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.
Using a Markov Chain
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Do curso por Stanford University
Probabilistic Graphical Models 2: Inference
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Sampling Methods
In this module, we discuss a class of algorithms that uses random sampling to provide approximate answers to conditional probability queries. Most commonly used among these is the class of Markov Chain Monte Carlo (MCMC) algorithms, which includes the simple Gibbs sampling algorithm, as well as a family of methods known as Metropolis-Hastings.
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Daphne KollerProfessor
School of Engineering
We previously defined the notion of a Markov chain that allows to generate
samples from an intractable distribution. But we left unanswered the question of,
assuming we have a Markov chain that has the desired stationary distributions.
How do we go about using it in the context of a concrete algorithmic
procedure? Well let's imagine that somebody has
provided us a Markov chain. for, a distribution, sorry.
The, imagine that our goal is to compute some probability relative to a
distribution P. But, for whatever reason, P is too hard
to sample from directly. And so how do I use the Markov chain to
get around that? First, we construct the Markov Chain, p,
whose unique stationary distribution is P. SO so it needs to be a regular Markov
chain, usually. And then we're going to generate samples
from this distribution. So we're going to start out by sampling
my, initial state from some arbitrary distribution, p0.
P0's not the same as P. So obviously I can't use the zero sample
as a way of estimating an event relative to P.
And so what I do now is I start walking along the chain.
Starting sampling xt + 1 from the transition model.
And because the chain converges to a stationary distribution eventually I'll
have a sample that is very close to to the to be in sample from the
distribution P that I care about. So, now, here's the sticky issue.
And this is really a sticky issue. when have we walked long enough that we
could actually use our samples? Because we don't want to use samples at
the beginning, because they're far from P.
and so we need to walk around long enough for the chain to get close to its
stationary distribution. And that state is called mixing.
So mix is when we, is when P of t is close enough.
Pi. So that your time T distribution is close
to the stationary, and that's the point that which I could say, good enough, I
can start collecting samples. So how do we know if a chain has mixed or
not? And the short answer is you don't.
and so this is, as I said, the sticky part of Markov chain methods.
So in general, you can never really prove that a chain has mixed.
But in some cases you can show that it hasn't mixed.
And then if you run lots and lots of tests, and none of them have convinced
you that the chain hasn't mixed, then you sort of resign yourself to assuming that
it has, in fact, mixed. So how do you convince yourself that a
chain hasn't mixed? You compute chained statistics, and we'll
show some examples of that in a moment, in different windows within a single run
of a chain. So for example, here I have a single run
of the chain. And I run for a while And then I look at
little windows, usually they won't be this small and I compare this window.
This window computing various statistics. For example, what is the probability of
hitting a particular state? And I say, is the probability of that in
this window close to the probability of that in this window?
And if it is, then maybe I've reached at least some kind of convergence point,
in terms of where the chain is reached. Now this of course is not a sufficiently
good answer because it could also be derived in a chain that has these sort of
two very high probability regions but are very hard to get from each other.
And if the chain is kind of ambling along in this part of the space and never
hitting the other part of the space. Then, you're still going to get very
similar probabilities across a window of a single run of the chain.
Because all of the samples are taken from this part, and none of the are ever taken
from that part. And so we don't know that we have them
mixed. So, a more reliable statistic is to take
these, more reliable evaluation criterion is to take these statistic across
different runs that are initialized in different parts of the space.
And then you might hope that one chain is traversing,
one run of the chain is traversing this region, and another run of the chain is
traversing this region. And so now the statistics would show a
difference and indicate that mixing hasn't taken place.
So, what statistics might, So how, what might we do it, more
concretely? So let's look at two examples.
Here is, two example runs of a chain that we'll describe a little bit later.
and what we measure here. And this is the first statistic to
measure in Markov chains, is the log of probability of of a sample.
So you compute, The log probability of sampling.
You can't always compute the log probability directly.
You might compute an un-normalized log probability, as we'll talk about, as
we'll talk about later. But basically, you compute the log
probability, or some constant factor thereof.
And now, you can compare two runs. And this is a run that's initialized from
an arbitrary state. And this is one that's initialized from a
high probability state. And you look at those, and you say, has
it mixed, relative to this criterion? The answer is maybe.
It looks okay, but you're not entirely sure.
But we can look at other statistics. Oh, sorry.
Let's look at another example. What about this one?
Here is exa, again an example of two runs, one of which is initalized from an
arbitrary state, and initialized from a high probability state and you can see,
that the log propability values are no where close to each other and so in this
case the answers is definetely not, these are two runs of the chain, and really,
neither is mixed. And so you need to run for a lot longer
which, you know, this comes out, this goes up to 600,000 so it kind of indicate
to you how much time this might take. A different statistic, a different way of
looking at this, is for a different kind of statistic.
So now we have for example, the probability relative to a window that we
compute in the chain. So remember, all of this is relative to a
window in the chain, after we hope that mixing has taken place.
And now we compute the probability that, within states in this region, what is the
probability of, that, the states are in some sets, so for
example the set of states where. X3 is equal to two.
And now we compute that statistic using the two initializations of the chain, so
this is chain one, or run one, and this is run two.
And now we do a scatter plot for for different statistics.
So each of these points. This is the probability say of X3 equals
X3 equals value two. This might be the probability that X1 is
equal to zero. This might be some other probability
that, you know, X5 is equal to seven. And so each of these is a statistic and
what you see here is a scatter plot. One is the estimate that they get from
the one chain and from the other, and looking at this It should be obvious that
this first chain, has not, that, we have not got mixing on the left hand side,
because you can see that there are all these points here that have high
probability in one of the two rungs, and a probability zero in the other, and vice
versa. Whereas here, most of the estimates are
clustered around the diagonals. So that you're getting similar estimates
from the two chains. And so again,
this one is, the first one is a clear no and the second is maybe.
And if I do a lot of these statics and they all come up with maybe then I'm
willing to then trust the answers and the that is taking place.
So now that I've started collecting samples, how do I use these samples?
Well, one important observation to keep in mind is that once the chain is mixed,
all of my samples are from stationary distribution.
That is xt is from pi, so as xt1, + 1, t2, + 2, t + 3 and so on and so forth.
And so we can use every single one of those samples.
Because they are all from the correct distribution.
So once I've determined that a cert-, that I've sampled long enough for mixing,
or believe that I've sampled long enough for mixing.
We should collect and use all of the samples.
And in fact, there are you might read some papers that say.
Oh, I'm only going to collect every hundredth sample.
There's actually papers that prove that using every single sample is better than
collecting every hundredth sample. But then you might ask, well why would
those papers tell me that I should only collect every 100 samples as opposed to
collecting all the samples if they're all from the correct distribution?
Because, and this is undoubtedly true, adjacent samples, ones that are near by
to each other in time are correlated with each other.
Because how, even if xt is from the right distribution Pi and so is xt + 1.
Xt + 1 is still going to be close to T, close to xt.
And so you're not really getting two different samples.
You're getting two that are very close relatives to each other.
Now as I said it's important to recognize that phenomenon.
Because it's important to realize that just because you've reached mixing and
collected a thousand samples, doesn't mean you have a thousand samples worth of
information. So you shouldn't go back and apply the,
apply the you know, one of the bounds that we saw in assuming that you have iad
samples. The samples are not.
I ID. So that, that doesn't mean you shouldn't
use them, using them is still better than not.
Now, this is where you get bitten twice by the same phenomenon.
The worse a chain is to mix. So the longer you need to wait for the
initial distri, for the initial samples to be good enough, the more correlated
the samples are because the slower you are moving around in the space in
general. And so if a chain is bad, it's bad in two
different ways. It's bad because it takes you longer to
mix, and it's bad because the samples you are collecting are not as useful.
because of the correlation structure between the samples.
So to summarize, first the algorithm and then the implications.
Here is how we would actually end up using a mark off chain.
So first of all I'm running C chain and parallel.
And I'm going to sample and initial state from each of them.
And then I'm going to repeat until I reach some determination that mixing is
taking place. And what I do is I forward march each of
the chains using a random walk process, where I generate the T plus first sample
from the Tth sample for that corresponding chain.
Then I compare windows statistics, of the type that we showed earlier in the
different chains, to try and determine whether mixing has taken place.
And then, I repeat until I am con, confident in the mixing properties.
With that, I now repeat until there's sufficient samples to collect data.
So I start out with an empty sample set, and then for each of my chains I generate
a sample. Then I put it in my sample sets.
And I repeat them until I have enough enough samples.
And then finally, when I've collected enough samples, I can go ahead and
compute the expectation in anything that I care about, whether it's an indicator
function or a more complicated function. Summarizing the implications, Markov
chains are a very general purpose class of methods for doing approximate
inference in a, in general probabilistic models, not even necessarily
probabilistic graphical models. they're often very easy to implement
because the local sampling that we're doing is often quite straightforward to
execute. And, it has good theoretical properties
as we sample, as we generate enough samples that are sufficiently far away
enough from our starting distribution. Unfortunately, these theoretical
guarantees are very theoretical in many cases.
And so this method also has some significant cons.
First, it has a very large number of tunable parameters and design choices.
What's my mixing time? Which statistics do I measure?
How close are the statistics to each other in order for me to declare that
mixing is taking place? how many samples do I collect?
do I what window size do I use to evaluate mixing.
All of these are design choices, they all make a difference, and so there's a lot
of, finicky tuning that needs to be done when running an MC and C method in
practice. The second is that bit depending on the
design of the chain, this can be quite slow to converge because it's very
difficult to design chains that have good mixing properties.
And finally, it's very hard to tell whether a chain is working.
That is it's not straight forward to determine whether the chain has mixed.
and how whether my samples are sufficiently uncorrelated to each other
that I'm getting a reliable estimate of whatever it is that I'm trying to figure
out. So, a lot of advantages but also some
significant disadvantages.
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