CS229 Lecture notes
Andrew Ng
Part IV
Generative Learning algorithms
So far, we’ve mainly been talking about learning algorithms that model
p
(
y

x
;
θ
), the conditional distribution of
y
given
x
.
For instance, logistic
regression modeled
p
(
y

x
;
θ
) as
h
θ
(
x
) =
g
(
θ
T
x
) where
g
is the sigmoid func
tion. In these notes, we’ll talk about a different type of learning algorithm.
Consider a classification problem in which we want to learn to distinguish
between elephants (
y
= 1) and dogs (
y
= 0), based on some features of
an animal.
Given a training set, an algorithm like logistic regression or
the perceptron algorithm (basically) tries to find a straight line—that is, a
decision boundary—that separates the elephants and dogs. Then, to classify
a new animal as either an elephant or a dog, it checks on which side of the
decision boundary it falls, and makes its prediction accordingly.
Here’s a different approach. First, looking at elephants, we can build a
model of what elephants look like.
Then, looking at dogs, we can build a
separate model of what dogs look like. Finally, to classify a new animal, we
can match the new animal against the elephant model, and match it against
the dog model, to see whether the new animal looks more like the elephants
or more like the dogs we had seen in the training set.
Algorithms that try to learn
p
(
y

x
) directly (such as logistic regression),
or algorithms that try to learn mappings directly from the space of inputs
X
to the labels
{
0
,
1
}
, (such as the perceptron algorithm) are called
discrim
inative
learning algorithms. Here, we’ll talk about algorithms that instead
try to model
p
(
x

y
) (and
p
(
y
)).
These algorithms are called
generative
learning algorithms. For instance, if
y
indicates whether a example is a dog
(0) or an elephant (1), then
p
(
x

y
= 0) models the distribution of dogs’
features, and
p
(
x

y
= 1) models the distribution of elephants’ features.
After modeling
p
(
y
) (called the
class priors
) and
p
(
x

y
), our algorithm
1
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can then use Bayes rule to derive the posterior distribution on
y
given
x
:
p
(
y

x
) =
p
(
x

y
)
p
(
y
)
p
(
x
)
.
Here, the denominator is given by
p
(
x
) =
p
(
x

y
= 1)
p
(
y
= 1) +
p
(
x

y
=
0)
p
(
y
= 0) (you should be able to verify that this is true from the standard
properties of probabilities), and thus can also be expressed in terms of the
quantities
p
(
x

y
) and
p
(
y
) that we’ve learned. Actually, if were calculating
p
(
y

x
) in order to make a prediction, then we don’t actually need to calculate
the denominator, since
arg max
y
p
(
y

x
)
=
arg max
y
p
(
x

y
)
p
(
y
)
p
(
x
)
=
arg max
y
p
(
x

y
)
p
(
y
)
.
1
Gaussian discriminant analysis
The first generative learning algorithm that we’ll look at is Gaussian discrim
inant analysis (GDA). In this model, we’ll assume that
p
(
x

y
) is distributed
according to a multivariate normal distribution. Lets talk brieﬂy about the
properties of multivariate normal distributions before moving on to the GDA
model itself.
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 Fall '09
 Algorithms, Normal Distribution, Machine Learning, Multivariate normal distribution, GDA

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