Chapter 13
Structured Probabilistic
Models for Deep Learning
Deep learning draws upon many modeling formalisms that researchers can use to
guide their design efforts and describe their algorithms. One of these formalisms is
the idea of structured probabilistic models. We have already discussed structured
probabilistic models briefly in Chapter 3.14. That brief presentation was sufficient
to understand how to use structured probabilistic models as a language to describe
some of the algorithms in part II of this book. Now, in part III, structured
probabilistic models are a key ingredient of many of the most important research
topics in deep learning. In order to prepare to discuss these research ideas, this
chapter describes structured probabilistic models in much greater detail. This
chapter is intended to be self-contained; the reader does not need to review the
earlier introduction before continuing with this chapter.
A structured probabilistic model is a way of describing a probability distribu-
tion, using a graph to describe which random variables in the probability distri-
bution interact with each other directly. Here we use “graph” in the graph theory
sense–a set of vertices connected to one another by a set of edges. Because the
structure of the model is defined by a graph, these models are often also referred
to as graphical models.
The graphical models research community is large and has developed many
different models, training algorithms, and inference algorithms. In this chap-
ter, we provide basic background on some of the most central ideas of graphical
models, with an emphasis on the concepts that have proven most useful to the
deep learning research community. If you already have a strong background in
graphical models, you may wish to skip most of this chapter. However, even a
graphical model expert may benefit from reading the final section of this chap-
ter, section 13.6, in which we highlight some of the unique ways that graphical
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models are used for deep learning algorithms. Deep learning practitioners tend to
use very different model structures, learning algorithms, and inference procedures
than are commonly used by the rest of the graphical models research community.
In this chapter, we identify these differences in preferences and explain the reasons
for them.
In this chapter we first describe the challenges of building large-scale proba-
bilistic models in section 13.1. Next, we describe how to use a graph to describe
the structure of a probability distribution in section 13.2. We then revisit the
challenges we described in section 13.1 and show how the structured approach to
probabilistic modeling can overcome these challenges in section 13.3. One of the
major difficulties in graphical modeling is understanding which variables need to
be able to interact directly, i.e., which graph structures are most suitable for a
given problem. We outline two approaches to resolving this difficulty by learning
about the dependencies in section 13.4. Finally, we close with a discussion of the
unique emphasis that deep learning practitioners place on specific approaches to
graphical modeling in section 13.6.
13.1 The Challenge of Unstructured Modeling
The goal of deep learning is to scale machine learning to the kinds of challenges
needed to solve artificial intelligence. This means being able to understand high-
dimensional data with rich structure. For example, we would like AI algorithms
to be able to understand natural images
1
, audio waveforms representing speech,
and documents containing multiple words and punctuation characters.
Classification algortihms can take such a rich high-dimensional input and sum-
marize it with a categorical label—what object is in a photo, what word is spoken
in a recording, what topic a document is about. The process of classification dis-
cards most of the information in the input and produces on a single output (or
a probability distribution over values of that single output). The classifier is also
often able to ignore many parts of the input. For example, when recognizing an
object in a photo, it is usually possible to ignore the background of the photo.
It is possible to ask probabilistic models to do many other tasks. These tasks
are often more expensive than classification. Some of them require producing
multiple output values. Most require a complete understanding of the entire
structure of the input, with no option to ignore sections of it. These tasks include
• Density estimation: given an input x, the machine learning system returns
an estimate of p(x). This requires only a single output, but it does require
1
A natural image is an image that might captured by a camera in a reasonably ordinary
environment, as opposed to synthetically rendered images, screenshots of web pages, etc.
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
a complete understanding of the entire input. If even one element of the
vector is unusual, the system must assign it a low probability.
• Denoising: given a damaged or incorrectly observed input
˜
x, the machine
learning system returns an estimate of the original or correct x. For exam-
ple, the machine learning system might be asked to remove dust or scratches
from an old photograph. This requires multiple outputs (every element of
the estimated clean example x) and an understanding of the entire input
(since even one damaged area will still reveal the final estimate as being
damaged).
• Missing value imputation: given the observations of some elements of x,
the model is asked to return estimates of or a probability distribution over
some or all of the unobserved elements of x. This requires multiple outputs,
and because the model could be asked to restore any of the elements of x,
it must understand the entire input.
• Sampling: the model generates new samples from the distribution p(x).
Applications include speech synthesis, i.e. producing new waveforms that-
sound like natural human speech. This requires multiple output values and
a good model of the entire input. If the samples have even one element
drawn from the wrong distribution, then the sampling process is wrong.
For an example of the sampling tasks on small natural images, see Fig. 13.1.
Modeling a rich distribution over thousands or millions of random variables
is a challenging task, both computationally and statistically. Suppose we only
wanted to model binary variables. This is the simplest possible case, and yet
already it seems overwhelming. For a small, 32 × 32 pixel color (RGB) image,
there are 2
3072
possible binary images of this form. This number is over 10
800
times larger than the estimated number of atoms in the universe.
In general, if we wish to model a distribution over a random vector x con-
taining n discrete variables capable of taking on k values each, then the naive
approach of representing P (x) by storing a lookup table with one probability
value per possible outcome requires k
n
parameters!
This is not feasible for several reasons:
• Memory: the cost of storing the representation : For all but very
small values of n and k, representing the distribution as a table will require
too many values to store.
• Statistical efficiency: As the number of parameters in a model increases,
so does the amount of training examples needed to choose the values of those
parameters using a statistical estimator. Because the table-based model has
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Figure 13.1: Probabilistic modeling of natural images. Top: Example 32 ×32 pixel color
images from the CIFAR-10 dataset (Krizhevsky and Hinton, 2009). Bottom: Samples
drawn from a structured probabilistic model trained on this dataset. Each sample appears
at the same position in the grid as the training example that is closest to it in Euclidean
space. This comparison allows us to see that the model is truly synthesizing new images,
rather than memorizing the training data. Contrast of both sets of images has been
adjusted for display. Figure reproduced with permission from (Courville et al., 2011).
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
an astronomical number of parameters, it will require an astronomically
large training set to fit accurately. Any such model will overfit the training
set very badly.
• Runtime: the cost of inference: Suppose we want to perform an infer-
ence task where we use our model of the joint distribution P (x) to compute
some other distribution, such as the marignal distribution P (x
1
) or the con-
ditional distribution P (x
2
| x
1
). Computing these distributions will require
summing across the entire table, so the runtime of these operations is as
high as the intractable memory cost of storing the model.
• Runtime: the cost of sampling: Likewise, suppose we want to draw a
sample from the model. The naive way to do this is to sample some value
u ∼ U(0, 1), then iterate through the table adding up the probability values
until they exceed u and return the outcome whose probability value was
added last. This requires reading through the whole table in the worst case,
so it has the same exponential cost as the other operations.
The problem with the table-based approach is that we are explicitly modeling
every possible kind of interaction between every possible subset of variables. The
probability distributions we encounter in real tasks are much simpler than this.
Usually, most variables influence each other only indirectly.
For example, consider modeling the finishing times of a team in a relay race.
Suppose the team consists of three runners, Alice, Bob, and Carol. At the start
of the race, Alice carries a baton and begins running around a track. After
completing her lap around the track, she hands the baton to Bob. Bob then runs
his own lap and hands the baton to Carol, who runs the final lap. We can model
each of their finishing times as a continuous random variable. Alice’s finishing
time does not depend on anyone else’s, since she goes first. Bob’s finishing time
depends on Alice’s, because Bob does not have the opportunity to start his lap
until Alice has completed hers. If Alice finishes faster, Bob will finish faster, all
else being equal. Finally, Carol’s finishing time depends on both her teammates.
If Alice is slow, Bob will probably finish late too, and Carol will have quite a late
starting time and thus is likely to have a late finishing time as well. However,
Carol’s finishing time depends only indirectly on Alice’s finishing time via Bob’s.
If we already know Bob’s finishing time, we won’t be able to estimate Carol’s
finishing time better by finding out what Alice’s finishing time was. This means
we can model the relay race using only two interactions: Alice’s effect on Bob,
and Bob’s effect on Carol. We can omit the third, indirect interaction between
Alice and Carol from our model.
Structured probabilistic models provide a formal framework for modeling only
direct interactions between random variables. This allows the models to have
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significantly fewer parameters which can in turn be estimated reliably from less
data. These smaller models also have dramatically reduced computation cost
in terms of storing the model, performing inference in the model, and drawing
samples from the model.
13.2 Using Graphs to Describe Model Structure
Structured probabilistc models use graphs (in the graph theory sense of “nodes”
or “vertices” connected by edges) to represent interactions between random vari-
ables. Each node represents a random variable. Each edge represents a direct
interaction. These direct interactions imply other, indirect interactions, but only
the direct interactions need to be explicitly modeled.
There is more than one way to describe the interactions in a probability dis-
tribution using a graph. In the following sections we describe some of the most
popular and useful approaches.
13.2.1 Directed Models
One kind of structured probabilistic model is the directed graphical model other-
wise known as the belief network or Bayesian network
2
(Pearl, 1985).
Directed graphical models are called “directed” because their edges are di-
rected, that is, they point from one vertex to another. This direction is repre-
sented in the drawing with an arrow. The direction of the arrow indicates which
variable’s probability distribution is defined in terms of the other’s. Drawing an
arrow from a to b means that we define the probability distribution over b via
a conditional distribution, with a as one of the variables on the right side of the
conditioning bar. In other words, the distribution over b depends on the value of
a.
Let’s continue with the relay race example from Section 13.1. Suppose we
name Alice’s finishing time t
0
, Bob’s finishing time t
1
, and Carol’s finishing time
t
2
. As we saw earlier, our estimate of t
1
depends on t
0
. Our estimate of t
2
depends directly on t
1
but only indirectly on t
0
. We can draw this relationship
in a directed graphical model, illustrated in Fig. 13.2.
Formally, a directed graphical model defined on variables x is defined by a
directed acyclic graph G whose vertices are the random variables in the model, and
a set of local conditional probability distributions p(x
i
| P a
G
(x
i
)) where P a
G
(x
i
)
2
Judea Pearl suggested using the term Bayes Network when one wishes to “emphasize the
judgmental” nature of the values computed by the network, i.e. to highlight that they usually
represent degrees of belief rather than frequencies of events.
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
t
0
t
1
t
2
Alice Bob Carol
Figure 13.2: A directed graphical model depicting the relay race example. Alice’s finishing
time t
0
influences Bob’s finishing time t
1
, because Bob does not get to start running until
Alice finishes. Likewise, Carol only gets to start running after Bob finishes, so Bob’s
finishing time t
1
influences Carol’s finishing time t
2
.
gives the parents of x
i
in G. The probability distribution over x is given by
p(x) = Π
i
p(x
i
| Pa
G
(x
i
)).
In our relay race example, this means that, using the graph drawn in Fig. 13.2,
p(t
0
, t
1
, t
2
) = p(t
0
)p(t
1
| t
0
)p(t
2
| t
1
).
This is our first time seeing a structured probabilistic model in action. We
can examine the cost of using it, in order to observe how structured modeling has
many advantages relative to unstructured modeling.
Suppose we represented time by discretizing time ranging from minute 0 to
minute 10 into 6 second chunks. This would make t
0
, t
1
, and t
2
each be discrete
variables with 100 possible values. If we attempted to represent p(t
0
, t
1
, t
2
) with a
table, it would need to store 999,999 values (100 values of t
0
× 100 values of t
1
×
100 values of t
2
, minus 1, since the probability of one of the configurations is made
redundant by the constraint that the sum of the probabilities be 1). If instead,
we only make a table for each of the conditional probability distributions, then
the distribution over t
0
requires 99 values, the table defining t
1
given t
0
requires
9900 values, and so does the table defining t
2
and t
1
. This comes to a total of
19,899 values. This means that using the directed graphical model reduced our
number of parameters by a factor of more than 50!
In general, to model n discrete variables each having k values, the cost of the
single table approach scales like O(k
n
), as we’ve observed before. Now suppose
we build a directed graphical model over these variables. If m is the maximum
number of variables appearing (on either side of the conditioning bar) in a single
conditional probability distribution, then the cost of the tables for the directed
model scales like O(k
m
). As long as we can design a model such that m << n,
we get very dramatic savings.
In other words, so long as each variable has few parents in the graph, the
distribution can be represented with very few parameters. Some restrictions on
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the graph structure (e.g. it is a tree) can also guarantee that operations like com-
puting marginal or conditional distributions over subsets of variables are efficient.
It’s important to realize what kinds of information can be encoded in the
graph, and what can’t be. The graph just encodes simplifying assumptions about
which variables are conditionally independent from each other. It’s also possible
to make other kinds of simplifying assumptions. For example, suppose we assume
Bob always runs the same regardless of how Alice performed. (In reality, Alice’s
performance probably influences Bob’s performance–depending on Bob’s person-
ality, if Alice runs especially fast in a given race, this might encourage Bob to
push hard and match her exceptional performance, or it might make him over-
confident and lazy). Then the only effect Alice has on Bob’s finishing time is
that we must add Alice’s finishing time to the total amount of time we think Bob
needs to run. This observation allows us to define a model with O(k) parameters
instead of O(k
2
). However, note that t
0
and t
1
are still directly dependent with
this assumption, because t
1
represents the absolute time at which Bob finishes,
not the total time he himself spends running. This means our graph must still
contain an arrow from t
0
to t
1
. The assumption that Bob’s personal running time
is independent from all other factors cannot be encoded in a graph over t
0
, t
1
,
and t
2
. Instead, we encode this information in the definition of the conditional
distribution itself. The conditional distribution is no longer a k × k − 1 element
table indexed by t
0
and t
1
but is now a slightly more complicated formula using
only k − 1 parameters. The directed graphical model syntax does not place any
constraint on how we define our conditional distributions. It only defines which
variables they are allowed to take in as arguments.
13.2.2 Undirected Models
Directed graphical models give us one language for describing structured proba-
bilistic models. Another popular language is that of undirected models, otherwise
known as Markov random fields (MRFs) or Markov networks (Kindermann,
1980). As their name implies, undirected models use graphs whose edges are
undirected.
Directed models are most naturally applicable to situations where there is
a clear reason to draw each arrow in one particular direction. Often these are
situations where we understand the causality, and the causality only flows in one
direction. One such situation is the relay race example. Earlier runners affects
the finishing times of later runners; later runners do not affect the finishing times
of earlier runners.
Not all situations we might want to model have such a clear direction to
their interactions. When the interactions seem to have no intrinsic direction, or
to operate in both directions, it may be more appropriate to use an undirected
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h
r
h
y
h
c
Figure 13.3: An undirected graph representing how your roommate’s health h
r
, your
health h
y
, and your work colleague’s health h
c
affect each other. You and your roommate
might infect each other with a cold, and you and your work colleague might do the same,
but assuming that your roommate and your colleague don’t know each other, they can
only infect each other indirectly via you.
model.
As an example of such a situation, suppose we want to model a distribution
over three binary variables: whether or not you are sick, whether or not your
coworker is sick, and whether or not your roommate is sick. As in the relay race
example, we can make simplifying assumptions about the kinds of interactions
that take place. Assuming that your coworker and your roommate do not know
each other, it is very unlikely that one of them will give the other a disease such
as a cold directly. This event can be seen as so rare that it is acceptable not
to model it. However, it is reasonably likely that either of them could give you
a cold, and that you could pass it on to the other. We can model the indirect
transmission of a cold from your coworker to your roommate by modeling the
transmission of the cold from your coworker to you and the transmission of the
cold from you to your roommate.
In this case, it’s just as easy for you to cause your roommate to get sick as
it is for your roommate to make you sick, so there is not a clean, uni-directional
narrative on which to base the model. This motivates using an undirected model.
As with directed models, if two nodes in an undirected model are connected by
an edge, then the random variables corresponding to those nodes interact with
each other directly. Unlike directed models, the edge in an undirected model has
no arrow, and is not associated with a conditional probability distribution.
Let’s call the random variable representing your health h
y
, the random variable
representing your roommate’s health h
r
, and the random variable representing
your colleague’s health h
c
. See Fig. 13.3 for a drawing of the graph representing
this scenario.
Formally, an undirected graphical model is a structured probabilistic model
defined on an undirected graph G. For each clique C in the graph
3
, a factor φ(C)
(also called a clique potential) measures the affinity of the variables in that clique
for being in each of their possible joint states. The factors are constrained to be
3
A clique of the graph is a subset of nodes that are all connected to each other by an edge of
the graph.
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
non-negative. Together they define an unnormalized probability distribution
˜p(x) = Π
C∈G
φ(C).
The unnormalized probability distribution is efficient to work with so long as
all the cliques are small. It encodes the idea that states with higher affinity are
more likely. However, unlike in a Bayesian network, there is little structure to the
definition of the cliques, so there is nothing to guarantee that multiplying them
together will yield a valid probability distribution. See Fig. 13.4 for an example
of reading factorization information from an undirected graph.
Our example of the cold spreading between you, your roommate, and your
colleague contains two cliques. One clique contains h
y
and h
c
. The factor for this
clique can be defined by a table, and might have values resembling these:
h
y
= 0 h
y
= 1
h
c
= 0 2 1
h
c
= 1 1 10
A state of 1 indicates good health, while a state of 0 indicates poor health
(having been infected with a cold). Both of you are usually healthy, so the
corresponding state has the highest affinity. The state where only one of you
is sick has the lowest affinity, because this is a rare state. The state where both of
you are sick (because one of you has infected the other) is a higher affinity state,
though still not as common as the state where both are healthy.
To complete the model, we would need to also define a similar factor for the
clique containing h
y
and h
r
.
13.2.3 The Partition Function
While the unnormalized probability distribution is guaranteed to be non-negative
everywhere, it is not guaranteed to sum or integrate to 1. To obtain a valid
probability distribution, we must use the corresponding normalized probability
distribution
4
:
p(x) =
1
Z
˜p(x)
where Z is the value that results in the probability distribution summing or
integrating to 1:
Z =
Z
˜p(x)dx.
4
A distribution defined by normalizing a product of clique potentials is also called a Gibbs
distribution.
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You can think of Z as a constant when the φ functions are held constant. Note
that if the φ functions have parameters, then Z is a function of those parameters.
It is common in the literature to write Z with its arguments omitted to save space.
Z is known as the partition function, a term borrowed from statistical physics.
Since Z is an integral or sum over all possible joint assignments of the state
x it is often intractable to compute. In order to be able to obtain the normalized
probability distribution of an undirected model, the model structure and the
definitions of the φ functions must be conducive to computing Z efficiently. In
the context of deep learning, Z is usually intractable, and we must resort to
approximations. Such approximate algorithms are the topic of Chapter 18.
One important consideration to keep in mind when designing undirected mod-
els is that it is possible for Z not to exist. This happens if some of the variables
in the model are continuous and the integral of ˜p over their domain diverges. For
example, suppose we want to model a single scalar variable x ∈ R with a single
clique potential φ(x) = x
2
. In this case,
Z =
Z
x
2
dx.
Since this integral diverges, there is no probability distribution corresponding
to this choice of φ(x). Sometimes the choice of some parameter of the φ func-
tions determines whether the probability distribution is defined. For example,
for φ(x; β) = exp
−βx
2
, the β parameter determines whether Z exists. Posi-
tive β results in a Gaussian distribution over x but all other values of β make φ
impossible to normalize.
One key difference between directed modeling and undirected modeling is that
directed models are defined directly in terms of probability distributions from the
start, while undirected models are defined more loosely by φ functions that are
then converted into probability distributions. This changes the intuitions one
must develop in order to work with these models. One key idea to keep in mind
while working with undirected models is that the domain of each of the variables
has dramatic effect on the kind of probability distribution that a given set of φ
functions corresponds to. For example, consider an n-dimensional vector-valued
random variable x and an undirected model parameterized by a vector of biases
b. Suppose we have one clique for each element of x, φ
i
(x
i
) = exp(b
i
x
i
). What
kind of probability distribution does this result in? The answer is that we don’t
have enough information, because we have not yet specified the domain of x. If
x ∈ R
n
, then the integral defining Z diverges and no probability distribution
exists. If x ∈ {0, 1}
n
, then p(x) factorizes into n independent distributions,
with p(x
i
= 1) = sigmoid (b
i
). If the domain of x is the set of elementary basis
vectors ({[1, 0, . . . , 0], [0, 1, . . . , 0], . . . , [0, 0, . . . , 1]} ) then p(x) = softmax(b), so a
large value of b
i
actually reduces p(x
j
= 1) for j 6= i. Often, it is possible to
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
A B C
D E F
Figure 13.4: This graph implies that p(A, B, C, D, E, F ) can be written as
1
Z
φ
A,B
(A, B)φ
B,C
(B, C)φ
A,D
(A, D)φ
B,E
(B, E)φ
E,F
(E, F ) for an appropriate choice of
the φ functions.
leverage the effect of a carefully chosen domain of a variable in order to obtain
complicated behavior from a relatively simple set of φ functions. We’ll explore a
practical application of this idea later, in Chapter 20.7.
13.2.4 Energy-Based Models
Many interesting theoretical results about undirected models depend on the as-
sumption that ∀x, ˜p(x) > 0. A convenient way to enforce this to use an energy-
based model (EBM) where
˜p(x) = exp(−E(x)) (13.1)
and E(x) is known as the energy function. Because exp(z) is positive for all
z, this guarantees that no energy function will result in a probability of zero for
any state x. Being completely free to choose the energy function makes learn-
ing simpler. If we learned the clique potentials directly, we would need to use
constrained optimization, and we would need to arbitrarily impose some specific
minimal probability value. By learning the energy function, we can use uncon-
strained optimization
5
, and the probabilities in the model can approach arbitrarily
close to zero but never reach it.
Any distribution of the form given by equation 13.1 is an example of a Boltz-
mann distribution. For this reason, many energy-based models are called Boltz-
mann machines. There is no accepted guideline for when to call a model an
energy-based model and when to call it a Boltzmann machines. The term Boltz-
mann machine was first introduced to describe a model with exclusively binary
variables, but today many models such as the mean-covariance restricted Boltz-
mann machine incorporate real-valued variables as well.
5
For some models, we may still need to use constrained optimization to make sure Z exists.
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
A B C
D E F
Figure 13.5: This graph implies that E(a, b, c, d, e, f) can be written as E
a,b
(a, b) +
E
b,c
(b, c) + E
a,d
(a, d) + E
b,e
(b, e) + E
e,f
(e, f) for an appropriate choice of the per-clique
energy functions. Note that we can obtain the φ functions in Fig. 13.4 by setting each φ
to the exp of the corresponding negative energy, e.g., φ
a,b
(a, b) = exp (−E(a, b)).
Cliques in an undirected graph correspond to factors of the unnormalized
probability function. Because exp(a) exp(b) = exp(a+b), this means that different
cliques in the undirected graph correspond to the different terms of the energy
function. In other words, an energy-based model is just a special kind of Markov
network: the exponentiation makes each term in the energy function correspond
to a factor for a different clique. See Fig. 13.5 for an example of how to read the
form of the energy function from an undirected graph structure.
One part of the definition of an energy-based model serves no functional pur-
pose from a machine learning point of view: the − sign in Eq. 13.1. This −
sign could be incorporated into the definition of E, or for many functions E the
learning algorithm could simply learn parameters with opposite sign. The − sign
is present primarily to preserve compatibility between the machine learning liter-
ature and the physics literature. Many advances in probabilistic modeling were
originally developed by statistical physicists, for whom E refers to actual, phys-
ical energy and does not have arbitrary sign. Terminology such as “energy” and
“partition function” remains associated with these techniques, even though their
mathematical applicability is broader than the physics context in which they were
developed. Some machine learning researchers (e.g., Smolensky (1986), who re-
ferred to negative energy as harmony) have chosen to emit the negation, but this
is not the standard convention.
13.2.5 Separation and D-Separation
The edges in a graphical model tell us which variables directly interact. We often
need to know which variables indirectly interact. Some of these indirect interac-
tions can be enabled or disabled by observing other variables. More formally, we
would like to know which subsets of variables are conditionally independent from
each other, given the values of other subsets of variables.
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
A S B
B A S B
(a) (b)
Figure 13.6: a) The path between random variable a and random variable b through s is
active, because s is not observed. This means that a and b are not separated. b) Here s
is shaded in, to indicate that it is observed. Because the only path between ra and b is
through s, and that path is inactive, we can conclude that a and b are separated given s.
A
B C
D
Figure 13.7: An example of reading separation properties from an undirected graph. Here
b is shaded to indicate that it is observed. Because observing b blocks the only path from
a to c, we say that a and c are separated from each other given b. The observation of b
also blocks one path between a and d, but there is a second, active path between them.
Therefore, a and d are not separated given b.
Identifying the conditional independences in a graph is very simple in the case
of undirected models. In this case, conditional independence implied by the graph
is called separation. We say that a set of variables A is separated from another set
of variables B given a third set of variables S if the graph structure implies that A
is independent from B given S. If two variables a and b are connected by a path
involving only unobserved variables, then those variables are not separated. If no
path exists between them, or all paths contain an observed variable, then they
are separated. We refer to paths involving only unobserved variables as “active”
and paths including an observed variable as “inactive.”
When we draw a graph, we can indicate observed variables by shading them
in. See Fig. 13.6 for a depiction of how active and inactive paths in an undirected
look when drawn in this way. See Fig. 13.7 for an example of reading separation
from an undirected graph.
Similar concepts apply to directed models, except that in the context of di-
rected models, these concepts are referred to as d-separation. The “d” stands for
“dependence.” D-separation for directed graphs is defined the same as separa-
tion for undirected graphs: We say that a set of variables A is d-separated from
another set of variables B given a third set of variables S if the graph structure
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
implies that A is independent from B given S.
As with undirected models, we can examine the independences implied by
the graph by looking at what active paths exist in the graph. As before, two
variables are dependent if there is an active path between them, and d-separated
if no such path exists. In directed nets, determining whether a path is active is
somewhat more complicated. See Fig. 13.8 for a guide to identifying active paths
in a directed model. See Fig. 13.9 for an example of reading some properties from
a graph.
It is important to remember that separation and d-separation tell us only
about those conditional independences that are implied by the graph. There is
no requirement that the graph imply all independences that are present. In
particular, it is always legitimate to use the complete graph (the graph with all
possible edges) to represent any distribution. In fact, some distributions contain
independences that are not possible to represent with existing graphical notation.
Context-specific independences are independences that are present dependent on
the value of some variables in the network. For example, consider a model of three
binary variables, a, b, and c. Suppose that when a is 0, b and c are independent,
but when a is 1, b is deterministically equal to c. Encoding the behavior when
a = 1 requires an edge connecting b and c. The graph then fails to indicate that
b and c are independent when a = 0.
In general, a graph will never imply that an independence exists when it does
not. However, a graph may fail to encode an independence.
13.2.6 Converting Between Undirected and Directed Graphs
In common parlance, we often refer to certain model classes as being undirected
or directed. For example, we typically refer to RBMs as undirected and sparse
coding as directed. This way of speaking can be somewhat leading, because no
probabilistic model is inherently directed or undirected. Instead, some models
are most easily described using a directed graph, or most easily described using
an undirected graph.
Ever probability distribution can be represented by either a directed model
or by an undirected model. In the worst case, one can always represent any
distribution by using a “complete graph.” In the case of a directed model, the
complete graph is any directed acyclic graph where we impose some ordering
on the random variables, and each variable has all other variables that precede
it in the ordering as its ancestors in the graph. For an undirected model, the
complete graph is simply a graph containing a single clique encompassing all of
the variables.
Of course, the utility of a graphical model is that the graph implies that some
variables do not interact directly. The complete graph is not very useful because
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
A S B
A S B
A S B
(a) (b)
A
S
B
A S B
C
(c) (d)
Figure 13.8: All of the kinds of active paths of length two that can exist between random
variables a and rb. a) Any path with arrows proceeding directly from a to b or vice versa.
This kind of path becomes blocked if s is observed. We have already seen this kind of
path in the relay race example. b) a and b are connected by a common cause s. For
example, suppose s is a variable indicating whether or not there is a hurricane and a and
b measure the wind speed at two different nearby weather monitoring outposts. If we
observe very high winds at station a, we might expect to also see high winds at b. This
kind of path can be blocked by observing s. If we already know there is a hurricane, we
expect to see high winds at b, regardless of what is observed at a. A lower than expected
wind at a (for a hurricane) would not change our expectation of winds at b (knowing
there is a hurricane). However, if s is not observed, then a and b are dependent, i.e., the
path is inactive. c) a and b are both parents of s. This is called a V-structure or the
collider case, and it causes a and a to be related by the explaining away effect. In this
case, the path is actually active when s is observed. For example, suppose s is a variable
indicating that your colleague is not at work. The variable a represents her being sick,
while b represents her being on vacation. If you observe that she is not at work, you
can presume she is probably sick or on vacation, but it’s not especially likely that both
have happened at the same time. If you find out that she is on vacation, this fact is
sufficient to explain her absence, and you can infer that she is probably not also sick. d)
The explaining away effect happens even if any descendant of s is observed! For example,
suppose that c is a variable representing whether you have received a report from your
colleague. If you notice that you have not received the report, this increases your estimate
of the probability that she is not at work today, which in turn makes it more likely that
she is either sick or on vacation. The only way to block a path through a V-structure is
to observe none of the descendants of the shared child.
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
A B
C
D E
Figure 13.9
From this graph, we can read out several d-separation properties. Examples
include:
• a and b are d-separated given the empty set.
• a and e are d-separated given c.
• d and e are d-separated given c.
We can also see that some variables are no longer d-separated when we observe
some variables:
• a and b are not d-separated given c.
• a and b are not d-separated given d.
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
it does not imply any independences. TODO figure complete graph
When we represent a probability distribution with a graph, we want to choose
a graph that implies as many independences as possible, without implying any
independences that do not actually exist.
From this point of view, some distributions can be represented more effi-
ciently using directed models, while other distributions can be represented more
efficiently using undirected models. In other words, directed models can encode
some independences that undirected models cannot encode, and vice versa.
Directed models are able to use one specific kind of substructure that undi-
rected models cannot represent perfectly. This substructure is called an immoral-
ity. The structure occurs when two random variables a and b are both parents
of a third random variable c, and there is no edge directly connecting a and b
in either direction. (The name “immorality” may seem strange; it was coined in
the graphical models literature as a joke about unmarried parents) To convert a
directed model with graph D into an undirected model, we need to create a new
graph U. For every pair of variables x and y, we add an undirected edge connect-
ing x and y to U if there is a directed edge (in either direction) connecting x and
y in D or if x and y are both parents in D of a third variable z. The resulting U
is known as a moralized graph. See Fig. 13.10 for examples of converting directed
models to undirected models via moralization.
Likewise, undirected models can include substructures that no directed model
can represent perfectly. Specifically, a directed graph D cannot capture all of
the conditional independences implied by an undirected graph U if U contains a
loop of length greater than three, unless that loop also contains a chord. A loop
is a sequence of variables connected by undirected edges, with the last variable
in the sequence connected back to the first variable in the sequence. A chord is
a connection between any two non-consecutive variables in this sequence. If U
has loops of length four or greater and does not have chords for these loops, we
must add the chords before we can convert it to a directed model. Adding these
chords discards some of the independence information that was encoded in U.
The graph formed by adding chords to U is known as a chordal or triangulated
graph, because all the loops can now be described in terms of smaller, triangular
loops. To build a directed graph D from the chordal graph, we need to also assign
directions to the edges. When doing so, we must not create a directed cycle in
D, or the result does not define a valid directed probabilistic model. One way
to assign directions to the edges in D is to impose an ordering on the random
variables, then point each edge from the node that comes earlier in the ordering
to the node that comes later in the ordering. TODO point to fig
IG HERE
TODO: started this above, need to scrap some some BNs encode indepen-
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
h
1
h
2
h
3
v
1
v
2
v
3
a b
c
a cb
h
1
h
2
h
3
v
1
v
2
v
3
a b
c
a cb
Figure 13.10: Examples of converting directed models to undirected models by construct-
ing moralized graphs. Left) This simple chain can be converted to a moralized graph
merely by replacing its directed edges with undirected edges. The resulting undirected
model implies exactly the same set of independences and conditional independences. Cen-
ter) This graph is the simplest directed model that cannot be converted to an undirected
model without losing some independences. This graph consists entirely of a single im-
morality. Because a and b are parents of c, they are connected by an active path when
c is observed. To capture this dependence, the undirected model must include a clique
encompassing all three variables. This clique fails to encode the fact that a⊥b. Right)
In general, moralization may add many edges to the graph, thus losing many implied
independences. For example, this sparse coding graph requires adding moralizing edges
between every pair of latent variables, thus introducing a quadratic number of new direct
dependences.
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
dences that MNs can’t encode, and vice versa example of BN that an MN can’t
encode: A and B are parents of C A is d-separated from B given the empty set
The Markov net requires a clique over A, B, and C in order to capture the active
path from A to B when C is observed This clique means that the graph cannot
imply A is separated from B given the empty set example of a MN that a BN
can’t encode: A, B, C, D connected in a loop BN cannot have both A d-sep D
given B, C and B d-sep C given A, D
In many cases, we may want to convert an undirected model to a directed
model, or vice versa. To do so, we choose the graph in the new format that implies
as many independences as possible, while not implying any independences that
were not implied by the original graph.
To convert a directed model D to an undirected model U, we re
TODO: conversion between directed and undirected models
13.2.7 Marginalizing Variables out of a Graph
TODO: marginalizing variables out of a graph
13.2.8 Factor Graphs
Factor graphs are another way of drawing undirected models that resolve an am-
biguity in the graphical representation of standard undirected model syntax. In
an undirected model, the scope of every φ function must be a subset of some
clique in the graph. However, it is not necessary that there exist any φ whose
scope contains the entirety of every clique. Factor graphs explicitly represent the
scope of each φ function. Specifically, a factor graph is a graphical representation
of an undirected model that consists of a bipartite undirected graph. Some of the
nodes are drawn as circles. These nodes correspond to random variables as in a
standard undirected model. The rest of the nodes are drawn as squares. These
nodes correspond to the factors φ of the unnormalized probability distribution.
Variables and factors may be connected with undirected edges. A variable and a
factor are connected in the graph if and only if the variable is one of the argu-
ments to the factor in the unnormalized probability distribution. No factor may
be connected to another factor in the graph, nor can a variable be connected to a
variable. See Fig. 13.11 for an example of how factor graphs can resolve ambiguity
in the interpretation of undirected networks.
13.3 Advantages of Structured Modeling
TODO– note that we have already shown that some things are cheaper in the
sections where we introduce the modeling syntax
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
A B
C
A
B
A B
C
f
1
A B
C
f
1
f
2
f
3
(a) (b) (c)
Figure 13.11: An example of how a factor graph can resolve ambiguity in the interpre-
tation of undirected networks. a) An undirected network with a clique involving three
variables a, b, and c. b) A factor graph corresponding to the same undirected model.
This factor graph has one factor over all three variables. c) Another valid factor graph
for the same undirected model. This factor graph has three factors, each over only two
variables. Note that representation, inference, and learning are all asymptotically cheaper
in (c) compared to (b), even though both require the same undirected graph to represent.
TODO: make sure figure respects random variable notation
TODO: revisit each of the three challenges from sec:unstructured TODO:
hammer point that graphical models convey information by leaving edges out
TODO: need to show reduced cost of sampling, but first reader needs to know
about ancestral and gibbs sampling.... TODO: benefit of separating representa-
tion from learning and inference
13.4 Learning About Dependencies
We consider here two types of random variables: observed or “visible” variables
v and latent or “hidden” variables h. The observed variables v correspond to the
variables actually provided in the data set during training. h consists of variables
that are introduced to the model in order to help it explain the structure in v.
Generally the exact semantics of h depend on the model parameters and are
created by the learning algorithm. The motivation for this is twofold.
13.4.1 Latent Variables Versus Structure Learning
Often the different elements of v are highly dependent on each other. A good
model of v which did not contain any latent variables would need to have very
large numbers of parents per node in a Bayesian network or very large cliques in a
Markov network. Just representing these higher order interactions is costly–both
in a computational sense, because the number of parameters that must be stored
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
in memory scales exponentially with the number of members in a clique, but also
in a statistical sense, because this exponential number of parameters requires a
wealth of data to estimate accurately.
There is also the problem of learning which variables need to be in such large
cliques. An entire field of machine learning called structure learning is devoted
to this problem . For a good reference on structure learning, see (Koller and
Friedman, 2009). Most structure learning techniques are a form of greedy search.
A structure is proposed, a model with that structure is trained, then given a score.
The score rewards high training set accuracy and penalizes model complexity.
Candidate structures with a small number of edges added or removed are then
proposed as the next step of the search, and the search proceeds to a new structure
that is expected to increase the score.
Using latent variables instead of adaptive structure avoids the need to perform
discrete searches and multiple rounds of training. A fixed structure over visible
and hidden variables can use direct interactions between visible and hidden units
to impose indirect interactions between visible units. Using simple parameter
learning techniques we can learn a model with a fixed structure that imputes the
right structure on the marginal p(v).
13.4.2 Latent Variables for Feature Learning
Another advantage of using latent variables is that they often develop useful
semantics.
As discussed in section 3.10.6, the mixture of Gaussians model learns a latent
variable that corresponds to which category of examples the input was drawn
from. This means that the latent variable in a mixture of Gaussians model can
be used to do classification.
In Chapter 15 we saw how simple probabilistic models like sparse coding learn
latent variables that can be used as input features for a classifier, or as coordinates
along a manifold. Other models can be used in this same way, but deeper models
and models with different kids of interactions can create even richer descriptions
of the input. Most of the approaches mentioned in sec. 13.4.2 accomplish feature
learning by learning latent variables. Often, given some model of v and h, it
turns out that E[h | v] TODO: uh-oh, is there a collision between set notation
and expectation notation? or argmax
h
p(h, v) is a good feature mapping for v.
TODO: appropriate links to Monte Carlo methods chapter spun off from here
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
13.5 Inference and Approximate Inference Over La-
tent Variables
As soon as we introduce latent variables in a graphical model, this raises the
question: how to choose values of the latent variables h given values of the visible
variables x? This is what we call inference, in particular inference over the latent
variables. The general question of inference is to guess some variables given others.
TODO: inference has definitely been introduced above... TODO: mention
loopy BP, show how it is very expensive for DBMs
TODO: briefly explain what variational inference is and reference approximate
inference chapter
13.5.1 Reparametrization Trick
Sometimes, in order to estimate the stochastic gradient of an expected loss over
some random variable h, with respect to parameters that influence h, we would
like to compute gradients through h, i.e., on the parameters that influenced the
probability distribution from which h was sampled. If h is continuous-valued,
this is generally possible by using the reparametrization trick, i.e., rewriting
h ∼ p(h | θ) (13.2)
as
h = f(θ, η) (13.3)
where η is some independent noise source of the appropriate dimension with
density p(η), and f is a continuous (differentiable almost everywhere) function.
Basically, the reparametrization trick is the idea that if the random variable to
be integrated over is continuous, we can back-propagate through the process that
gave rise to it in order to figure how to change that process.
For example, let us suppose we want to estimate the expected gradient
∂
∂θ
Z
L(h)p(h | θ)dh (13.4)
where the parameters θ influences the random variable h which in term influence
our loss L. A very efficient (Kingma and Welling, 2014b; Rezende et al., 2014)
way to achieve
6
this is to perform the reparametrization in Eq. 13.3 and the
corresponding change of variable in the integral of Eq. 13.4, integrating over η
rather than h:
∂
∂θ
Z
L(f(θ, η))p(eta)dη. (13.5)
6
compared to approaches that do not back-propagate through the generation of h
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
We can now more easily enter the derivative in the integral, getting
g =
Z
∂L(f(θ, η))
∂θ
p(eta)dη.
Finally, we get a stochastic gradient estimator
ˆg =
∂L(f(θ, η))
∂θ
where we sampled η ∼ p(η) and E[ˆg] = g.
This trick was used by Bengio (2013b); Bengio et al. (2013a) to train a neu-
ral network with stochastic hidden units. It was described at the same time
by Kingma (2013), but see the further developments in Kingma and Welling
(2014b). It was used to train generative stochastic networks (GSNs) (Bengio
et al., 2014a,b), described in Section 20.11, which can be viewed as recurrent
networks with noise injected both in input and hidden units (with each time step
corresponding to one step of a generative Markov chain). The reparametriza-
tion trick was also used to estimate the parameter gradient in variational auto-
encoders (Kingma and Welling, 2014a; Rezende et al., 2014; Kingma et al., 2014),
which are described in Section 20.9.3.
13.6 The Deep Learning Approach to Structured Prob-
abilistic Models
Deep learning practictioners generally use the same basic computational tools
as other machine learning practitioners who work with structured probabilistic
models. However, in the context of deep learning, we usually make different design
decisions about how to combine these tools, resulting in overall algorithms and
models that have a very different flavor from more traditional graphical models.
The most striking difference between the deep learning style of graphical model
design and the traditional style of graphical model design is that the deep learn-
ing style heavily emphasizes the use of latent variables. Deep learning models
typically have more latent variables than observed variables. Moreover, the prac-
titioner typically does not intend for the latent variables to take on any specific
semantics ahead of time— the training algorithm is free to invent the concepts it
needs to model a particular dataset. The latent variables are usually not very easy
for a human to interpret after the fact, though visualization techniques may allow
some rough characterization of what they represent. Complicated non-linear in-
teractions between variables are accomplished via indirect connections that flow
through multiple latent variables. By contrast, traditional graphical models usu-
ally contain variables that are at least occasionally observed, even if many of the
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
variables are missing at random from some training examples. Complicated non-
linear interactions between variables are modeled by using higher-order terms,
with structure learning algorithms used to prune connections and control model
capacity. When latent variables are used, they are often designed with some spe-
cific semantics in mind—the topic of a document, the intelligence of a student,
the disease causing a patient’s symptoms, etc. These models are often much more
interpretable by human practitioners and often have more theoretical guarantees,
yet are less able to scale to complex problems and are not reuseable in as many
different contexts as deep models.
Another obvious difference is the kind of graph structure typically used in
the deep learning approach. This is tightly linked with the choice of inference
algorithm. Traditional approaches to graphical models typically aim to maintain
the tractability of exact inference. When this constraint is too limiting, a popular
exact inference algorithm is loopy belief propagation. Both of these approaches
often work well with very sparsely connected graphs. By comparison, very few
interesting deep models admit exact inference, and loopy belief propagation is al-
most never used for deep learning. Most deep models are designed to make Gibbs
sampling or variational inference algorithms, rather than loopy belief propagation,
efficient. Another consideration is that deep learning models contain a very large
number of latent variables, making efficient numerical code essential. As a result
of these design constraints, most deep learning models are organized into regu-
lar repeating patterns of units grouped into layers, but neighboring layers may
be fully connected to each other. When sparse connections are used, they usu-
ally follow a regular pattern, such as the block connections used in convolutional
models.
Finally, the deep learning approach to graphical modeling is characterized by
a marked tolerance of the unknown. Rather than simplifying the model until
all quantities we might want can be computed exactly, we increase the power of
the model until it is just barely possible to train or use. We often use models
whose marginal distributions cannot be computed, and are satisfied simply to
draw approximate samples from these models. We often train models with an
intractable objective function that we cannot even approximate in a reasonable
amount of time, but we are still able to approximately train the model if we can
efficiently obtain an estimate of the gradient of such a function. The deep learning
approach is often to figure out what the minimum amount of information we
absolutely need is, and then to figure out how to get a reasonable approximation
of that information as quickly as possible.
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
h
1
h
2
h
3
v
1
v
2
v
3
h
4
Figure 13.12: An example RBM drawn as a Markov network
13.6.1 Example: The Restricted Boltzmann Machine
TODO: rework this section. Add pointer to Chapter 20.2. TODO what do we
want to exemplify here?
The restricted Boltzmann machine (RBM) (Smolensky, 1986) or harmonium
is an example of a model that TODO what do we want to exemplify here?
It is an energy-based model with binary visible and hidden units. Its energy
function is
E(v, h) = −b
>
v − c
>
h −v
>
W h
where b, c, and W are unconstrained, real-valued, learnable parameters. The
model is depicted graphically in Fig. 13.12. As this figure makes clear, an impor-
tant aspect of this model is that there are no direct interactions between any two
visible units or between any two hidden units (hence the “restricted,” a general
Boltzmann machine may have arbitrary connections).
The restrictions on the RBM structure yield the nice properties
p(h | v) = Π
i
p(h
i
| v)
and
p(v | h) = Π
i
p(v
i
| h).
The individual conditionals are simple to compute as well, for example
p(h
i
= 1 | v) = σ
v
>
W
:,i
+ b
i
.
Together these properties allow for efficient block Gibbs sampling, alternating
between sampling all of h simultaneously and sampling all of v simultaneously.
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
Since the energy function itself is just a linear function of the parameters, it
is easy to take the needed derivatives. For example,
∂
∂W
i,j
E
v,h
E(v, h) = −v
i
h
j
.
These two properties–efficient Gibbs sampling and efficient derivatives– make
it possible to train the RBM with stochastic approximations to ∇
θ
log Z.
13.6.2 The Computational Challenge with High-Dimensional Dis-
tributions
TODO: this whole section should probably just be cut, IG thinks YB has written
the same thing in 2-3 other places (ml.tex for sure, and maybe also manifolds.tex
and prob.tex, possibly others IG hasn’t read yet) YB doesn’t seem to have read
the intro part of this chapter which discusses these things in more detail, double
check to make sure there’s not anything left out above If this section is kept, it
needs cleanup, i.e. a instead A, etc. If this section is cut, need to search for refs
to it and move them to one of the other versions of it
High-dimensional random variables actually bring two challenges: a statistical
challenge and a computational challenge.
The statistical challenge was introduced in Section 5.12 and regards gener-
alization: the number of configurations we may want to distinguish can grow
exponentially with the number of dimensions of interest, and this quickly be-
comes much larger than the number of examples one can possibly have (or use
with bounded computational resources).
The computational challenge associated with high-dimensional distributions
arises because many algorithms for learning or using a trained model (especially
those based on estimating an explicit probability function) involve intractable
computations that grow exponentially with the number of dimensions.
With probabilistic models, this computational challenge arises because of in-
tractable sums (summing over an exponential number of configurations) or in-
tractable maximizations (finding the best out of an intractable number of config-
urations), discussed mostly in the third part of this book.
• Intractable inference: inference is discussed mostly in Chapter 19. It
regards the question of guessing the probable values of some variables A,
given other variables B, with respect to a model that captures the joint
distribution between A, B and C. In order to even compute such conditional
probabilities one needs to sum over the values of the variables C, as well as
compute a normalization constant which sums over the values of A and C.
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CHAPTER 13. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
• Intractable normalization constants (the partition function): the
partition function is discussed mostly in Chapter 18. Normalizing constants
of probability functions come up in inference (above) as well as in learning.
Many probabilistic models involve such a constant. Unfortunately, the pa-
rameters (which we want to tune) influence that constant, and computing
the gradient of the partition function with respect to the parameters is
generally as intractable as computing the partition function itself. Monte-
Carlo Markov chain (MCMC) methods (Chapter 14) are often used to deal
with the partition function (computing it or its gradient) but they may also
suffer from the curse of dimensionality, when the number of modes of the
distribution of interest is very large, and these modes are well separated
(Section 14.2).
One way to confront these intractable computations is to approximate them,
and many approaches have been proposed, discussed in the chapters listed above.
Another interesting way would be to avoid these intractable computations alto-
gether by design, and methods that do not require such computations are thus
very appealing. Several generative models based on auto-encoders have been
proposed in recent years, with that motivation, and are discussed at the end of
Chapter 20.
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