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 first in a sequence of three. It describes the two basic PGM representations: Bayesian Networks, which rely on a directed graph; and Markov networks, which use an undirected graph. The course discusses both the theoretical properties of these representations as well as their use in practice. The (highly recommended) honors track contains several hands-on assignments on how to represent some real-world problems. The course also presents some important extensions beyond the basic PGM representation, which allow more complex models to be encoded compactly.
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From the course by Stanford University
Probabilistic Graphical Models 1: Representation
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From the lesson
Bayesian Network (Directed Models)
In this module, we define the Bayesian network representation and its semantics. We also analyze the relationship between the graph structure and the independence properties of a distribution represented over that graph. Finally, we give some practical tips on how to model a real-world situation as a Bayesian network.
Meet the Instructors
Daphne KollerProfessor
School of Engineering
You now know a bunch about machine learning.
In this video, I'd like to teach you a programming language, Octave, in which
you'll be able to very quickly implement the learning algorithms we've seen
already and the learning algorithms we'll see later in this course.
In the past, I've tried to teach machine learning using a live variety of
different programming languages including C+++, Java
Python NumPy, R and also Octave. And what I've found is that students were
able to learn the most productively learn the most quickly and prototype the
algorithms most quickly, using a relatively high level language like
Octave. In fact .
what I often see in Silicon Valley is that
if, even if you need to build, if you want to build a large scale deployment of
a learning algorithm, what people will often do is prototype in
the language like Octave, which is a great prototyping language.
So you can sort of get your learning algorithms working quickly.
And then, only if you need a very large scale deployment of it,
only then spend your time reimplementing the algorithms in C+++ or Java, or some
other language like that. Because one of the lessons we've learned
is that program a time or developer time, that is your time,
you know, the machine learning experience time is incredibly valuable.
And if you can get your learning happens to work more quickly in Octave, then
overall, you have a huge time savings by first developing the algorithms in
Octave, and then implementing in like maybe C++ or Java only after we have the
ideas working. The most common prototyping languages I
see people use for machine learning are Octave,
Matlab, Python NumPy, and R. Octave is nice since it's free open
source and Matlab works well, too, but is expensive to many people.
But if you have access to a copy of Matlab, you can also use Matlab for this
class. If you know Python NumPy, or if you know
R, I do see some people use it. But what I see is that people usually end
up developing somewhat more slowly in, you know, mul, these languages because
the Python NumPy syntax is just slightly concur than the Octave syntax. and so
because of that and because we're releasing starter code in Octave I
strongly recommend that you not try to do the programming exercises in this class
and NumPy R, but that I do recommend that you instead do the programming exercises
for this class in Octave instead. What we're going to do in this video is
go through a list of commands fairly quickly, and the goal is to show you the
range of commands and the range of things you can do in Octave.
The course website will have a transcript of everything I do.
And so after watching this video, you can refer to the, to the transcript posted on
the course website when you want to find a command.
Briefly, what I recommend you do is first watch the tutorial videos.
And after watching, you know, to the end, then install Octave
on your computer. And finally, go to the course website,
download the transcript of the things you see in this session and type in whatever
commands seem interesting to you in to the octaves of running on your own
computer so that you can see it work for yourself.
And with that, let's get started. Here's my Windows desktop, and I'm going
to start an Octave. And I'm now in octave and, and there's my
octave prompt. Let me first show you the elementary
operations you can do in octave. So you can type in 56, + 6, that gives
you the answer of 11. 3 - 2,
5 * 8, 1/2, 2^6 is 64. So, those are the elementary math
operations. You can also do logical operations. So, 1
= 2 this evaluates to false. The percent command here is means a
comment. So, 1 = 2 evaluates to false which is represented by zero.
One not equals to two, this is true so that returns one, note that the not equal
sign is tilde equals symbol and not than equals which is what some other program
languages use. Let's see logical operations one and
zero, use a double ampersand sign to denote the logical and,
and that evaluates the false one over zero.
z or operation, and that evaluates to true.
And I can xor one and zero, and that evaluates to one.
This thing over on the left is octave 3.2.4.exe:11. This is the default octave
prompt. So it shows the what?
a version in Octave, and so on. if you don't want that problem, there's a
somewhat cryptic command, PS, quote, greater than, greater than, and so on,
that you can use to change the prompt. And I guess this quote of string in the
middle your quote greater than, greater than space,
that's what I prefer my octave prompt to look like.
So, if I hit Enter oops, excuse me. Like so,
PS1 like so, now my octave prompt is changed to the
greater than, greater than sign which, you know, looks quite a bit better.
Next, let's talk about octave variables. I can take the variable a and assign it
to three, and hit Enter. And now, a is equal to three.
If you want to assign a variable but you don't want it to print out the result, if
you put a semicolon, the semicolon suppresses the print the, the print
output. So, if we do that, Enter, it doesn't print anything.
Whereas a equals three, you know, makes it prints it out, whereas a equals three
semicolon doesn't print anything. I can do string assignments, b equals hi.
now, if I just enter b, it prints out the variable b,
so b is the string hi. c3, equals three greater than or equal to
one. So now, c evaluates to true.
If you want to print out or display a variable, here's how you go about it.
Let me set api. equals pi.
And if I want to print a, I can just type a like so, and it'll print it out.
For more complex printing, there's also the disp command which sense the
displays, when display a, just prints out a like so.
You can also display strings, so disp print f, two decimals.
The percent of 0.2f, a like so. And this print out the string, 2
decimals: 3.14. This is kind of a old, old style C syntax
for those of you that have coded in C before, this is essentially the syntax
you used to print strings. So the, sprint f generator generates a,
generates a string that is this, you know, two decimals 3.14 one for string.
This percent 0.2f means substitute a into here, sharing it with, you know, two
digits after the decimal point. And the disp takes this string that's generated
by this print f command, s print f or the string print f command, and disp actually
displays the string. And to show you another example,
sprint f six decimals percent 0.6f, a and this should print pi with six decimal to,
to six decimal places. Finally, I was explaining a like so, a
lot like this. There, there are useful short cuts that
you type format long. It causes strings to by default to be
assuage to a lot more decimal places to and format short is a command that
resources default of just printing a small number of digits.
Okay. That's how you work with variables.
Now, let's look at vectors and matrices. Let's say, I want to assign a to a
matrix, let me show you an example. 12; 34; 56, this generates a three by two
matrix a whose first row is one two, second row is three, four, the third row
is five, six. What the semicolon does is essentially
say, go to the next row of the matrix. There are other ways to type this, a=12;1
34;. 56 like so,
and that's another equivalent way of assigning a to be the values of this
three by two matrix. Similarly, you can assign vectors.
So, v = 1,2,3. this is actually a row vector,
or this is a three by one vector, right?
This is a fat y vector, as a, excuse me, not, this is
a one by three matrix, right?
not three by one. If I want to assign this to a column
vector, what I would do is instead do v1;2;3.
= 1; 2; 3, and this will give me a three by one, instead of one by three vector.
So this would be a column vector. So here's some more useful notation.
I can all set v1:0.1:2. = 1:0.1:2.
What this does is it sets v to the bunch of elements that start from one and
increments in steps of 0.1 until you get up to two.
So, I might do this. v is going to be this, you know, row
vector or this is what? A one by eleven matrix really that's 1.1,
one, 1.1, 1.2, 1.3 and so on until we get up to two.
Now and I can also say that v1:6, = 1:6 and that set's v to v, you know, these
numbers one through six. Okay.
Now, here are some other ways to generating matrices.
One's two by three, is a command that generates a matrix that is a two by three
matrix that is the matrix of all ones. So, if I set c equals 2 times ones, two
by three, this generates a two by three matrix that is all twos.
And this is, you can think of it as a shorter way of writing this and c equals
two, two, two, semicolon two, two, two, which would also give you the same
result. The set w equals ones, one by three, so
this is going to be a row vector or a you know, a row of three ones.
And similarly, you can also say w equals zeroes one by three, and, this generates
a matrix, a one by three matrix of all zeroes.
Just a couple more ways to generate matrices.
if I do w equals rand one by three, this gives me a one by three matrix of all
random numbers. If I do rand three by three,
this gives me a three by three matrix of all random numbers drawn from the uniform
distribution between zero and one. So, every time I do this, I get a
different set of random numbers drawn uniformly between zero and one.
For those of that who know what a Gaussian random variable is or for those
of you who know what a normal random variable is,
you can also set w equals rand n one by three.
and so these are going to be three values drawn from a Gaussian distribution with
mean zero and variant or standard deviation equal to one. And you can set
more complex things like w-6 = -6 plus square root of ten times let's say, rand
n one by ten thousand, and I'm going to put a semicolon at the end because I
don't really want this printed out. This is going to be what, well, it's going to
be a vector with a hundred thousand excuse me, ten thousand elements. So,
well, actually, you know what? Lets, lets print it out.
So this will generate a matrix like this right? With 10,000 elements, so that's
what w is. And if I now plot the histogram of w with
the hist command, I can now, and octaves print hist command,
you know, it takes a couple seconds to bring this up, but this is a histogram of
my random variable for w. That was -6 plus zero, ten times this
Gaussian random variable. And I can plot a histogram with more
buckets with more bins with say, 50 bins and
this is my histogram of Gaussian with mean minus six, because I have minus six
there, plus square root ten times this. So, the the varians or, this, of, of this
Gaussian random variable is ten over standard diviation, this square root of
ten, which is about what? 3.1. Finally, one special command to generate
a matrix, which is the eye command. So, eye stands for this is maybe a pun on
the word identity, but so I'm going to set eye 4, this is the four by four
identity matrix. So, I equals eye 4, this gives me a four
by four identity matrix. And I equals eye 5 eye 6, that gives me a
six by six identity matrix, and eye 3 is the three by three identity matrix.
Lastly, to wrap up this video there's one more useful command which is the help
command. So you can type help eye, and this brings
up the help function for the identity matrix.
hit q to quit. And you can also type help rand, brings
up documentation for the rand or the random number generation function, or
even help, help which shows you, you know, help on the help function.
So, those are the basic operations in Octave.
and with this you should be able to generate a few matrices multiply, add
things and, and use the basic operations in Octave.
In the next video, I'd like to start talking about more sophisticated commands
and how to move data around and to start to process data in Octave.
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