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.
Overview: MAP Inference
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Del curso dictado por Stanford University
Probabilistic Graphical Models 2: Inference
322 calificaciones
De la lección
Inference Overview
This module provides a high-level overview of the main types of inference tasks typically encountered in graphical models: conditional probability queries, and finding the most likely assignment (MAP inference).
Conoce a los instructores
Daphne KollerProfessor
School of Engineering
As we said, there's many different kinds of queries that one can use a graphical
model to answer. Conditional probability queries are very
commonly used, but another commonly used type of inference is called MAP
Inference. So, what is MAP inference?
MAP stands as a shorthand for what's, for Maximum a Posteriori, and it's defined as
follows. Here, we have a set of evidence of
observations, e over some variables to E, and we have a query Y.
And it turns out that for computational reasons, that we're not going to discuss
for the moment it's important that the set of Y's is everything, all of the
variables other than E. [SOUND] So now, our task is to compute
what's called the MAP assignment, which is the Y, the silent little y, to the
variables y that maximizes the conditional probability of the variables
y given the evidence. Now so for those of you who haven't seen
the notation Argo Max. Argo max is the y that provides the
maximal value to this expression, over here.
Now, note that in some cases, this maximizing value might not be unique.
That is there might be several different assignments say, A1, Y1, and Y2 that give
the exact same probability. And so, the MAP is not necessarily a
unique assignment. This has many applications, some of which
we've discussed before. So, in the context of message decoding,
for example, where we have a set of noisy bits that are passed over the channel of
a noisy communication channel, what we often want to get is the most
likely message that was transmitted. That is, an assignment to the transmitted
bits that is the most likely given or evidence.
In the context of image segmentation, we would like to take the pixels and figure
out the most likely assignment of pixels to se, to semantic categories.
So both of these can be viewed as MAP assignment problems.
And an important thing to understand about MAP is that it really is a
different problem than conditional probability queries.
So to understand that, let's look at the following very simple
example over just two random variables. So here, we have a Bayesian network over
the variables A and B. And if you multiply the to CPDs, it's you
get the joint distribution shown over here.
And it's not and it's fairly immediate to see that the MAP assignment is this one,
because it has the highest probability. Can we get at the map assignment by
looking separately at the variable A and at the variable B?
So if we look at that, we see that if we look separately at the variable A, we
can, the most likely assignment to the variable A is A1.
As opposed to in the MAP assignment, the variable A took the value, A0.
And so, one can't look separately at the marginal over A and over B, and use that
to infer the MAP assignment. And the reason is that we're looking for
a single assignment over all of the variables that together has has the
highest probability. Unfortunately, just like the conditional
probability inference however, this problem too is empty heart.
So, again let's formalize what exact problem is empty heart in this context.
And it and so here is, again an, one example of an empty heart problem in the
context of MAP which is just define the joint assignment with the highest
probability. It's not the only NP-hard problem.
Here is another one. Figuring out what, for a given
probabilistic graphical model and some threshold little p,
whether there exists an assignment, little x, whose probability is greater
than p. That problem too NP-hard.
So should we give up. Well, just like in the context of
conditional probability queries, the answer is no. And there is algorithms
that can solve this problem very efficiently. In fact, in a even broader
set of problems then, for conditional probability queries.
So, let's again look deeper into this problem and understand the foundations of
what might make it tractable. So, going back to our example of a Bayesian
network, once again we're going to view CPDs as
factors. So here a P of C again translates into a
factor over C just like before. And whereas, in the case of conditional
probability queries, we want it to sum out some of the variables marginalize
them. Now, we're going to what to find the arg
max which is the assignment of these variables which maximizes the product.
And so, here we have a max of a product, which is why this is often called a max
product problem. Let's break down the max product problem.
So, imagine that we, in more general have, in the more general case have
probability over Y given E equals little e and re, let's remind ourselves that Y
is is the set of all variables other than the ones in E.
And so, by the definition of conditional probability, we have this ratio whose
where we're trying to find the maximal Y that maximizes this ratio.
And notice that the denominator is constant relative to Y.
[SOUND] Sorry yes, with respect to Y. Which means that for the purpose of
finding the maximum Y, we don't really care about the denominator only the
numerator which is the probability of Y, Ee = e, the unnormalized a normalized
measure. The prob, this numerator in the general
case is a product of factors normalized by the partition function, where the
factors here are the reduced factors, reduced relative to the evidence.
Once again, we notice that this is the partition function and its constant
relative to y. Which means that we can ignore it. And
so, once again, we have that this expression is now proportional to a
product of the same reduced factors. And so, what we're trying to do is we're
trying to maximize a product of factors in this more general case, as well.
Which gives us the following as the optimization problem.
There's many algorithms that can solve the MAP problem.
The first is analogist two algorithms for some product.
these algorithms take the maximization operation and pushes that into the factor
product, giving rise to a max product variable of a nation algorithm.
We also have message-passing algorithms, which again are a direct analog to
algorithms for some product. And they give rise to a class of
algorithms called max-product belief propagation.
However, because the MAP problem is intrinsically an optimization problem,
one can also call in techniques that are specific to optimization.
And the class of such methods include methods that are based on integer
programming, which is a general class of optimization
over discreet spaces. It turns out that this class of methods
that build on integer programming techniques, is one of the most popular
methods in the last few years and has given rise to, a whole new range of math
algorithms that are considerably better than many of the previous algorithms
developed after that point, especially for the approximate case.
It also turns out that for certain types of networks that some of which we'll
discuss, there are specific algorithms that are very efficient for that
particular class of of graphical model. And one of those, perhaps the most
commonly used if not the only one, is methods based on class of algorithms
called graph cuts and. And,
and finally, because it's an optimization problem, one can also use standard search
techniques over communatorial search spaces,
and they're problems for which this is also a very useful and successful
solution. So, to summarize the MAP problem, aims to
find a single coherent assignment of highest probability.
And, that means that it is not the same as maximizing individual marginal
probabilities as we saw in the example. one can reformulate this problem as one
of finding the max over a factor product. And, this is a communitorial optimization
problem which admits a whole range of different solutions on which some are
exact and others approximate.
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