This presentation is delivered by the Stanford Center for Professional
Development.
So what I want to do today is talk about a different type of learning algorithm, and, in particular,
start to talk about generative learning algorithms
and the specific algorithm called Gaussian Discriminant Analysis.
Take a slight digression, talk about Gaussians, and I'll briefly discuss
generative versus discriminative learning algorithms,
and then hopefully wrap up today's lecture with a discussion of Naive Bayes and
the Laplace Smoothing.
So just to motivate our
discussion on generative learning algorithms, right, so by way of contrast,
the source of classification algorithms we've been talking about
I think of algorithms that do this. So you're given a training set,
and
if you run an algorithm right, we just see progression on those training sets.
The way I think of logistic regression is that it's trying to find " look at the date and is
trying to find a straight line to divide the crosses and O's, right? So it's, sort of,
trying to find a straight line. Let
me " just make the days a bit noisier.
Trying to find a straight line
that separates out
the positive and the negative classes as well as pass the law, right? And,
in fact, it shows it on the laptop. Maybe just use the screens or the small
monitors for this.
In fact,
you can see there's the data
set with
logistic regression,
and so
I've initialized the parameters randomly, and so logistic regression is, kind
of, the outputting " it's
the, kind of, hypothesis that
iteration zero is that straight line shown in the bottom right.
And so after one iteration and creating descent, the straight line changes a bit.
After two iterations, three,
four,
until logistic regression converges
and has found the straight line that, more or less, separates the positive and negative class, okay? So
you can think of this
as logistic regression,
sort of, searching for a line that separates the positive and the negative classes.
What I want to do today is talk about an algorithm that does something slightly
different,
and to motivate us, let's use our old example of trying to classifythe
team malignant cancer and benign cancer, right? So a patient comes in
and they have a cancer, you want to know if it's a malignant or a harmful cancer,
or if it's a benign, meaning a harmless cancer.
So rather than trying to find the straight line to separate the two classes, here's something else
we could do.
We can go from our training set
and look at all the cases of malignant cancers, go through, you know, look for our training set for all the
positive examples of malignant cancers,
and we can then build a model for what malignant cancer looks like.
Then we'll go for our training set again and take out all of the examples of benign cancers,
and then we'll build a model for what benign cancers look like, okay?
And then
when you need to
classify a new example, when you have a new patient, and you want to decide is this cancer malignant
or benign,
you then take your new cancer, and you
match it to your model of malignant cancers,
and you match it to your model of benign cancers, and you see which model it matches better, and
depending on which model it matches better to, you then
predict whether the new cancer is malignant or benign,
okay?
So
what I just described, just this cross
of methods where you build a second model for malignant cancers and
a separate model for benign cancers
is called a generative learning algorithm,
and let me just, kind of, formalize this.
So in the models that we've
been talking about previously, those were actually
all discriminative learning algorithms,
and studied more formally, a discriminative learning algorithm is one
that either learns P of Y
given X directly,
or even
learns a hypothesis
that
outputs value 0, 1 directly,
okay? So logistic regression is an example of
a discriminative learning algorithm.
In contrast, a generative learning algorithm of
models P of X given Y.
The probability of the features given the class label,
and as a technical detail, it also models P of Y, but that's a less important thing, and the
interpretation of this is that a generative model
builds a probabilistic model for
what the features looks like,
conditioned on the
class label, okay? In other words, conditioned on whether a cancer is
malignant or benign, it models probability distribution over what the features
of the cancer looks like.
Then having built this model " having built a model for P of X
given Y and P of Y,
then by Bayes rule, obviously, you can compute P of Y given 1,
conditioned on X.
This is just
P of X
given Y = 1
times P of X
divided by P of X,
and, if necessary,
you can calculate the denominator
using this, right?
And so by modeling P of X given Y
and modeling P of Y, you can actually use Bayes rule to get back to P of Y given
X,
but a generative model -
generative learning algorithm starts in modeling P of X given Y, rather than P of Y
given X, okay?
We'll talk about some of the tradeoffs, and why this may be a
better or worse idea than a discriminative model a bit later.
Let's go for a specific example of a generative learning algorithm,
and for
this specific motivating example, I'm
going to assume
that your input feature is X
and RN
and are
continuous values, okay?
And under this assumption, let
me describe to you a specific algorithm called Gaussian Discriminant Analysis,
and the, I
guess, core assumption is that we're going to assume in the Gaussian discriminant analysis
model of that P of X given Y
is Gaussian, okay? So
actually just raise your hand, how many of you have seen a multivariate Gaussian before -
not a 1D Gaussian, but the higher range though?
Okay, cool, like maybe half of you, two-thirds of
you. So let me just say a few words about Gaussians, and for those of you that have seen
it before, it'll be a refresher.
So we say that a random variable Z is distributed Gaussian, multivariate Gaussian as - and the
script N for normal
with
parameters mean U and covariance sigma squared. If
Z has a density 1
over 2 Pi, sigma
2,
okay?
That's the formula for the density as a generalization of the one dimension of Gaussians and no
more the familiar bell-shape curve. It's
a high dimension vector value random variable
Z.
Don't worry too much about this formula for the density. You rarely end up needing to
use it,
but the two key quantities are this
vector mew is the mean of the Gaussian and this
matrix sigma is
the covariance matrix -
covariance,
and so
sigma will be equal to,
right, the definition of covariance of a vector valued random variable
is X - U, X - V conspose,
okay?
And, actually, if this
doesn't look familiar to you,
you might re-watch the
discussion section that the TAs held last Friday
or the one that they'll be holding later this week on, sort of, a recap of
probability, okay?
So
multi-grade Gaussians is parameterized by a mean and a covariance, and let me just -
can I
have the laptop displayed, please?
I'll just go ahead and actually show you,
you know, graphically, the effects of
varying a Gaussian -
varying the parameters of a Gaussian.
So what I have up here
is the density of a zero mean Gaussian with
covariance matrix equals the identity. The covariance matrix is shown in the upper right-hand
corner of the slide, and
there's the familiar bell-shaped curve in two dimensions.
And so if I shrink the covariance matrix, instead of covariance your identity, if
I shrink the covariance matrix, then the Gaussian becomes more peaked,
and if I widen the covariance, so like same = 2, 2,
then
the distribution - well, the density becomes more spread out, okay?
Those vectors stand at normal, identity
covariance one.
If I increase
the diagonals of a covariance matrix, right,
if I make the variables correlated, and the
Gaussian becomes flattened out in this X = Y direction, and
increase it even further,
then my variables, X and Y, right - excuse me, it goes Z1 and Z2
are my two variables on a horizontal axis become even more correlated. I'll just show the same thing in
contours.
The standard normal of distribution has contours that are - they're actually
circles. Because of the aspect ratio, these look like ellipses.
These should actually be circles,
and if you increase the off diagonals of the Gaussian covariance matrix,
then it becomes
ellipses aligned along the, sort of,
45 degree angle
in this example.
This is the same thing. Here's an example of a Gaussian density with negative covariances.
So now the correlation
goes the other way, so that even strong [inaudible] of covariance and the same thing in
contours. This is a Gaussian with negative entries on the diagonals and even
larger entries on the diagonals, okay?
And other parameter for the Gaussian is the mean parameters, so if this is - with mew0,
and as he changed the mean parameter,
this is mew equals 0.15,
the location of the Gaussian just moves around, okay?
All right. So that was a quick primer on what Gaussians look like, and here's
as a roadmap or as a picture to keep in mind, when we described the Gaussian discriminant
analysis algorithm, this is what we're going to do. Here's
the training set,
and in the Gaussian discriminant analysis algorithm,
what I'm going to do is I'm going to look at the positive examples, say the crosses,
and just looking at only the positive examples, I'm gonna fit a Gaussian distribution to the
positive examples, and so
maybe I end up with a Gaussian distribution like that,
okay? So there's P of X given Y = 1.
And then I'll look at the negative examples, the O's in this figure,
and I'll fit a Gaussian to that, and maybe I get a
Gaussian
centered over there. This is the concept of my second Gaussian,
and together -
we'll say how later -
together these two Gaussian densities will define a separator for these two classes, okay?
And it'll turn out that the separator will turn out to be a little bit
different
from what logistic regression
gives you.
If you run logistic regression,
you actually get the division bound to be shown in the green line, whereas Gaussian discriminant
analysis gives you the blue line, okay? Switch back to chalkboard, please. All right. Here's the
Gaussian discriminant analysis model, put
into model P of Y
as a Bernoulli random variable as usual, but
as a Bernoulli random variable and parameterized by parameter phi; you've
seen this before.
Model P of X given Y = 0 as a Gaussian -
oh, you know what? Yeah,
yes, excuse me. I
thought this looked strange.
This
should be a sigma,
determined in a sigma to the one-half of the denominator there.
It's no big deal. It was - yeah,
well, okay. Right.
I
was listing the sigma to the determining the sigma to the one-half on a previous
board, excuse me. Okay,
and so I model P of X given Y = 0 as a Gaussian
with mean mew0 and covariance sigma to the sigma to
the minus one-half,
and
-
okay?
And so the parameters of this model are
phi,
mew0,
mew1, and sigma,
and so
I can now write down the likelihood of the parameters
as - oh, excuse
me, actually, the log likelihood of the parameters as the log of
that,
right?
So, in other words, if I'm given the training set, then
they can write down the log likelihood of the parameters as the log of, you know,
the probative probabilities of P of XI, YI, right?
And
this is just equal to that where each of these terms, P of XI given YI,
or P of YI is
then given
by one of these three equations on top, okay?
And I just want
to contrast this again with discriminative learning algorithms, right?
So
to give this a name, I guess, this sometimes is actually called
the Joint Data Likelihood - the Joint Likelihood,
and
let me just contrast this with what we had previously
when we're talking about logistic
regression. Where I said with the log likelihood of the parameter's theater
was log
of a product I = 1 to M, P of YI
given XI
and parameterized
by a theater, right?
So
back where we're fitting logistic regression models or generalized learning
models,
we're always modeling P of YI given XI and parameterized by a theater, and that was the
conditional
likelihood, okay,
in
which we're modeling P of YI given XI,
whereas, now,
regenerative learning algorithms, we're going to look at the joint likelihood which
is P of XI, YI, okay?
So let's
see.
So given the training sets
and using the Gaussian discriminant analysis model
to fit the parameters of the model, we'll do maximize likelihood estimation as usual,
and so you maximize your
L
with respect to the parameters phi, mew0,
mew1, sigma,
and so
if we find the maximum likelihood estimate of parameters, you find that phi is
-
the maximum likelihood estimate is actually no surprise, and I'm writing this down mainly as a practice for
indicating notation,
all right? So the maximum likelihood estimate for phi would be Sum over
I, YI Ăˇ M,
or written alternatively as Sum over -
all your training examples of indicator YI = 1 Ăˇ M, okay?
In other words, maximum likelihood estimate for a
newly parameter phi is just the faction of training examples with
label one, with Y equals 1.
Maximum likelihood estimate for mew0 is this, okay?
You
should stare at this for a second and
see if it makes sense.
Actually, I'll just write on the next one for mew1 while you do that.
Okay?
So what this is is
what the denominator is sum of your training sets indicated YI = 0.
So for every training example for which YI = 0, this will
increment the count by one, all right? So the
denominator is just
the number of examples
with
label zero, all right?
And then the numerator will be, let's
see, Sum from I = 1 for M, or every time
YI is equal to 0, this will be a one, and otherwise, this thing will be zero,
and so
this indicator function means that you're including
only the times for
which YI is equal to one - only the turns which Y is equal to zero
because for all the times where YI is equal to one,
this sum and will be equal to zero,
and then you multiply that by XI, and so the numerator is
really the sum
of XI's corresponding to
examples where the class labels were zero, okay?
Raise your hand if this makes sense. Okay, cool.
So just to say this fancifully,
this just means look for your training set,
find all the examples for which Y = 0,
and take the average
of the value of X for all your examples which Y = 0. So take all your negative fitting
examples
and average the values for X
and
that's mew0, okay? If this
notation is still a little bit cryptic - if you're still not sure why this
equation translates into
what I just said, do go home and stare at it for a while until it just makes sense. This is, sort
of, no surprise. It just says to estimate the mean for the negative examples,
take all your negative examples, and average them. So
no surprise, but this is a useful practice to indicate a notation.
[Inaudible]
divide the maximum likelihood estimate for sigma. I won't do that. You can read that in
the notes yourself.
And so having fit
the parameters find mew0, mew1, and sigma
to your data,
well, you now need to make a prediction. You
know, when you're given a new value of X, when you're given a new cancer, you need to predict whether
it's malignant or benign.
Your prediction is then going to be,
let's say,
the most likely value of Y given X. I should
write semicolon the parameters there. I'll just give that
- which is the [inaudible] of a Y
by Bayes rule, all right?
And that is, in turn,
just that
because the denominator P of X doesn't depend on Y,
and
if P of Y
is uniform.
In other words, if each of your constants is equally likely,
so if P of Y
takes the same value for all values
of Y,
then this is just arc X over Y, P of X
given Y, okay?
This happens sometimes, maybe not very often, so usually you end up using this
formula where you
compute P of X given Y and P of Y using
your model, okay? Student:Can
you give
us arc x? Instructor (Andrew Ng):Oh, let's see. So if you take - actually
let me.
So the min of
- arcomatics means the value for Y that maximizes this. Student:Oh, okay. Instructor (Andrew Ng):So just
for an example, the min of X - 5
squared is 0 because by choosing X equals 5, you can get this to be zero,
and the argument over X
of X - 5 squared is equal to 5 because 5 is the value of X that makes this minimize, okay?
Cool. Thanks
for
asking that. Instructor (Andrew Ng):Okay. Actually any other questions about this? Yeah? Student:Why is
distributive removing? Why isn't [inaudible] - Instructor (Andrew Ng):Oh, I see. By uniform I meant - I was being loose
here.
I meant if
P of Y = 0 is equal to P of Y = 1, or if Y is the
uniform distribution over
the set 0 and 1. Student:Oh. Instructor (Andrew Ng):I just meant - yeah, if P of Y = 0
zero = P of Y given 1. That's all I mean, see? Anything else?
All
right. Okay. So
it
turns out Gaussian discriminant analysis has an interesting relationship
to logistic
regression. Let me illustrate that.
So let's say you have a training set
- actually let me just go ahead and draw 1D training set, and that will
kind of work, yes, okay.
So let's say we have a training set comprising a few negative and a few positive examples,
and let's say I run Gaussian discriminate analysis. So I'll fit Gaussians to each of these two
densities - a Gaussian density to each of these two - to my positive
and negative training
examples,
and so maybe my
positive examples, the X's, are fit with a Gaussian like this,
and my negative examples I will fit, and you have a
Gaussian that looks like that, okay?
Now, I
hope this [inaudible]. Now, let's
vary along the X axis,
and what I want to do is I'll
overlay on top of this plot. I'm going to plot
P of Y = 1 - no, actually,
given X
for a variety of values X, okay?
So I actually realize what I should have done.
I'm gonna call the X's the negative examples, and I'm gonna call the O's the positive examples. It just
makes this part come in better.
So let's take a value of X that's fairly small. Let's say X is this value here on a horizontal
axis.
Then what's the probability of Y being equal to one conditioned on X? Well,
the way you calculate that is you write P of Y
= 1 given X, and then you plug in all these formulas as usual, right? It's P of X
given Y = 1, which is
your Gaussian density,
times P of Y = 1, you know, which is
essentially - this is just going to be equal to phi,
and then divided by,
right, P of X, and then this shows you how you can calculate this.
By using
these two Gaussians and my phi on P of Y, I actually compute what P of Y
= 1 given X is, and
in this case,
if X is this small, clearly it belongs to the left Gaussian. It's very unlikely to belong to
a positive class, and so
it'll be very small; it'll be very close to zero say, okay?
And then we can increment the value of X a bit, and study a different value of X, and
plot what is the P of Y given X - P of Y
= 1 given X, and, again, it'll be pretty small. Let's
use a point like that, right? At this point,
the two Gaussian densities
have equal value,
and if
I ask
if X is this value, right, shown by the arrow,
what's the probably of Y being equal to one for that value of X? Well, you really can't tell, so maybe it's about 0.5, okay? And if
you fill
in a bunch more points, you get a
curve like that,
and then you can keep going. Let's say for a point like that, you can ask what's the probability of X
being one? Well, if
it's that far out, then clearly, it belongs to this
rightmost Gaussian, and so
the probability of Y being a one would be very high; it would be almost one, okay?
And so you
can repeat this exercise
for a bunch of points. All right,
compute P of Y equals one given X for a bunch of points,
and if you connect up these points,
you find that the
curve you get [Pause] plotted
takes a form of sigmoid function, okay? So,
in other words, when you make the assumptions under the Gaussian
discriminant analysis model,
that P of X given Y is Gaussian,
when you go back and compute what P of Y given X is, you actually get back
exactly the same sigmoid function
that we're using which is the progression, okay? But it turns out the key difference is that
Gaussian discriminant analysis will end up choosing a different
position
and a steepness of the sigmoid
than would logistic regression. Is there a question? Student:I'm just
wondering,
the Gaussian of P of Y [inaudible] you do? Instructor (Andrew Ng):No, let's see. The Gaussian - so this Gaussian is
P of X given Y = 1, and
this Gaussian is P of X
given Y = 0; does that make sense? Anything else? Student:Okay. Instructor (Andrew Ng):Yeah? Student:When you drawing all the dots, how did you
decide what Y
given
P of X was? Instructor (Andrew Ng):What - say that again. Student:I'm sorry. Could you go over how you
figured out where
to draw each dot? Instructor (Andrew Ng):Let's see,
okay. So the
computation is as follows, right? The steps
are I have the training sets, and so given my training set, I'm going to fit
a Gaussian discriminant analysis model to it,
and what that means is I'll build a model for P of X given Y = 1. I'll
build
a model for P of X given Y = 0,
and I'll also fit a Bernoulli distribution to
P of Y, okay?
So, in other words, given my training set, I'll fit P of X given Y and P of Y
to my data, and now I've chosen my parameters
of find mew0,
mew1,
and the sigma, okay? Then
this is the process I went through
to plot all these dots, right? It's just I pick a point in the X axis,
and then I compute
P of Y given X
for that value of X,
and P of Y given 1 conditioned on X will be some value between zero and one. It'll
be some real number, and whatever that real number is, I then plot it on the vertical
axis,
okay? And the way I compute P of Y = 1 conditioned on X is
I would
use these quantities. I would use
P of X given Y
and P of Y, and, sort of, plug them into Bayes rule, and that allows me
to
compute P of Y given X
from these three quantities; does that make
sense? Student:Yeah. Instructor (Andrew Ng):Was there something more that -
Student:And how did you model P of X; is that - Instructor (Andrew Ng):Oh, okay. Yeah, so
-
well,
got this right
here. So P of X can be written as,
right,
so
P of X given Y = 0 by P of Y = 0 + P of X given Y = 1, P of Y =
1, right?
And so each of these terms, P of X given Y
and P of Y, these are terms I can get out of, directly, from my Gaussian discriminant
analysis model. Each of these terms is something that
my model gives me directly,
so plugged in as the denominator,
and by doing that, that's how I compute P of Y = 1 given X, make sense? Student:Thank you. Instructor (Andrew Ng):Okay. Cool.
So let's talk a little bit about the advantages and disadvantages of using a
generative
learning algorithm, okay? So in the particular case of Gaussian discriminant analysis, we
assume that
X conditions on Y
is Gaussian,
and the argument I showed on the previous chalkboard, I didn't prove it formally,
but you can actually go back and prove it yourself
is that if you assume X given Y is Gaussian,
then that implies that
when you plot Y
given X,
you find that - well, let me just write logistic posterior, okay?
And the argument I showed just now, which I didn't prove; you can go home and prove it
yourself,
is that if you assume X given Y is Gaussian, then that implies that the posterior
distribution or the form of
P of Y = 1 given X
is going to be a logistic function,
and it turns out this
implication in the opposite direction
does not hold true,
okay? In particular, it actually turns out - this is actually, kind of, cool. It
turns out that if you're
seeing that X given Y = 1 is
Hessian with
parameter lambda 1,
and X given Y = 0,
is Hessian
with parameter lambda 0.
It turns out if you assumed this,
then
that also
implies that P of Y
given X
is logistic, okay?
So there are lots of assumptions on X given Y
that will lead to P of Y given X being logistic, and,
therefore,
this, the assumption that X given Y being Gaussian is the stronger assumption
than the assumption that Y given X is logistic,
okay? Because this implies this,
right? That means that this is a stronger assumption than this because
this, the logistic posterior holds whenever X given Y is Gaussian but not vice versa.
And so this leaves some
of the tradeoffs between Gaussian discriminant analysis and logistic
regression,
right? Gaussian discriminant analysis makes a much stronger assumption
that X given Y is Gaussian,
and so when this assumption is true, when this assumption approximately holds, if you plot the
data,
and if X given Y is, indeed, approximately Gaussian,
then if you make this assumption, explicit to the algorithm, then the
algorithm will do better
because it's as if the
algorithm is making use of more information about the data. The algorithm knows that
the data is Gaussian,
right? And so
if the Gaussian assumption, you know,
holds or roughly holds,
then Gaussian
discriminant analysis may do better than logistic regression.
If, conversely, if you're actually not sure what X given Y is, then
logistic regression, the discriminant algorithm may do better,
and, in particular, use logistic regression,
and
maybe you see [inaudible] before the data was Gaussian, but it turns out the data
was actually Poisson, right?
Then logistic regression will still do perfectly fine because if
the data were actually Poisson,
then P of Y = 1 given X will be logistic, and it'll do perfectly
fine, but if you assumed it was Gaussian, then the algorithm may go off and do something
that's not as good, okay?
So it turns out that - right.
So it's slightly different.
It
turns out the real advantage of generative learning algorithms is often that it
requires less data,
and, in particular,
data is never really exactly Gaussian, right? Because data is often
approximately Gaussian; it's never exactly Gaussian.
And it turns out, generative learning algorithms often do surprisingly well
even when
these modeling assumptions are not met, but
one other tradeoff
is that
by making stronger assumptions
about the data,
Gaussian discriminant analysis
often needs less data
in order to fit, like, an okay model, even if there's less training data. Whereas, in
contrast,
logistic regression by making less
assumption is more robust to your modeling assumptions because you're making a weaker assumption; you're
making less assumptions,
but sometimes it takes a slightly larger training set to fit than Gaussian discriminant
analysis. Question? Student:In order
to meet any assumption about the number [inaudible], plus
here
we assume that P of Y = 1, equal
two
number of.
[Inaudible]. Is true when
the number of samples is marginal? Instructor (Andrew Ng):Okay. So let's see.
So there's a question of is this true - what
was that? Let me translate that
differently. So
the marving assumptions are made independently of the size
of
your training set, right? So, like, in least/great regression - well,
in all of these models I'm assuming that these are random variables
flowing from some distribution, and then, finally, I'm giving a single training set
and that as for the parameters of the
distribution, right?
Student:So
what's the probability of Y = 1?
Instructor (Andrew Ng):Probability of Y + 1?
Student:Yeah, you used the - Instructor (Andrew Ng):Sort
of, this like
-
back to the philosophy of mass molecular estimation,
right? I'm assuming that
they're P of Y is equal to phi to the Y,
Y - phi to the Y or Y - Y. So I'm assuming that there's some true value of Y
generating
all my data,
and then
-
well, when I write this, I guess, maybe what I should write isn't -
so when I write this, I
guess there are already two values of phi. One is
there's a true underlying value of phi
that guards the use to generate the data,
and then there's the maximum likelihood estimate of the value of phi, and so when I was writing
those formulas earlier,
those formulas are writing for phi, and mew0, and mew1
were really the maximum likelihood estimates for phi, mew0, and mew1, and that's different from the true
underlying values of phi, mew0, and mew1, but - Student:[Off mic]. Instructor (Andrew Ng):Yeah, right. So maximum
likelihood estimate comes from the data,
and there's some, sort of, true underlying value of phi that I'm trying to estimate,
and my maximum likelihood estimate is my attempt to estimate the true value,
but, you know, by notational and convention
often are just right as that as well without bothering to distinguish between
the maximum likelihood value and the true underlying value that I'm assuming is out
there, and that I'm
only hoping to estimate.
Actually, yeah,
so for the sample of questions like these about maximum likelihood and so on, I hope
to tease to the Friday discussion section
as a good time to
ask questions about, sort of,
probabilistic definitions like these as well. Are there any
other questions? No, great. Okay.
So,
great. Oh, it
turns out, just to mention one more thing that's, kind of, cool.
I said that
if X given Y is Poisson, and you also go logistic posterior,
it actually turns out there's a more general version of this. If you assume
X
given Y = 1 is exponential family
with parameter A to 1, and then you assume
X given Y = 0 is exponential family
with parameter A to 0, then
this implies that P of Y = 1 given X is also logistic, okay? And
that's, kind of, cool.
It means that Y given X could be - I don't
know, some strange thing. It could be gamma because
we've seen Gaussian right
next to the - I
don't know, gamma exponential.
They're actually a beta. I'm
just rattling off my mental list of exponential family extrusions. It could be any one
of those things,
so [inaudible] the same exponential family distribution for the two classes
with different natural parameters
than the
posterior
P of Y given 1 given X - P of Y = 1 given X would be logistic, and so this shows
the robustness of logistic regression
to the choice of modeling assumptions because it could be that
the data was actually, you know, gamma distributed,
and just still turns out to be logistic. So it's the
robustness of logistic regression to modeling
assumptions.
And this is the density. I think,
early on I promised
two justifications for where I pulled the logistic function out of the hat, right? So
one was the exponential family derivation we went through last time, and this is, sort of, the second one.
That all of these modeling assumptions also lead to the logistic function. Yeah? Student:[Off
mic]. Instructor (Andrew Ng):Oh, that Y = 1 given as the logistic then this implies that, no. This is also
not true, right?
Yeah, so this exponential
family distribution
implies Y = 1 is logistic, but the reverse assumption is also not true.
There are actually all sorts of really bizarre distributions
for X that would give rise to logistic function, okay? Okay. So
let's talk about - those are first generative learning algorithm. Maybe I'll talk about the second
generative learning algorithm,
and the motivating example, actually this is called a Naive Bayes algorithm,
and the motivating example that I'm gonna use will be spam classification. All right. So let's
say that you want to build a spam classifier to take your incoming stream of email and decide if
it's spam or
not.
So let's
see. Y will be 0
or
1, with 1 being spam email
and 0 being non-spam, and
the first decision we need to make is, given a piece of email,
how do you represent a piece of email using a feature vector X,
right? So email is just a piece of text, right? Email
is like a list of words or a list of ASCII characters.
So I can represent email as a feature of vector X.
So we'll use a couple of different
representations,
but the one I'll use today is
we will
construct the vector X as follows. I'm gonna go through my dictionary, and, sort of, make a listing of
all the words in my dictionary, okay? So
the first word is
RA. The second word in my dictionary is Aardvark, ausworth,
okay?
You know, and somewhere along the way you see the word buy in the spam email telling you to buy
stuff.
Tell you how you collect your list of words,
you know, you won't find CS229, right, course number in a dictionary, but
if you
collect a list of words via other emails you've gotten, you have this list somewhere
as well, and then the last word in my dictionary was
zicmergue, which
pertains to the technological chemistry that deals with
the fermentation process in
brewing.
So say I get a piece of email, and what I'll do is I'll then
scan through this list of words, and wherever
a certain word appears in my email, I'll put a 1 there. So if a particular
email has the word aid then that's 1.
You know, my email doesn't have the words ausworth
or aardvark, so it gets zeros. And again,
a piece of email, they want me to buy something, CS229 doesn't occur, and so on, okay?
So
this would be
one way of creating a feature vector
to represent a
piece of email.
Now, let's throw
the generative model out for this. Actually,
let's use
this.
In other words, I want to model P of X given Y. The
given Y = 0 or Y = 1, all right?
And my feature vectors are going to be 0, 1
to the N. It's going to be these split vectors, binary value vectors. They're N
dimensional.
Where N
may
be on the order of, say, 50,000, if you have 50,000
words in your dictionary,
which is not atypical. So
values from - I don't
know,
mid-thousands to tens of thousands is very typical
for problems like
these. And, therefore,
there two to the 50,000 possible values for X, right? So two to
50,000
possible bit vectors
of length
50,000, and so
one way to model this is
the multinomial distribution,
but because there are two to the 50,000 possible values for X,
I would need two to the 50,000, but maybe -1 parameters,
right? Because you have
this sum to 1, right? So
-1. And this is clearly way too many parameters
to model
using the multinomial distribution
over all two to 50,000 possibilities.
So
in a Naive Bayes algorithm, we're
going to make a very strong assumption on P of X given Y,
and, in particular, I'm going to assume - let
me just say what it's called; then I'll write out what it means. I'm going to assume that the
XI's
are conditionally independent
given Y, okay?
Let me say what this means.
So I have that P of X1, X2, up to X50,000,
right, given the
Y. By the key rule of probability, this is P of X1 given Y
times P of X2
given
Y,
X1
times PF - I'll just put dot, dot, dot. I'll just write 1, 1 Ă dot, dot, dot up to, you know, well -
whatever. You get the idea, up to P of X50,000, okay?
So this is the chain were of probability. This always holds. I've not
made any assumption yet, and now, we're
gonna
meet what's called the Naive Bayes assumption, or this assumption that X
defies a conditionally independent given Y. Going
to assume that -
well, nothing changes for the first term,
but I'm gonna assume that P of X3 given Y, X1 is equal to P of X2 given the Y. I'm gonna assume that that term's equal to P of X3
given
the
Y,
and so on, up
to P of X50,000 given Y, okay?
Or just written more compactly,
means assume that P of X1, P of X50,000 given Y is
the product from I = 1 to 50,000 or P of XI
given the Y,
okay?
And stating informally what this means is that I'm, sort of, assuming that -
so unless you know the cost label Y, so long as you know whether this is spam or not
spam,
then knowing whether the word A appears in email
does not affect
the probability
of whether the word
Ausworth appears in the email, all right?
And, in other words, there's assuming - once you know whether an email is spam
or not spam,
then knowing whether other words appear in the email won't help
you predict whether any other word appears in the email, okay?
And,
obviously, this assumption is false, right? This
assumption can't possibly be
true. I mean, if you see the word
- I don't know, CS229 in an email, you're much more likely to see my name in the email, or
the TA's names, or whatever. So this assumption is normally just false
under English, right,
for normal written English,
but it
turns out that despite
this assumption being, sort of,
false in the literal sense,
the Naive Bayes algorithm is, sort of,
an extremely effective
algorithm for classifying text documents into spam or not spam, for
classifying your emails into different emails for your automatic view, for
looking at web pages and classifying
whether this webpage is trying to sell something or whatever. It
turns out, this assumption
works very well for classifying text documents and for other applications too that I'll
talk a bit about later.
As a digression that'll make sense only to some of you.
Let me just say that
if you're familiar with Bayesian X world, say
graphical models, the Bayesian network associated with this model looks like this, and you're assuming
that
this is random variable Y
that then generates X1, X2, through
X50,000, okay? If you've not seen the
Bayes Net before, if
you don't know your graphical model, just ignore this. It's not important to our purposes, but
if you've seen it before, that's what it will look like. Okay.
So
the parameters of the model
are as follows
with phi FI given Y = 1, which
is probably FX = 1 or XI = 1
given Y = 1,
phi I
given Y = 0, and phi Y, okay?
So these are the parameters of the model,
and, therefore,
to fit the parameters of the model, you
can write down the joint likelihood, right,
is
equal to, as usual, okay?
So given the training sets,
you can write down the joint
likelihood of the parameters, and
then when
you
do maximum likelihood estimation,
you find that the maximum likelihood estimate of the parameters are
- they're really, pretty much, what you'd expect.
Maximum likelihood estimate for phi J given Y = 1 is
sum from I = 1 to
M,
indicator
XIJ =
1, YI = 1, okay?
And this is just a,
I guess, stated more simply,
the numerator just says, Run for
your entire training set, some [inaudible] examples,
and count up the number of times you saw word Jay
in a piece of email
for which the label Y was equal to 1. So, in other words, look
through all your spam emails
and count the number of emails in which the word
Jay appeared out of
all your spam emails,
and the denominator is, you know,
sum from I = 1 to M,
the number of spam. The
denominator is just the number of spam emails you got.
And so this ratio is
in all your spam emails in your training set,
what fraction of these emails
did the word Jay
appear in -
did the, Jay you wrote in your dictionary appear in?
And that's the maximum likelihood estimate
for the probability of seeing the word Jay conditions on the piece of email being spam, okay? And similar to your
maximum likelihood estimate for phi
Y
is pretty much what you'd expect, right?
Okay?
And so
having estimated all these parameters,
when you're given a new piece of email that you want to classify,
you can then compute P of Y given X
using Bayes rule, right?
Same as before because
together these parameters gives you a model for P of X given Y and for P of Y,
and by using Bayes rule, given these two terms, you can compute
P of X given Y, and
there's your spam classifier, okay?
Turns out we need one more elaboration to this idea, but let me check if there are
questions about this so far.
Student:So does this model depend
on
the number of inputs? Instructor (Andrew Ng):What do
you
mean, number of inputs, the number of features? Student:No, number of samples. Instructor (Andrew Ng):Well, N is the number of training examples, so this
given M training examples, this is the formula for the maximum likelihood estimate of the parameters, right? So other questions, does it make
sense? Or M is the number of training examples, so when you have M training examples, you plug them
into this formula,
and that's how you compute the maximum likelihood estimates. Student:Is training examples you mean M is the
number of emails? Instructor (Andrew Ng):Yeah, right. So, right.
So it's, kind of, your training set. I would go through all the email I've gotten
in the last two months
and label them as spam or not spam,
and so you have - I don't
know, like, a few hundred emails
labeled as spam or not spam,
and that will comprise your training sets for X1 and Y1 through XM,
YM,
where X is one of those vectors representing which words appeared in the email and Y
is 0, 1 depending on whether they equal spam or not spam, okay? Student:So you are saying that this model depends on the number
of examples, but the last model doesn't depend on the models, but your phi is the
same for either one. Instructor (Andrew Ng):They're
different things, right? There's the model which is
- the modeling assumptions aren't made very well.
I'm assuming that - I'm making the Naive Bayes assumption.
So the probabilistic model is an assumption on the joint distribution
of X and Y.
That's what the model is,
and then I'm given a fixed number of training examples. I'm given M training examples, and
then it's, like, after I'm given the training sets, I'll then go in to write the maximum
likelihood estimate of the parameters, right? So that's, sort
of,
maybe we should take that offline for - yeah, ask a question? Student:Then how would you do this, like,
if this [inaudible] didn't work? Instructor (Andrew Ng):Say that again. Student:How would you do it, say, like the 50,000 words - Instructor (Andrew Ng):Oh, okay. How to do this with the 50,000 words, yeah. So
it turns out
this is, sort of, a very practical question, really. How do I count this list of
words? One common way to do this is to actually
find some way to count a list of words, like go through all your emails, go through
all the -
in practice, one common way to count a list of words
is to just take all the words that appear in your training set. That's one fairly common way
to do it,
or if that turns out to be too many words, you can take all words that appear
at least
three times
in your training set. So
words that
you didn't even see three times in the emails you got in the last
two months, you discard. So those are - I
was talking about going through a dictionary, which is a nice way of thinking about it, but in
practice, you might go through
your training set and then just take the union of all the words that appear in
it. In some of the tests I've even, by the way, said select these features, but this is one
way to think about
creating your feature vector,
right, as zero and one values, okay? Moving on, yeah. Okay. Ask a question? Student:I'm getting, kind of, confused on how you compute all those parameters. Instructor (Andrew Ng):On
how I came up with the parameters?
Student:Correct. Instructor (Andrew Ng):Let's see.
So in Naive Bayes, what I need to do - the question was how did I come up with the parameters, right?
In Naive Bayes,
I need to build a model
for P of X given Y and for
P of Y,
right? So this is, I mean, in generous of learning algorithms, I need to come up with
models for these.
So how'd I model P of Y? Well, I just those to model it using a Bernoulli
distribution,
and so
P of Y will be
parameterized by that, all right? Student:Okay.
Instructor (Andrew Ng):And then how'd I model P of X given Y? Well,
let's keep changing bullets.
My model for P of X given Y under the Naive Bayes assumption, I assume
that P of X given Y
is the product of these probabilities,
and so I'm going to need parameters to tell me
what's the probability of each word occurring,
you know, of each word occurring or not occurring,
conditions on the email being spam or not spam email, okay? Student:How is that
Bernoulli? Instructor (Andrew Ng):Oh, because X is either zero or one, right? By the way I defined the feature
vectors, XI
is either one or zero, depending on whether words I appear as in the email,
right? So by the way I define the
feature vectors, XI -
the XI is always zero or one. So that by definition, if XI, you know, is either zero or
one, then it has to be a Bernoulli distribution, right?
If XI would continue as then
you might model this as Gaussian and say you end up
like we did in Gaussian discriminant analysis. It's
just that the way I constructed my features for email, XI is always binary
value, and so you end up
with a
Bernoulli here, okay? All right. I
should move on. So
it turns out that
this idea
almost works.
Now, here's the problem.
So let's say you
complete this class and you start to do, maybe do the class project, and you
keep working on your class project for a bit, and it
becomes really good, and you want to submit your class project to a conference, right? So,
you know, around - I don't know,
June every year is the conference deadline for the next conference.
It's just the name of the conference; it's an acronym.
And so maybe
you send your project partners or senior friends even, and say, Hey, let's
work on a project and submit it to the NIPS conference. And so you're getting these emails
with the word NIPS in them,
which you've probably never seen before,
and so a
piece of email comes from your project partner, and so you
go, Let's send a paper to the NIPS conference.
And then your stamp classifier
will say
P of X -
let's say NIPS is the 30,000th word in your dictionary, okay?
So X30,000 given
the 1, given
Y =
1
will be equal to 0.
That's the maximum likelihood of this, right? Because you've never seen the word NIPS before in
your training set, so maximum likelihood of the parameter is that probably have seen the word
NIPS is zero,
and, similarly,
you
know, in, I guess, non-spam mail, the chance of seeing the word NIPS is also
estimated
as zero.
So
when your spam classifier goes to compute P of Y = 1 given X, it will
compute this
right here P of Y
over - well,
all
right.
And so
you look at that terms, say, this will be product from I =
1 to 50,000,
P of XI given Y,
and one of those probabilities will be equal to
zero because P of X30,000 = 1 given Y = 1 is equal to zero. So you have a
zero in this product, and so the numerator is zero,
and in the same way, it turns out the denominator will also be zero, and so you end
up with -
actually all of these terms end up being zero. So you end up with P of Y = 1
given X is 0 over 0 + 0, okay, which is
undefined. And the
problem with this is that it's
just statistically a bad idea
to say that P of X30,000
given Y is
0,
right? Just because you haven't seen the word NIPS in your last
two months worth of email, it's also statistically not sound to say that,
therefore, the chance of ever seeing this word is zero, right?
And so
is this idea that just because you haven't seen something
before, that may mean that that event is unlikely, but it doesn't mean that
it's impossible, and just saying that if you've never seen the word NIPS before,
then it is impossible to ever see the word NIPS in future emails; the chance of that is just zero.
So we're gonna fix this,
and
to motivate the fix I'll talk about
- the example we're gonna use is let's say that you've been following the Stanford basketball
team for all of their away games, and been, sort of, tracking their wins and losses
to gather statistics, and, maybe - I don't know, form a betting pool about
whether they're likely to win or lose the next game, okay?
So
these are some of the statistics.
So on, I guess, the 8th of February
last season they played Washington State, and they
did not win.
On
the 11th of February,
they play Washington, 22nd
they played USC,
played UCLA,
played USC again,
and now you want to estimate
what's the chance that they'll win or lose against Louisville, right?
So
find the four guys last year or five times and they weren't good in their away games, but it
seems awfully harsh to say that - so it
seems awfully harsh to say there's zero chance that they'll
win in the last - in the 5th game. So here's the idea behind Laplace smoothing
which is
that we're estimate
the probably of Y being equal to one, right?
Normally, the maximum likelihood [inaudible] is the
number of ones
divided by
the number of zeros
plus the number of ones, okay? I
hope this informal notation makes sense, right? Knowing
the maximum likelihood estimate for, sort of, a win or loss for Bernoulli random
variable
is
just the number of ones you saw
divided by the total number of examples. So it's the number of zeros you saw plus the number of ones you saw. So in
the Laplace
Smoothing
we're going to
just take each of these terms, the number of ones and, sort of, add one to that, the number
of zeros and add one to that, the
number of ones and add one to that,
and so in our example,
instead of estimating
the probability of winning the next game to be 0 Ăˇ
5 + 0,
we'll add one to all of these counts, and so we say that the chance of
their
winning the next game is 1/7th,
okay? Which is
that having seen them lose, you know, five away games in a row, we aren't terribly -
we don't think it's terribly likely they'll win the next game, but at
least we're not saying it's impossible.
As a historical side note, the Laplace actually came up with the method.
It's called the Laplace smoothing after him.
When he was trying to estimate the probability that the sun will rise tomorrow, and his rationale
was in a lot of days now, we've seen the sun rise,
but that doesn't mean we can be absolutely certain the sun will rise tomorrow.
He was using this to estimate the probability that the sun will rise tomorrow. This is, kind of,
cool. So,
and more generally,
if Y
takes on
K possible of values,
if you're trying to estimate the parameter of the multinomial, then you estimate P of Y = 1.
Let's
see.
So the maximum likelihood estimate will be Sum from J = 1 to M,
indicator YI = J Ăˇ M,
right?
That's the maximum likelihood estimate
of a multinomial probability of Y
being equal to - oh,
excuse me, Y = J. All right.
That's the maximum likelihood estimate for the probability of Y = J,
and so when you apply Laplace smoothing to that,
you add one to the numerator, and
add K to the denominator,
if Y can take up K possible values, okay?
So for Naive Bayes,
what that gives us is -
shoot.
Right? So that was the maximum likelihood estimate, and what you
end up doing is adding one to the numerator and adding two to the denominator,
and this solves the problem of the zero probabilities, and when your friend sends
you email about the NIPS conference,
your spam filter will still be able to
make a meaningful prediction, all right? Okay.
Shoot. Any questions about this? Yeah? Student:So that's what doesn't makes sense because, for instance, if you take the
games on the right, it's liberal assumptions that the probability
of
winning is very close to zero, so, I mean, the prediction should
be equal to PF, 0. Instructor (Andrew Ng):Right.
I would say that
in this case the prediction
is 1/7th, right? We don't have a lot of - if you see somebody lose five games
in a row, you may not have a lot of faith in them,
but as an extreme example, suppose you saw them lose one game,
right? It's just not reasonable to say that the chances of winning the next game
is zero, but
that's what maximum likelihood
estimate
will say. Student:Yes. Instructor (Andrew Ng):And -
Student:In such a case anywhere the learning algorithm [inaudible] or - Instructor (Andrew Ng):So some questions of, you
know, given just five training examples, what's a reasonable estimate for the chance of
winning the next game,
and
1/7th is, I think, is actually pretty reasonable. It's less than 1/5th for instance.
We're saying the chances of winning the next game is less
than 1/5th. It turns out, under a certain set of assumptions I won't go into - under a certain set of
Bayesian assumptions about the prior and posterior,
this Laplace smoothing actually gives the optimal estimate,
in a certain sense I won't go into
of what's the chance of winning the next game, and so under a certain assumption
about the
Bayesian prior on the parameter.
So I don't know. It actually seems like a pretty reasonable assumption to me.
Although, I should say, it actually
turned out -
No, I'm just being mean. We actually are a pretty good basketball team, but I chose a
losing streak
because it's funnier that way.
Let's see. Shoot. Does someone want to - are there other questions about
this? No, yeah. Okay. So there's more that I want to say about Naive Bayes, but
we'll do that in the next lecture. So let's wrap it.