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 Prediction
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来自 Stanford University 的课程
Probabilistic Graphical Models 3: Learning
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从本节课中
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.
与讲师见面
Daphne KollerProfessor
School of Engineering
We previously defined the notion of Bayesian estimation.
In which we have a prior over the parameters, and we continue to maintain a
posterior over the parameters as we accumulate new data.
What we haven't discussed though is how one might use a model in which we have a
distribution over the parameters. That is how do we take such a model and
use it to make predictions about new instances?
So, now let's think about how we use a Dirichlet distribution once we have it.
So assume that we have a param-, a, a model where our parameter theta is
distributed Dirichlet with some set of hyperparameters.
Now, if we're trying to make a prediction over the value of a variable X, that
depends on the parameter theta. Well, we're just, this is just now a
problem of inference problem. So the probability of x,
is simply, the probability of x given theta.
Marginali times the prior over theta. Marginalizing, in this case corresponding
to an integration over the value of zero. And I give this, this interval over here.
So, if we plug through the integral, what we're going to get is the following form.
And I'm not going to go through the integration by parts that's required to
actually show this. But it's really a, a straight-forward
consequence of the properties of integrals of polynomials.
in this case we have that the probability that x takes the particular value xi
is one is 1 / Z times the integral over all of the parameters theta of theta i
which is the, probability given the parameterization theta, that x takes the
value of little xi times this thing over here, which is the prior.
And we multiply the two together, integrate out over the parameter vector,
theta which, in this case, is a k dimensional parameter vector.
And it turns out that when one does that, you end up with alpha i over the sum of
all J's alpha J, a quantity typically known as alpha.
And so we end up with a case where, the prediction over the next instance
represents the fraction of the instances that we've seen, as represented in the
hyperparameter of Dirichlet where we have x, little xi.
So if alpha i represents the number of instances that we've seen where x eq-.
Where the variable took the value, little xi.
The, prediction very naturally represents.
It, it is simply the fraction of the instances with that property.
And so once again, we see that there is a natural intuition for the hyper
parameters as representing the motion of counts.
Now, let's put these two results together and think about Bayesian prediction as a
function of, as the number of data instances that we have grows.
So here, we have a parameter theta, which initially was distributed as a Dirichlet,
with some set of hyper-parameters. And let's imagine that we've seen m beta
instances, x1 up to xm. And now we have the M plus first data
instance. And want to make a prediction about that.
So, the problem that we're trying to solve is now the probability of the M
plus first data instance, given the M first, the M instances that we've seen
previously. And so, we can once again plug that into
a probabilistic inference equation. So this is going to be the probability of
the M plus first data instance given everything, including theta times the
probability of theta given x[1] up to x[m] So we've introduced the variable
theta into this probability and we're marginalizing out over the variable
theta. Well, one thing that, immediately follows
is because of the structure of the, probabilistic graphical model here.
We have that xm + one is conditionally independent of all of these previous xes,
given theta. And so we can, cancel these from the
right-hand side of the conditioning bar. Which gives us, over here, probability of
xm + one given theta. And over here, we have the probability of
theta. Given x1 of xm.
And, so now let's think about the blue equation, the blue expression over here,
which is just the posterior. Over theta, given D.
Which are X1 off the XM. And we've already seen what that looks
like. That, as we showed just on the previous
slide is simply Dirichlet who's hyperparameters are Alpha one plus m1 up
the Alpha 1 plus mk. And so now, we're making a prediction of
a single random variable from a Dirichlet that has a certain set of
hyperparameters. And that was a thing we showed on the
slide just before that. Which is simply the fraction of the out.
The fraction of the hyperparameter corresponding to the outcome, xi.
As a fraction of all of, the sum of all of the hyperparameters.
Where, again, just to introduce notation. Alpha is equal to the sum of the sum of
alpha I. And M to the sum of the MI's.
Now notice what happens here. This parameter alpha that we just defined
which is the sum over all of the alpha I's that I have is a parameter known as
the equivalent sample size. And it represents the number of if you,
if you will imaginary samples that I would have seen prior to receiving the
new data, x1 of xm. Now look what happens if we multiply
alpha by a constant. So say we double all of our alpha I, then
we have we're going to let the MI's effect our estimate a lot less than for
smaller values of alpha. And so the larger the alpha, the more
confidence we have in our prior, and the less we let our data move us away from
that prior. So let's look at an example of the
influence that this, might have. So let's go back to, binomial data, or
Bernoulli random variable. And let's take the simplest example where
a prior is uniform for theta in 01. And we've previously seen that, that
corresponds to Dirichlet with hyperparameters 1-1.
So, this is, are deascht so this a general purpose derscht slave
distribution, in this case the hyper parameters are one, one and let's imagine
we get, five data incidences of which we have four ones.
And one zero. And, if you actually.
Think about the differences between what the Bayesian estimate gives you for the
sixth next coin toss relative in, in, when
doing maximum likely estimation versus the Bayesian estimation.
For maximum likely estimation we have, four heads, four tails.
Maximum likely is is four fifths, so that's going to be the prediction for the
sixth instance. The Bayesian prediction on the other
hand, remember is going to do the hyper-parameter alpha one plus M1 divided
by alpha plus M which in this case is going to be one plus four divided by two
plus five and that's suppose to give us 5/7.
Now let's look more qualitatively at the effect of the predictions, on a next
instance, after seeing certain amounts of data.
And for the moment, we're going to assume that the ratio between the number of 1s
and the number of 0s is fixed, so that we have one 1 for every four 0.
And that's the data that we are getting. And now let's see what happens as a
function of the sample size. So as we get more and more data, all of
which satisfy this particular ratio. So here we're playing around with a
different strength, our equivalent sample size but we're fixing the ratio of alpha
one to alpha zero to represent in this case the 50% level.
So our prior is a uniform fire but of greater and greater changing strength.
And so this little green line down at the bottom represents a low alpha.
Because we can see that the data gets pulled our, posterior.
So sorry. The line is drawing the posterior over on
the parameter or rather equivalency, the prediction of the next data instance over
time. And you can see here that alpha is low
and that means that even for fairly small amounts of data say twenty data points
are fairly close to the data estimates. On the other hand, this bluish line here
We can see that the alpha is high. And that means it takes more time for the
data to pull us, to the empirical fraction of heads versus tails.
Now let's look at varying the other parameter.
We're going to now fix the equivalent sample size.
And we're going to just start out with different prior.
And we can see that now we get pulled down to the 0.2 value that we see in the,
in the empirical data. and the further away from it.
We start, though. It takes us a little bit longer to
actually get pulled down to the data estimate.
But in all cases, we eventually get convergence to the value in the actual
data set. But, from a pragmatic perspective it
turns out that Bayesian estimates provide us with a smoothness where the random
fluctuations in the data don't don't cause quite as much random jumping
around as they do for example in maximum likelihood estimates.
So if what we have here is the actual value of the coin toss at different
points in the process, you can see that the blue line, this
light blue line corresponds to maximum likely data estimation basically bops
around the pheromone, especially in the low data regime.
Whereas the ones that use a prior, estimate to be the prior are considerably
smoother, and less subject to random noise.
In summary, Bayesian prediction combines two types of, you might call them
sufficient statistics. There is the sufficient statistics from
the real data. But there's also sufficient statistics,
from the imaginary samples, that, contribute, eh, to the derscht laid
distribution, these alpha hyper parameters, and the basion prediction
effectively makes the prediction about the new data instance by combining both
of these. Now, as the amount of data increases,
that is, at the asymptotic limit of many beta instances.
The term that corresponds to the real data samples is going to dominate.
And therefore, the prior is going to become vanishingly small in terms of the
contribution that it makes. So at the limit, the Bayesian prediction
is the same as maximum likelihood destination.
But initially in the early stages of destination, before we have a lot of data
the the priors actually make a significant difference.
and we see that the Dirichlet hyper parameters basically determine both our
prior beliefs, initially before we have a lot of data as well as the strength of
these beliefs that is how long it takes for the data to outweigh the prior, and
move us towards what we see in the empirical distribution.
But importantly, even as we've seen here in the very simple examples, and as we'll
see later on when we talk about learning with Bayesian networks,
it turns out that this Bayesian learning paradigm is considerably more robust in
the sparse data regime, in terms of its generalization ability.
