problems including learning CPDs for continuous variables and learning
tree-structured graphs among body parts. Finally, you will use the learned models to classify
unseen pose data as either humans or aliens.
2 Notation and Model Choices
2.1 Representation of Body Poses
We model the body parts with ten parts, as illustrated in Figure 1. The configuration of each
body part is parameterized by 3 real numbers: (y, x, α), where (y, x) is the position of the anchor
point of the body part (shown as filled black squares) and α is the orientation of the body part
(shown as arrows). The y value increases from top to bottom; x increases from left to right; α
increases clockwise and has the origin (zero value) pointing upright.
Given this setting, a pose of the full body can be specified by a 10 × 3 matrix, where the
10 rows correspond to the 10 different body parts, and the columns refer to (in order) y, x, α,
respectively. A large number of example poses represented in this format will be provided to
you for model learning.
There are two classes, humans and aliens, in the dataset. Humans and aliens have different
body structures, therefore they display different pose variations. Our goal is to learn models
that capture the variations of the poses in each of the two classes. The class labels are provided
to you in the training set, so the problem is a typical example of supervised learning.
2.2 Variables and State Spaces
Let’s consider a Bayesian network, where each node represents a body part. We denote the body
part variable as {Oi}
10
i=1, each of which take on continuous values in R
3
: Oi = (yi
, xi
, αi)
1
. We
denote the class variable as C, which takes on discrete values {1, 2} for the two classes we have
in this programming assignment, humans and aliens, respectively.
1You might notice that the domain of α is actually [−π, π]. Cutting the circle at any point might introduce
artifacts in modeling. Therefore, for simplicity, in this programming assignment we will just assume that α can
take on any real value and ignore the fact that α and α + 2π are actually the same.
CS228 Programming Assignment #8 2
x
y
(1) Torso
(2) Head
(3) Left-arm
(4) Left-forearm
(5) Right-arm
(6) Right-forearm
(7) Left-thigh
(8) Left-leg
(9) Right-thigh
(10) Right-leg
0
pi/2 -pi/2
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Figure 1: Representation of the body pose. Note that the boxes for the body parts are for
visualization purposes only, and are not part of our model specification, i.e., we do not attempt
to model the sizes of the body parts.
2.3 Naive Bayes Model
As a first attempt, we adopt the naive Bayes assumption: all variables {Oi}
10
i=1 are independent
given the class label C. This assumption gives rise to the graph structure shown in Figure 2a.
Recall that this is the plate representation we have encountered earlier in this course. In the
naive Bayes model, each body part only has the class label as its parent. The variables can then
be parametrized as follows,
P(C = k) = ck, (1)
yi
|C = k ∼ N (µ
y
ik, σ
y
ik
2
), (2)
xi
|C = k ∼ N (µ
x
ik, σx
ik
2
), (3)
αi
|C = k ∼ N (µ
α
ik, σα
ik
2
), (4)
where i = 1, 2, . . . , 10 indexes the body parts, and k = 1, 2 indexes the two classes. The notation
Y |X = x ∼ N (µ, σ2
) means the conditional distribution of random variable Y given X is a
Gaussian distribution with mean µ and variance σ
2
. Given the class label, each continuous
variable is modeled by a Gaussian distribution. Therefore each body variable will have two sets
of Gaussian parameters, corresponding to the two classes. We will learn the parameters from
the training data.
Since the class label C is always observed in the training data, we can collect the sufficient
statistics for each of the classes from the training data.
• FitGaussianParameters.m (5 points): In this function, you will implement the algorithm for fitting maximum likelihood parameters µ = EPˆ[X] and σ =
p
VarPˆ[X] of the
Gaussian distribution given the samples. EPˆ is the expectation with respect to the empirical distribution Pˆ, which is specified by the dataset D. The variance can be computed as
VarPˆ[X] = EPˆ[X2
] − EPˆ[X]
2
. Note that this function needs to return the standard deviation σ instead of the variance σ
2
. You should fill in the code to compute the parameters.
CS228 Programming Assignment #8 3
C
Oi
10
(a)
C
O1
O2
O3
O4 O5
O6
(b)
Figure 2: Graph structures. (a) The naive Bayes model. (b) Introducing skeletal edges among
body part variables. For clarity we only show 6 upper body part variables.
2.4 Learning with Known Skeletal Structure
While the naive Bayes assumption results in simple model that is easy to learn, it makes unrealistic assumptions. It assumes that body parts are independent given the class label. But
we know that is not the case, since body parts are connected by joints and therefore dependent
based on how they are connected. Our second attempt will exploit the kinematic structure of
the human body by adding edges among the body part variables.
Let’s first assume that humans and aliens share the same skeletal structure, and that structure
is known. Specifically, we take torso as the root variable, and head, left-arm, right-arm, left-thigh,
right-thigh are its children nodes. We then add left-forearm as a child of left-arm, and left-leg as
a child of left-thigh. Similarly, we add right-forearm as a child of right-arm, and right-leg as a
child of right-thigh. Without the class variable C, the body parts form a tree-structured model.
Recall that a tree-structured model is a Bayes Net where each variable has at most one parent.
With the class variable C as a parent to all the body part variables, the graph structure is shown
in Figure 2b, where we only show O1 to O6 for clarity.
The next problem is how to parameterize the local CPDs. For the root body part O1 (torso),
it only has the class variable C as its parent, so its parameterization is exactly the same as before.
However, all the other body parts have two parents, the class variable and a parent body part.
How should we handle this?
Earlier in this course, we learned about the conditional linear Gaussian (CLG) model to
parameterize variables with both discrete and continuous variables as parents. Intuitively, for
each assignment of the discrete parent, we have a different linear Gaussian model. In our case,
we would learn a linear Gaussian model for humans, and a linear Gaussian model for aliens,
Formally, the configuration of each child body part Oi = (yi
, xi
, αi) is modeled as a linear
Gaussian in the configurations of its parent body part conditioned on each possible assignment
of the class label:2
yi
|Op(i)
, C = k ∼ N (θ
(1)
ik + θ
(2)
ik yp(i) + θ
(3)
ik xp(i) + θ
(4)
ik αp(i)
, σ
y
ik
2
), (5)
xi
|Op(i)
, C = k ∼ N (θ
(5)
ik + θ
(6)
ik yp(i) + θ
(7)
ik xp(i) + θ
(8)
ik αp(i)
, σx
ik
2
), (6)
2
In fact, the position of a body part doesn’t have a linear dependency on the orientation angle of its parent
body part, but rather the sine and cosine values of it. But for simplicity, we will use a linear model in this
programming assignment.
CS228 Programming Assignment #8 4
αi
|Op(i)
, C = k ∼ N (θ
(9)
ik + θ
(10)
ik yp(i) + θ
(11)
ik xp(i) + θ
(12)
ik αp(i)
, σα
ik
2
), (7)
where i = 2, . . . , 10 indexes the body parts, and k = 1, 2 indexes the two classes. The notation
p(i) denotes the parent body part of the ith body part.
Now consider the problem of learning the parameters in the CLG model. As before, the class
labels are given. The only difference is that we now fit a linear Gaussian model instead of a
Gaussian model for each body part variable (except the root body part, which only has the class
variable as its parent). Let’s now implement the learning process for conditional linear Gaussian
models.
• FitLinearGaussianParameters.m (15 points): In this function you will implement the
algorithm for fitting linear Gaussian parameters in the general form (such that it can be
used in different scenarios):
X|U ∼ N (β1U1 + β2U2 + · · · + βnUn + βn+1, σ2
) (8)
Note that we use βn+1 instead of the usual notation β0 to be consistent with the indexing
method of Matlab and Octave, and you should keep this in mind when you invoke this
function in subsequent tasks. By taking derivatives of the log-likelihood function3
, we can
see that the parameters (β1, · · · , βn+1) satisfy the following system of linear equations:
EPˆ[X] = β1EPˆ[U1] + β2EPˆ[U2] + · · · + βnEPˆ[Un] + βn+1 (9)
EPˆ[X · Ui
] = β1EPˆ[U1 · Ui
] + β2EPˆ[U2 · Ui
] + · · · + βnEPˆ[Un · Ui
] + βn+1EPˆ[Ui
] (10)
where i = 1, . . . , n. Now we have n + 1 variables and n + 1 equations, so we can use
the backslash operator \ to solve them. More specifically, you can solve a linear equation
Ax = b by setting x = A\b. Having obtained the parameters (β1, · · · , βn+1), the standard
deviation σ can be computed by:
σ =
vuutCovPˆ[X; X] −
Xn
i=1
Xn
j=1
βiβjCovPˆ[Ui
;Uj ] (11)
where the covariance can be computed as CovPˆ[X; Y ] = EPˆ[X · Y ] − EPˆ[X] · EPˆ[Y ].
3 Data Format and Code Structure
3.1 Data Structure
Before we start implementing all the learning procedures above, let’s first take a look at the
data structures we will use in this assignment. To simplify the interface of the program, we use
a unified data structure to represent the parameters and graph structures we encountered in
all previous subsections. Specifically, the parameters are encoded in a structural array P. The
vector P.c encodes the prior on class probabilities in Equation (1), where humans have class
label 1 and aliens have class label 2. The graph parameterization is encoded in a 10 × 2 matrix
G, where each row of G corresponds to a body part (in the order shown in Figure 1). The first
column of G represents the graph parameterization choice:
3
If you are interested in the details of the derivation, please refer to 17.2.4 Gaussian Bayesian Networks
of the text book.
CS228 Programming Assignment #8 5
• G(i, 1) = 0 indicates that body part i only has the class variable as its parent. In this
case, G(i, 2) can take on an arbitrary value. Its parameterization follows the naive Bayes
model described in Equations (2,3,4). The parameters can be found in the following
1 × 2 vectors: P.clg(i).mu_x, P.clg(i).mu_y, P.clg(i).mu_angle, P.clg(i).sigma_x,
P.clg(i).sigma_y, and P.clg(i).sigma_angle. For example, P.clg(2).sigma_y(2) is
the standard deviation of the y-position of the head given that the class label is 2 (aliens).
• G(i, 1) = 1 indicates that body part i has, besides the class variable, another parent
G(i, 2), and the CPDs are parameterized following Equations (5,6,7). The parameters can
be found in the 2 × 12 matrix P.clg(i).theta and 1 × 2 vectors P.clg(i).sigma_x,
P.clg(i).sigma_y, and P.clg(i).sigma_angle. For example, P.clg(i).theta(k, 9)
corresponds to θ
(9)
ik in Equation (7).
Later in the assignment, we will learn different graph structures for each of the two classes. In
that case, a 10 × 2 × 2 matrix G will be used, where G(:, :, k) specifies the graph structure and
parameterization for the kth class (in the same way as described above).
3.2 Provided Data
We provide you with all the data you need in a file PA8Data.mat, which contains the following
data structures:
• trainData: This is a structure that contains two fields. The data field is a N × 10 × 3
matrix that contains N training examples. The labels field is a N ×2 vector that provides
the true class labels for the training examples. The entry in row i and column j is 1 if the
ith example belongs to class j, and 0 elsewhere.
• testData: Similarly, the test data is provided in this structure. It has the same format
as trainData. You will evaluate your learned models on this test set.
• G1, G2: We have constructed these 10 × 2 graph structures for you. G1 is the graph
structure for the naive Bayes model, and G2 is the graph structure for the CLG model.
They follow the graph parameterization choices as describes in 3.1.
To test your code, a separate file PA8SampleCases.mat contains the input and correct output for
each of the sample cases that we test for in the submit script. For argument j of the function call
in part i, you should use exampleINPUT.t#ia#j (replacing the #i with i). For output, look at
exampleOUTPUT.t#i for the output to part i, or exampleOUTPUT.t#io#j for output j if
there are multiple outputs.
3.3 Infrastructure Code
The following functions are provided to you, and you do not need to modify them.
• ShowPose.m: This function outputs a bitmap image of the human figure given a pose
configuration specified by a 10 × 3 matrix.
• SamplePose.m: This function takes as input the graph structure G and the parameter
structure array P in the format specified above, samples from the distribution and outputs
a pose configuration represented by a 10×3 matrix. You can also fix the class label and sample from the conditional distribution. It calls SampleGaussian.m and SampleMultinomial.m,
which are as simple as their names suggest.
CS228 Programming Assignment #8 6
• VisualizeModels.m: This function gives you an intuitive assessment of the quality of your
learned model by visualizing it. The code calls SamplePose.m and ShowPose.m, and shows
a window with N sub-windows. Each sub-window shows a series of samples from the model
for each of the N classes.
• VisualizeDataset.m: This function shows each example pose in a dataset one-by-one in
an animation.
• MaxSpanningTree.m: This function finds the maximum spanning tree given the symmetric
weight matrix. Output of this function is represented as an (non-symmetric) adjacency
matrix A (where one of A(i, j) and A(j, i) is 1 if i and j are connected by an edge, both
are 0 otherwise).
• ConvertAtoG.m: Convert the adjacent matrix representation into our representation G of
the tree structured model. In this function we always use the first body part (torso) as
root node (such that the conversion is unique), and we always use the parameterization in
(5,6,7) (by setting G(i, j) = 1 for i 6= 1).
• lognormpdf.m: This function computes the Gaussian probability density function in log
space at the input value, given the mean and standard deviation.
• GaussianMutualInformation.m: This function computes the mutual information between
two multi-dimensional Gaussian variables (they can have different dimensionality). You
might find it useful when learning the graph structure.
4 Model Learning
We are now ready to implement the learning procedures. You will subsequently implement and
experiment with different model choices. Let’s first load the data that we will be working with,
and visualize the dataset by running the following,
load PA8Data.mat;
VisualizeDataset(trainData.data);
You should see an animation of humans and aliens with different poses. To evaluate how
well the models are capturing these pose variations in the provided data, we first implement the
following function.
• ComputeLogLikelihood.m (20 points): This function computes, given the model P, coupled with the graph structure G, and the dataset, the log likelihood of the model over the
dataset:
X
N
i=1
log P(O1 = o
(i)
1
, · · · , O10 = o
(i)
10 ) (12)
where N is the number of examples in the dataset. The log function here refers to the
natural logarithm function, which we use throughout this programming assignment. The
likelihood of each individual observed pose configuration {Oi}
10
i=1 can be computed as
follows.
P(O1, · · · , O10) = X
2
k=1
P(C = k, O1, · · · , O10) = X
2
k=1
P(C = k)
Y
10
i=1
P(Oi
|C = k, Op(i)).
(13)
CS228 Programming Assignment #8 7
where we sum over the joint probabilities of each possible class label. Recall that Op(i)
is
the parent of Oi
, and p(i) can be found in the given graph structure G(i, 2). While the
Matlab/Octave function normpdf can be used to evaluate the Gaussian density function,
numerical issues introduced by extremely small numbers returned by normpdf can cause
problems. Therefore, instead of multiplying probabilities, we will take the logarithms of
the probabilities and sum the log-probabilities. You function should compute everything in
log-space, and return log-probabilities instead of probabilities. You may find the function
lognormpdf.m we provided useful. Notice that your code should handle the case where
we have multiple graph structures for each class.
That is, G can be either 10 × 2 (shared
structure) or 10 × 2 × 2 (different structure for each class), and your function should be
able to choose the right graph based on the dimension of G. Please follow the instructions
in the comments in this file to complete the implementation. You may refer to the usage
of this function in LearnCPDsGivenGraph.m.
The log likelihood of the model over the dataset measures how the model fits the observed data.
This measure allows us to compare different model choices quantitatively.
4.1 Learning with Known Graph Structures
We are now ready to learn the model parameters. You should write code to fit the model given
the graph structure.
• LearnCPDsGivenGraph.m (25 points): This is the function where we learn the parameters.
This function takes the dataset and a known graph structure G as input, looks into G to
decide which parameterization is used for each of the body parts. So this function handles
all the model choices we discussed in a unified interface. We have already written some
infrastructure code in this function for you, and all you need to do is to invoke the functions
you just implemented by feeding them with the correct parameters, and write the output
in the correct format into the parameter structure array P. Please follow the instructions
given as comments in this file to complete the implementation. You may look into the
sample output exampleOUTPUT.t4o1, which contains the model P learned for the given
graph exampleOUTPUT.t4a2, to understand the structure of P.
We evaluate the learned models on the test data provided. Fill in the following function:
• ClassifyDataset.m (15 points): This function takes a dataset, a learned model P, and a
graph structure G. It then predicts the class labels for all test examples in the dataset, and
computes the accuracy of the model on the entire dataset. To perform the classification
task, you need to find the most likely class label assignment for each test example, i.e., the
class label that has the highest joint probability with the observed instance. The accuracy
is then computed as {number of correctly classified instances}/{total number of instances}.
Now, you can learn the model parameters for the difference model choices. In particular, run
the following,
[P1 likelihood1] = LearnCPDsGivenGraph(trainData.data, G1, trainData.labels);
which learns the parameters for the naive Bayes model. We can run the classification task
for the naive Bayes model you just learned,
accuracy1 = ClassifyDataset(testData.data, testData.labels, P1, G1);
How does the naive Bayes model perform? Now let’s visualize the learned model by sampling
from the learned parameters,
VisualizeModels(P1, G1);
CS228 Programming Assignment #8 8
Now it should be clear why the model is not performing so well – body parts are disconnected
in this model. Now let’s try to improve the model by putting the CLG model to work:
[P2 likelihood2] = LearnCPDsGivenGraph(trainData.data, G2, trainData.labels);
Repeat the classification task for the new model, (P2, G2). Compare the log-likelihood
scores and the accuracies. You should be able to see a higher likelihood score and a better
accuracy for the new model. If your implementation is correct, you should get accuracies of 79%
and 84% respectively. Visualize the model, and see where the improvement is coming from. The
structure for humans should now be clearly seen.
4.2 Learning Graph Structures
In our previous attempt, we tried to learn model parameters for humans and aliens with the
same skeletal structure. When visualizing the models, you should be able to see that the model
captures the human body very well, but fails on aliens. Some body parts are detached from the
rest of body, and the body structure is not clear. This is because we are using a graph structure
different from the true skeletal structure of the aliens. Therefore, instead of fitting the data to
a given structure, we will try to learn the structures from data.
We want to learn a tree-structured model for each class. Given a scoring function that
satisfies decomposability and score-equivalence, we can compute the score, or weight, between
all pairs of variables, and find the maximum spanning tree. Using the likelihood score, we set
the edge weights to be:
wi→j = ScoreL(Oi
|Oj : D) − ScoreL(Oi
: D) = M · IPˆ(Oi
, Oj ) (14)
where M is the number of instances and IPˆ is the mutual information with respect to distribution
Pˆ. We have learned in class the definition of mutual information between discrete variables and
how to compute it. It turns out that the same concept applies to continuous variables as well,
and it has a closed form expression for Gaussian variables. In general, the mutual information
between two multidimensional Gaussian variables X and Y can be computed by:
I(X, Y) = 1
2
log
|ΣXX| · |ΣYY|
|Σ|
(15)
and
Σ =
ΣXX ΣXY
ΣYX ΣYY
(16)
where | · | denotes the determinant of a matrix, and Σ denotes the covariance matrix.
Now we are set to use it to learn the graph structure for each class. This is done in the
following functions that you will implement.
• LearnGraphStructure.m (10 points): Given the functions GaussianMutualInformation.m
and MaxSpanningTree.m that we provide, learning the tree-structured graph should be
quite straightforward. You need to fill in the code to compute the weights between every pair of body part nodes. Computing the maximum spanning tree from the weights is
already implemented for you. Follow the instructions given in the code to complete the
implementation.
• LearnGraphAndCPDs.m (10 points): This function learns the parameters, as well as the
graph structure. For learning the graph structure, you should call the function LearnGraphStructure.m
CS228 Programming Assignment #8 9
you just implemented, and use the provided function ConvertAtoG.m to convert the maximum spanning tree to the desired graph structure G. You can copy most of the code from
LearnCPDsGivenGraph.m. The only difference is that instead of using any of the provided
graph structures, we estimate the graph structure for each of the classes.
When you are done, try:
[P G likelihood3] = LearnGraphAndCPDs(trainData.data, trainData.labels);
ClassifyDataset(testData.data, testData.labels, P, G);
If your implementation is correct, your model should achieve 93% accuracy! Compare the
log-likelihood score with previous results. Visualize the learned graph structure for aliens, and
you will see the body structure of the aliens from the samples of the model.
5 Conclusion
Congratulations! You have successfully used graphical models to learn classifiers to distinguish
between two poses drawing from different distributions and graph structures. Even though it
is a simple, synthetic dataset, we still saw how one can improve the models by incorporating
knowledge about the dataset. In the next programming assignment, you will extend what you
have done for this assignment to design models for activity recognition on real data.
