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 third in a sequence of three. Following the first course, which focused on representation, and the second, which focused on inference, this course addresses the question of learning: how a PGM can be learned from a data set of examples. The course discusses the key problems of parameter estimation in both directed and undirected models, as well as the structure learning task for directed models. The (highly recommended) honors track contains two hands-on programming assignments, in which key routines of two commonly used learning algorithms are implemented and applied to a real-world problem.
Bayesian Estimation for Bayesian Networks
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Из курса от партнера Stanford University
Probabilistic Graphical Models 3: Learning
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Parameter Estimation in Bayesian Networks
This module discusses the simples and most basic of the learning problems in probabilistic graphical models: that of parameter estimation in a Bayesian network. We discuss maximum likelihood estimation, and the issues with it. We then discuss Bayesian estimation and how it can ameliorate these problems.
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Daphne KollerProfessor
School of Engineering
When we previously defined the notion of Bayesian estimation and showed how it
could be applied in the context of a single random variable say, a multinomial
random variable. Now we're going step back to the world of
probabilistic graphical models and think about the application of these ideas to
the problem of estimating parameters in the Bayesian network.
So, let's draw again the probabilistic graphical model that represents Bayesian
estimation in a Bayesian network. So just as before, in the single variable
case, we're going to inject into the model explicitly the parameters that
characterize sorry,
random variables that define the parameters.
And so here we have two random variables theta X which represent the CPD of X,
and theta Y given X, which represents CPD P of Y given X.
Now notice that each of these actually, vector value, because there's going to be
multiple actual numbers in these in each of these CPDs,
but we're going to draw them as single circles.
Now once again we can look at this network and read out certain important
conclusions. So the first important conclusion is that
the instances, these XY pairs, are independent given the parameters and we
can see that by noticing that if I condition on both theta X and theta Y
given X, then the XY pairs become conditionally become d-separated from
each other and so we have conditional independence following as a consequence
of the structure of the graphical model. We also have, and this is another
explicit property that we can read from this diagram is that theta X and theta Y
given X are marginally independent. So a priori, we have that the parameter
prior over all of the, all of the parameters.
theta can be written as the product over i, and in this case, i is being the
random variables in the network of the prior over the CPD for Xi and so the
prior is the product of little priors one through each CPD.
Now, it follows from this, from, by just writing down the the
graphical model and looking at what the implications of the expressions are, that
the posteriors of this data are also independent given complete data.
And the reason for that is that complete data d-separates the parameters for the
two CPDs. If you look at at this network over here
and assume that we have all of these variables conditioned observed,
then you can see that there is no active path between theta X and theta Y given X.
because, for example, if we look at potentially, this trail, we can see that
X[2] blocks the trail from theta X to theta Y given X.
And so again, following directly from the structure of this network, we can see
that the posterior distribution theta X, theta Y given X given d decomposes as a
product of the posterior over theta x given d times the posterior of theta Y, Y
given X given D. Which means that just as in maximum
likelihood estimation where you could break up the estimation problem into one
of estimating each CPD separately, we can do the same here.
Only now, we can do it using Bayesian estimation, where instead of just picking
a single parameter setting for the CPD, we compute the separate posteriors
separately and then put them together into a single posterior.
Now, it turns out that we can do even finer breakdown in the context of table
CPDs. So here we have and we're now looking at
the binary case where X is a binary valued random variable.
So now we have two multinomials in our CPD, one for Y given X, one corresponding
to the case of Y given X1 and the other Y given X0.
And it turns out that if, again in this model we're assuming, that these are
independent a priori, which is what this diagram says, because you notice there's
no edges between so you notice that they are marginally independent, theta Y given
X1 and theta Y given X0. If we're postulating that that holds as
we are in this diagram, it turns out that they are also
independent in the posterior. Now, that is a little bit trickier to
show, because it turns out that you can read it directly from the diagram because
in fact even given complete data we have an active trail that goes from theta Y
given X1 through Y1, which since it's observed, it activates the V structure
into theta Y given X0. [COUGH] But it turns out that if we go
back to some of the examples of context-specific independence, that we
had in the case of specifically a multiplexer CPD, we can derive that these
are in fact, despite the appearance of an activated V structure, conditionally
dependent in the posterior as well. And so, once again,
we can compute the posterior as a product of posteriors of the form p of theta X
given d times the probability of theta Y given X1 given d times the probability of
theta one given X0. Okay.
So we can generalize this to a general Bayesian network and let's assume that we
have a Bayesian network with table CPDs that's specified in terms of multinomial
parameters of the form theta X given u, where u is some assignment to X's
parents, u, Then if, for each such multinomial
parameter, we have a Dirichlet prior with appropriate hyperparameters, then, we can
show using the kind of analysis that we just did, combine with the analysis of
the posterior for a single multinomial, that the posterior is now a Dirichlet,
with hyperparameters that represents the prior that we had for that multinomial
plus the sufficient statistics that represent the count in the [INAUDIBLE]
particular combinations of the parent and the child.
And so for example, for the, entry in the multinomial representing the value little
x1 for X and the [INAUDIBLE] little u for the parents u,
we have, this prior parameter plus the count in the data for that combination of
x and u. So,
now we know how to take a set of priors, and use the data to update them, to form
posteriors. Now, let's think about where the priors
might come from. And a priori, it might seem very daunting
to construct a set of priors for all of the nodes in a Bayesian network.
It turns out however that there is a general purpose, scheme for doing that,
that, is both, easy and has some good theoretical properties, and, that, scheme
works as follows, so, what we're going to do, is we're going to define a prior
basion network, that has some set of parameters state of zero.
And we're going to define a single equivalent sample size.
Which is going to be applicable or applied to all of the nodes in the
Bayesian network, and so in order to specify the parameter, the hyper
parameter alpha X given u for for an assignment X equals little x, and u
equals little u. We're simply going to compute the
probability in this parameterized network of X and u and we're going to multiply it
by the equivalent sample size alpha. Now, in many cases, you are just going to
use theta zero to be the uniform parameters which makes it all a very easy
computation. But this provides a simple, coherent way
to specific all of the hyperparameters simultaneously.
And so, let's look at an example, here is a network xy that, that has no
edge and and let's imagine that, that is our
that is our prior network. And, so we're going to in fact assume a
uniform distribution over the values xy and now let's, so theta zero is uniform.
And now, let's look at what we would get for different situations in a network
where we have X being a parent of Y which is the network with parameters we
actually want to estimate. And so what we would get for.
And let's assume they're both binary, X and Y.
And so X is going to be distributed as a parameter with Dirichlet with
hyperparameters, alpha over two, alpha over two.
And Y is going to be distributed, remember, so,
not what. Not X theta of X is going to be
distributed this way. And theta of Y given X0 was going to be
distributed in the following way. So Dirichlet with hyperparameters alpha
times the probability of X, Y, which is the uniform distribution is a
quarter. And similarly for theta Y given X1 is
going to have the same, here say distribution.
And if you think about this, this makes perfect sense, because it tells us that
we have seen the same number of examples of X as we have of Y.
It's just that in Y, we had to partition the examples of X, of Y over those where
we had Xxo. = X0 and those where we had Xx1.
= X1. If on the other hand, we had say
Dirichlet of alpha over two alpha over two for the two except for the two
multinomials corresponding to, the two corresponding to Y,
this one and this one. It would, it would imply that we've seen
twice as many Ys as we've seen Xs. So let's see what kind of effect using
the Bayesian estimation has on a pseudo real world example.
So, this is actually a real network. It was developed for monitoring patients
in an ICU and we call it the ICU-Alarm network.
And it turns out that the ICU-Alarm network has 37 different variables that
represent things like whether the patient was intubated the patient's blood
pressure heart rate, and various other medical events that might happen.
And, it turns out that overall, the network has 504 parameters.
Now, there aren't actually data cases here, this was a hand constructed
network, and so what we're going to do is we're going to sample instances from the
network. And, then, we're going to pretend that we don't know the network
parameters and see the extent of which we can recover the network parameters via
learning from the instances that we sampled from it.
I should say that this is a pseudorealistic learning problem, because
the instances that one samples from a network are, are always cleaner than the
instances that one gets in the context of a real world data case, data set, because
it, in a real world scenario, it is rarely the case that the network whose
structure you have the network whose that you're trying to learn has the exact same
structure as the true underlying distribution from which the data were
generated. And so this is a much cleaner scenario,
but still it's useful and indicative. So what we see here, are, the results of
learning, as a function of the x-axis, which is the number of samples and the
y-axis is a distance function between the true distribution and the learn
distribution, and that distance function we're not going to talk about this at the
moment, it's the notion called the relative entropy, it's also called KL
divergence. But what we need to know about this for
the purposes of the current discussion is that when distributions are identical
it's zero, and otherwise it's non-negative.
So, what we see here is that the blue line, corresponds to maximum life data
information. And we can see several things about the
poline. First of all it's very jagged, there's a
lot of bumps in it, and second, it's consistently higher then all of the other
lines. Which means that max and likelihood
estimation, although it does continue to get lower as we get more data, with as
high as five thousand data points, we still haven't gotten close, to the true
underlying distribution. Conversely let's see what happens with a
Bayesian estimation. This is all Bayesian estimation with a
uniform prior. And different, equivalent sample size.
So this is using a prior network with a uniform network in different values of
alpha. And what we see here is that, for alpha
equals five. That's the green line.
Alph equals ten are almost sitting directly on top of each other and they're
both considerably lower then all of the other lines and also the maximum likely
destination. As we increase the prior strength so that
we are have a firmer belief in in. A uniform prior.
We can see that we move a little bit away.
and now the performance becomes a little worse.
But notice that by around 2,000 data points we're already pretty close to the
case that we were for an equivalent sample size of five.
For 50, which is this dark blue line. It takes a little bit longer to converge,
and it doesn't quite make it, But even with an equivalent sample size of 50,
which is pretty high. you get convergence to the correct
distribution much faster than you do from maximum likelihood destination.
So, to summarize. in Bayesian networks, if we're doing
Bayesian parameter estimation. If we're willing to stipulate that the
parameters are independent, a priori. Then they're also independent in the
posterior. Which allows us to maintain the posterior
as a product of posteriors over individual parameters.
For multinomial Bayesian networks, we can go ahead and, con-, perform Bayesian
estimation using the exact same sufficient statistics that we used for
maximum likelihood destination. Which are the counts corresponding to a
value of the variable and a value of its parents.
And whereas, in the context of maximum likelihood estimation, we would simply
use the formula on the left. In the case of Bayesian estimation, we
are going to use the formula on the right.
Which has exactly the same form. Only, it also accounts for the
hyperparameters. And, in order to do this kind of process,
we need a choice of prior, and we show how that can be effectively elisteted
using both a prior distribution specified say, as Bazy network as well as an
equivalent sample size.
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