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 second in a sequence of three. Following the first course, which focused on representation, this course addresses the question of probabilistic inference: how a PGM can be used to answer questions. Even though a PGM generally describes a very high dimensional distribution, its structure is designed so as to allow questions to be answered efficiently. The course presents both exact and approximate algorithms for different types of inference tasks, and discusses where each could best be applied. The (highly recommended) honors track contains two hands-on programming assignments, in which key routines of the most commonly used exact and approximate algorithms are implemented and applied to a real-world problem.
Properties of Cluster Graphs
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En provenance du cours de Stanford University
Probabilistic Graphical Models 2: Inference
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À partir de la leçon
Belief Propagation Algorithms
This module describes an alternative view of exact inference in graphical models: that of message passing between clusters each of which encodes a factor over a subset of variables. This framework provides a basis for a variety of exact and approximate inference algorithms. We focus here on the basic framework and on its instantiation in the exact case of clique tree propagation. An optional lesson describes the loopy belief propagation (LBP) algorithm and its properties.
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Daphne KollerProfessor
School of Engineering
We described the belief propagation algorithm as passing message over cluster
graph but we left unspecified how that cluster graph would be constructed and
what properties it needs to satisfy in order to support reasonable message
passing. So let's remind ourselves what cluster
graphs are. A cluster graph is an undirected graph
whose nodes are clusters that involve subsets of variables and edges involve a
subset SIJ, which is a subset of the two clusters on
the twin points. What are some properties that the cluster
graph has to satisfy? The first property is the one called
family preservation an obvious value. where remember that we need, given a set
of factors phi, we need to be able to assign each phi K to some cluster,
C alpha K, such that the fact, such that the cluster can accommodate.
The scope. So can accommodate 5K.
oops accommodate.. Well, in order for the cluster graph to
allow this to be done, we need to have an appropriate cluster in the cluster graph.
So, this imposes a constraint on the cluster graph that says that for every
factor phi K in my set of factors phi, there exists some cluster CI that, such
that CI accommodates phi K, which means that you can put phi k inside
this cluster. That's the family preservation property.
The second property is a little bit trickier to understand.
It's called the Running Intersection Property.
The Running Intersection Property, let's first read the definition and then
understand what it says. It says that let's assume that we have a
pair of clusters CI and CJ and a variable X that sub, that belongs to both of them.
So for example we might have the variable X sitting here in C7, and we might have
the variable X sitting here, in C5. This property says that there exists a,
exists a unique path between Ci and Cj for which all clusters and subsets along
the path contain x. What does that mean?
It means that for example should I choose this to be my unique path.
It means that X needs to sit here, here and here.
So, there is a connecting path that involves X and that path is along the
entire route and that path is unique. So let's try and understand the intuition
between both sides of this definition, the existence and the uniqueness.
Imagine that I, let's do the existence first.
And imagine that for whatever reason I decide that there is not going to be a
path over here. So there is no way for C7 to communicate
to C3 about the variable X. Well, that means that we now have these
two separate, isolated communities, each of which has some information about X and
they can never talk to each other about X.
So they're never going to get to agree about X, there's never going to be any
information transfer, of these two pieces of information.
Well, so that's not very good, which is why we need, that path to exist.
Okay, so that's the exists part. What about the unique part?
The unique part is a little bit trickier to understand but let's but let's try and
provide some intuition for it. Let's imagine that I have two paths that
involved X. Let's so now let's think about this
message passing algorithm. So C3 can send the message.
C5 with information about X. For example, I think X is taking the value one, I have
a, I have a factor that suggests that X takes the value 1.
Well, C5, you know, integrates that with it's own information and sends it on to
C2. Which sends it back to C3.
And now C3 hears from C2 that ooh, X needs to take the value one.
Huh? That reinforces its beliefs that X needs
to take the value one and so the probability goes up.
It now goes back and sends that information on to C5, which sends it on
to C2, which sends it back to C3, and then once again the probability in X
taking the value 1 goes up. And so we have this sort of
self-reinforcing loop that's going to give rise to very extreme and very skewed
probabilities in many examples. And so a way of avoiding that is to or at
least reducing that risk is to is to prevent these kinds of feedback loops.
Now, it's important to realize that by preventing these loops that only reduces
opposed to eliminates the problem. And specifically this is kind of a little
bit of a digression but it's important to know,
is that, imagine for example that we have X, and Y, that are very strongly
correlated. So here, we have for example X and Y and
here we have a path. That involves Y.
Okay? So now what's going to happen is that C3
sends information to C5 about X. C5, translates that to information about
Y, Y should take the value one. That information about Y goes back to C3
and increases the probability in X taking the value one.
Now this is a little bit of a forward looking hint about some of the issues
that we'll see with belief propagation later on, which is that belief
propagation does very poorly. When we have strong correlations.
And that's because of these feedback loops.
The more skewed the probabilities in your graphical model, the harder time belief
propagation has in terms of the results that it gets.
So, with that digression, aside, let's go back to the running interception
property, and let's provide an alternative definition of the running
interception property, just to give ourselves some additional intuition.
The running interception property is equivalent to saying that for any X.
The set of clusters and subsets that contain X form a tree, so for example if
we have X here, here, here, here, here and here.
We can see that the set of a cluster is in sub sets that contain x form a tree.
It has to be connected, because of the existence of the path.
And it can't be, a non tree. Because, because that would give us two
different paths. And so that's a different view of this.
And so you can think of each variable inducing its own little tree.
across which information about that variable flows in the graph Now, let's go
back with that definition and consider some cluster graphs that we might adopt
for for this example that we've seen before.
So, here we have our five clusters so let's check whether it satisfies the
running interception property. We've already done family preservation
for a particular set of factors. So, let's consider for example the
variable, B, and we can see that we have B here, here, here, here,
here, here and here, and we can see that, that is a tree.
Here's my tree. Note, very carefully, that B isn't here.
If B were here that wouldn't satisfy the running intersection property.
That would be an illegal cluster graph, that's actually on the next, on a
subsequent slide. Here is an illegal cluster graph.
It violates the running intersection property Y.
Because here is B, and here is a bunch of other B's and there's no way to connect
cluster two to any of the others. So this one violates the existance.
This one, which we just talked about, violates the uniqueness because here we
have the, all of the clusters and subsets that involve b, and there's this nice
little loop over here that has two paths between say, cluster one and cluster
four. If we wanted to take the cluster graph
and still allow communication between B and C, so we want to have, for example we
want cluster one and cluster two to be able to transmit to each other
information about how B and C are correlated with each other, which they
can't do in this graph over here. So, one way to do that is to we have, we
have eliminated in this case, this edge that we have her that involves B, and now
once again we have B. Being a tree in the graph now focused on
be here but its not difficult to convince yourself that other that other nodes also
satisfied that we satisfy the running intersection property also with respect
to other variables so just as an example here is a set of clusters and subsets
involved in D and too is a tree and here is the one's involving E and that too is
a tree and so on. So and running intersection property
needs to hold for every for every one of the variables.
How do we construct a cluster graph that has, that has a desired properties.
One very simple and in some ways degenerate but still because of its
simplicity very often used is a cluster graph called the beta cluster graph.
And that's a term from statistical physics where people use this kind of
approximation to energy functions in in some, in some calculations, in, in
Statistical Physics. So here, we have in the beta cluster
graph, we have two types of clusters. We have big clusters and little clusters.
The big clusters, these are the big clusters correspond, to factors in phi.
So for each phi k, we have a cluster, a factor cluster whose scope is exactly the
scope of phi k, the big clusters.
The little clusters correspond into individual variables.
So for each Xi, we have a cluster whose scope is just the single variable Xi
itself. Now we're going to connect Ck to Xi
exactly when Xi is a subset of Ck, is a member of Ck.
So if we consider the set of factors that we had before, we can now produce a
different cluster graph then one that we had previously.
This is a beta cluster graph. Notice that we have these big clusters.
These are these four, I'm sorry, these five.
And we have these, little clusters corresponding to A, B, C, D, E, and f.
And we can see that we have an edge for example, between the ABC cluster and the
A cluster, the B cluster, and the C cluster.
And that's how messages are passed. And you can see how this graph is
degenerate in the sense that it only allows information about Singleton's to
be passed. It loses, in every message passing step,
any information about the correlation between variables.
But, nevertheless, it's simple to construct and it's guaranteed to satisfy
the running interception property. Why is that?
So let's consider for example, all of the factors that involve the variable, umm,
D. So here is, what are the clusters that
involve the variable D? Well, there's this one.
And then there's this, that, that. Those are the set-, where we have the
subsets, which I didn't mark. And then these ones.
And notice that it's a tree by definition.
Because d doesn't appear on any other subset, except these ones.
And so the trees is a star. It's the variable cluster plus the big
factors that contain the variable. So d, and, in this case, the three
clusters that contain it. Look to summarize.
We have, we know, we looked at the properties of the coaster graph, we see
there is two of them that it needs to satisfy, the first is family preservation
which allows the set of factors to be encoded and the second is the running
intersection property which serves two purposes, the first is to connect all
information of any single variables that it can all be transmitted through the
graph but without creating type feedback loops that are going to create the self
reinforcement and highly inaccurate answers.
And the beta cluster graph is often the first default that people use.
Because it's easy to define. And it's guaranteed to be correct.
But richer cluster graph structures of the type that we talked about can offer
very different and sometimes significantly better trade-offs.
With respect to, on the one hand, the computational cost.
And on the other hand. which of course, as you start increasing
the amount of, of the sizes of the messages that are passed.
that can grow more expensive. But at the same time, allow us to
improve. dip the preservation of dependencies as
messages are passed in the graph so that more information is actually kept and not
lost in this message passing process.
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