Probabilistic graphical models (PGMs) are a rich framework for encoding probability distributions over complex domains: joint (multivariate) distributions over large numbers of random variables that interact with each other. These representations sit at the intersection of statistics and computer science, relying on concepts from probability theory, graph algorithms, machine learning, and more. They are the basis for the state-of-the-art methods in a wide variety of applications, such as medical diagnosis, image understanding, speech recognition, natural language processing, and many, many more. They are also a foundational tool in formulating many machine learning problems. This course is the first in a sequence of three. It describes the two basic PGM representations: Bayesian Networks, which rely on a directed graph; and Markov networks, which use an undirected graph. The course discusses both the theoretical properties of these representations as well as their use in practice. The (highly recommended) honors track contains several hands-on assignments on how to represent some real-world problems. The course also presents some important extensions beyond the basic PGM representation, which allow more complex models to be encoded compactly.
Semantics & Factorization
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From the course by Stanford University
Probabilistic Graphical Models 1: Representation
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From the lesson
Bayesian Network (Directed Models)
In this module, we define the Bayesian network representation and its semantics. We also analyze the relationship between the graph structure and the independence properties of a distribution represented over that graph. Finally, we give some practical tips on how to model a real-world situation as a Bayesian network.
Meet the Instructors
Daphne KollerProfessor
School of Engineering
So, now we're getting into Bayes Net Lab. And we're finally going to start talking
about the actual representations that are going to be the bread and butter of what
we're going to describe in this class. And, so we're going to start by defining
the basic semantics of a Bayesian network and how it's constructed from a set of
from a set of factor. So let's start by looking at a running
example that will see us throughout a large part of at least the first part of
this course and this is what we call the student example.
So in the student example, we have a student whose taking the class for a
grade and we're going to use the first letter of of the word to denote the name
of the random variable just like we did in previous examples.
So, here, the random variable is going to be G.
now the weight of the student obviously depends on how difficult the course that
he or she is taking and the intelligence of the student.
So that gives us in addition to G, we also have D and I. And we're going to
add a couple of extra random variables just to make things a little bit more
interesting. So we're going to assume that the student has taken the SAT,
so, he may or may have not scored well on the SAT,
so that's another random variable, S. And then finally, we have, that this case
of the disappearing line, we also have the recommendation letter, L, that the
student gets from the instructor of the class.
Okay? And we're going to grossly oversimplify this problem by basically
binarizing everything except for grades. So everything has only has two values
except for grade that has three and this is only so I can write things compactly.
This is not a limitation of the framework, it's just so that the
probability distribution don't become unmanageable.
Okay. So now, let's think about how we can construct the dependencies of of
this, in this probability distribution. Okay.
So, let's start with the random variable grade.
I'm going to put G in the middle and ask ourselves what the grade of the student
depends on. And it seems, just, you know, from a
completely intuitive perspective, it seems clear that the grade of the student
depends on the difficulty of the course and on the intelligence of the student.
And so we already have a little baby Bayesian network with three random
variables. Let's now take the other random variable
and introduce them into the mix. so for example, the SAT score of the
student doesn't seem to depend on the difficulty of the course or on the grade
that the student gets in the course. The only thing it's likely to depend on
in the context of this model is the intelligence of the student.
And finally, caricaturing the way in which instructors write recommendation
letters. We're going to assume that the quality of
the letter depends only on the student's grade,
but professor's teaching, you know, 600 students or maybe 100,000 online
students. And so, the only thing that one can say about the student is by looking
at their actual grade record and so the and so, regardless of anything else, the
quality of the letter depends only on the grade.
Now, this is a model of the dependencies, it's only one model that one can
construct through these dependencies. So for example I could easily imagine
other models, for instance, ones that have students who
are brighter taking harder courses in which case, there might be potentially an
edge between I and D. But we're not going to use that model,
so let's erase that because we're going to stick with a simpler model for the
time being. But, this is only to highlight the fact
that a model is not set in stone, it's a representation of how we believe the
world works. So, here is the model drawn out a little
bit more nicely than than the picture before.
And now let's think about what we need to do in order to turn this into our
presentation of probability distribution, because right now, all it is is a bunch
of you know nodes stuck together with edges and so how do we actually get this
to represent the probability distribution?
And the way which we're going to do that is we're going to annotate each of the
nodes in the network with what's called, with a CPD.
So, we previously defined CPD. CPD is just as a reminder,
is a conditional probability distribution hm,
using the abbreviation here. And, each of theses is a CPD,
so we have five nodes, we have five CPDs. Now, if you look at some of these CPDs,
they're kind of degenerate, so for example, the difficulty CPD isn't
actually conditioned on anything. It's just a unconditional probability
distribution that tells us, for example, that courses are only 40%.
likely to be difficult and 60% to be easy.
here is a similar unconditional probability for intelligence.
Now this gets more interesting when you look at the actual conditional
probability distributions. So here, for example, is a CPD that we've
seen before this is the CPD of the grades A, B, and C.
So, here is the conditional probability date distribution that we've already seen
before for the probability of grade given intelligence, and difficulty, and we've
already discussed how each of these rows necessarily sums to one because the
probability distribution over the variable grade and we have two other
CPD's here. In this case, the probability of SAT
given intelligence and the probability of letter given grade.
So, just to write this out completely, we have P of D,
P of I, P of G given I, D, P of L given G, and P of F given I.
And that now is a fully parameterized Bayesian network and what we'll show next
is how this Bayesian network produces a joint probability distribution over these
five variables. So, here are my CPDs and what we're going
to define now is the chain rule for Bayesian networks and that chain rule
basically takes these different CPDs, these, all these little CPDs and
multiplies them together, like that.
Now, before we think of what that means, let us first note that this is actually a
factor product in exactly the same way that we just defined.
So here, we have five factors, they have overlapping scopes and what we
end up with is a factor product that gives us a big, big factor whose scope is
five variables. So what does that translate into when we
apply the chain rule for Bayesian networks in the context of the particular
example? So let's look at this particular
assignment and remember there's going to be a bunch of these different assignments
and I'm just going to compute this the probability of this one.
So the probability of d0, i1, g3, s1, and l1, well, so the first thing we need is
the probability of d0 and the probability of d0 is 0.6.
The next thing from the next factor is the probability of i1,
which is 0.3. What about the next one?
The next one now is a conditional factor and for that, we need the probability of
g3, because we have g3 here, so we want from this column over here, and which row
do we want, we want the probability, the row that corresponds to d0 and i1, which
is this row, so 0.02.
Continuing on, we know now that we, we now have g3, so
we want from this row and we want the probability of l1, so we want this entry,
0.01. And finally, here, we have the
probability s1 given given i1, so that would be this one over here and 0.8.
And in this exact same fashion, we're going to end up defining all of the
entries in this joint probability distribution.
Great. So, what does that give us as a
definition? a Bayesian network is a directed acyclic
graph. Acyclic means that has no cycles that is
you don't, you can't reverse the edges and get back you started.
This is typically abbreviated as a DAG and we're are going use the letter G to
denote to denote directed acyclic graphs. And the nodes of this directed acyclic
graph represent the random variable that we're interested in reasoning about, the
X1 up to Xn. And for each node in the graph, Xi, we
have a CPD that demotes the dependance of Xi on its parents in the graph G. Okay?
So, this is a set of variables, so this would be like the probability of
G given I and D. So, this would be G,
the Xi would be G and, and the parents of G, of, of Xi would be I and D.
And the BN, this Bayesian network, which represents a joint probability
distribution via the chain rule for Bayesian networks,
which is the rule that's written over here.
And this is just a generalization of the rule that we wrote on the previous slide,
where we multiplied 1CPD for each variable.
Here, we have, again, we're multiplying, we're multiplying over all the variables
I and multiplying the CPD for each of the Xis and this is once again a factor
product. So, that's great, and I just defined the
joint probability distribution, but who's to say that that joint
probability distribution is even a joint probability distribution?
So what do you have to show in order to demonstrate that something is a joint
probability distribution? First thing you have to show is that it's
greater than or equal to zero, because probability distributions better be
non-negative. And so, in order to do that, we need to
show that for this distribution P. And the way, and this as it happens is,
is quite trivial, because P is a product of factors.
It actually is a product of CPD and CPDs are non-negative.
And you if multiply a bunch of non-negative factors, you get a
non-negative factor. That's fairly fairly, straightforward.
Next one's a little bit trickier. The other thing that we have to show in
order to demonstrate that something is a legal distribution is to prove that it
sums to one. So how do we prove that something sums to
that this probability distribution sums to one?
So, let's actually work through this and one of the reasons for working through
this is not just to convince you that I'm not lying to you, but more importantly,
the techniques that we're going to use for this simple argument are going to
accompany us throughout much of the first part of the course.
and so it's worth going through this in a little bit of detail.
but I'm going to show this in the context of the example of the distribution that
we just showed, we just end up cluttered with notation, but the, but the idea is
still exactly the same. So,
let's sum up over all possible assignments D, I, G, S, L of this
probability, of this hopefully probability distribution that I just
defined the P of D, I, G, S, and L. Okay?
So we are going to brake this up using the chain rule for Bayesian networks we
because that's how we defined this distribution P.
Okay. So here is the magic trick that we use
whenever we deal with distributions like that.
It's to notice that when we have the summation over several random variables
here, and here inside the summations, we have a bunch of functions inside here
that, only some, of the, the, each, factor only involves a small subset of
the variables. And when you do, when you have a situation like that, you can push
in the summations. So specifically, we're going to start out
by pushing in the summation over, L, and which factors involve L? Only the last
one. So, we can move in the summation over L
to this position ending up with the following expression.
So notice that now the summation is over here,
that's the first trick. The second trick is the definition of
CPDs and we observe here that the summation over L is summing up over all
possible mutually exclusive and exhausted values of the variable L in this,
conditional probability distribution P of L given G.
And that means, basically, we're summing up over a row of
the CPD and that means that the sum has to be one.
And so this term effectively cancels out, which gives us this,
and now we can do exactly the same thing. We can take S and move it over here, and
that gives us that, and this two is a sum over a row of a CPD, so this two
cancelled and can be written as one. We could do the same with G, and again,
this has to be equal to one and so we can cancel this and so on and so forth.
And so, by the end of this, you're going to have, you're, you're going to
have cancelled all of the variables in the summation and what remains is one.
Now, as I said, this is a very simple proof, but these tools of pushing in
summations and removing variables that are not relevant are going to turn out to
be very important. Okay.
So now, let's just define a little bit of a terminology that will, again, accompany
us later on. We're going to say that a distribution P
factorizes over G, means we can represent it over the graph G if we can encode it
using the chain rule for Bayesian networks.
So the distribution factorizes over big, over a graph G if I can represent it in
this way as a product of these conditional probabilities.
So let me just conclude this with a little example which is arguably, the
very first example of Bayesian networks ever.
it was an example of Bayesian networks before anybody even called them Bayesian
networks. and it was defined actually by
statistical geneticists who were trying to model the notion of genetic
inheritance in in a population. And, so here is a an example of genetic
inheritance of blood types, so let me just give a little bit of background on
the genetics of this. So, in this very simple, oversimplified example a person's
genotype is defined by two values, because we have two copies of of each of
those, of each of those chromosomes in this case.
And, so it, so for each chromosome, we have three possible values,
so we have either the A value, the B Value, or the O value.
These are familiar to many of you from blood types,
which is what we're modelling here. And so the total number of genotypes that
we have is listed over here, we have the AA genotype,
AB, AO, BO, BB, and OO, so we have a total of six distinct
genotypes. However, we have few more phenotypes
because it turns out that AA and AO manifest exactly the same way as blood
type A and BO and BD manifest as blood type B, and so we have only four
phenotypes. So how do we model genetic inheritance in
a Bayesian network? here is a little Bayeseian network for
genetic inheritance, we can see that for each individual, say Marge.
We have a genotype variable, which takes on six values, one of the six values that
we saw before. And the, the phenotype, in this case the
blood type, depends only on the person's genotype.
We also see that a person's genotype depends on the genotype of her parents,
because after all, she inherits one chromosome from her mother and one
chromosome from her father. And so and so the,
a person's individual genotype just depends on those two variables.
And so this is very nice and very compelling Bayesian network, because
everything just fits in beautifully in terms of
things really do depend only on only on the variables that that we stipulate in
the model that they depend on.
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