PATRICK WINSTON: Here we are,
down to the final sprint. Three to go. And we're going to take some of
the last three, maybe two of the last three, to talk a
little bit about stuff having to do with probabilistic
approaches-- use of probability in artificial
intelligence. Now, for many of you, this
will be kind of a review, because I know many of you
learned about probability over the [? sand ?] table and
every year since then. But maybe we'll put another
little twist into it, especially toward the end of
the hour when we get into a discussion of that which has
come to be called belief nets. But first, I was driving in this
morning, and I was quite astonished to see, as I drove
in, this thing here. And my first reaction was, oh
my god, it's the world's greatest hack. And then I decided, well, maybe
it's a piece of art. So I'd like to address the
question of how I could come to grips with that issue. There's a distinct possibility
that this thing is a consequence of a hat, possibly
the result of some kind of art show. And in any event, some sort of
statue appeared, and statues don't usually appear
like that. So I got the possibility of
thinking about how all these things might occur together
or not occur together. So the natural thing is to
build myself some sort of table to keep track of
my observations. So I have three columns
in my table. I've got the possibility of
a statue appearing, a hack having occurred, and some
sort of art show. And so I can make a table of all
the combinations of those things that might appear. And I happen to have already
guessed that there are going to be eight rows in my table. So it's going to
look like this. And this is the set of
combinations in this row where none of that occurs at all. And down here is the situation
where all of those things occur. After all, it's possible that
we can have an art show and have a hack be a legitimate
participant in the art show. That's why we have
that final row. So we have all manner of
combinations in between. So those are those
combinations. Then we have F, F, T, T, F, F,
T, T, F, T, F, T, F, T, F, T. So it's plain that the number of
rows in the table, or these binary possibilities, is 2 to
the number of variables. And that could be
a big number. In fact, I'd love to do a bigger
example, but I don't have the patience to do it. But anyhow, what we might do is
in order to figure out how likely any of these combinations
are, is we might have observed the area outside
the student center and rest of campus over a long period of
time and keep track of what happens on 1,000 days. Or maybe 1,000 months
or 1,000 years. I don't know. The trouble is, these events
don't happen very often. So the period of time that I use
for measurement needs to be fairly long. Probably a day is not
short enough. But in any case, I can keep a
tally of how often I see these various combinations. So this one might be, for
example, 405, this one might be 45, this one might be
225, this one might be 40, and so on. And so having done all those
measurements, kept track of all that data, then I could
say, well, the probability that at any given time period
one of these things occurs will just be the frequency-- the number of tallies
divided by the total number of tallies. So that would be a number
between 0 and 1. So that's the probability for
each of these events. And it's readily calculated
from my data. And once I do that, then I can
say that I got myself a joint probability table, and I could
perform all manner of miracles using that joint probability
table. So let me perform a few
of those miracles, while we're at it. There's the table. And now, what I want to do
is I want to count up the probability in all the rows
where the statue appears. So that's going to be
the probability of the statue appearing. So I'll just check off those
four boxes there. And it looks like the
probability of the statue appearing is about 0.355
in my model. I don't think it's quite that
frequent, but this is a classroom exercise, right? So I can make up whatever
numbers I want. Now, I could say, well, what's
the probability of a statue occurring given that there's
an art show? Well, I can limit my tallies to
those in which art show is true, like so. And in that case,
the probability has just zoomed up. So if I know there's an art
show, there's a much higher probability that a statue
will appear. And if I know there's a hack as
well as an art show going on, it goes up higher
still to 0.9. We can also do other
kinds of things. For example, we can go back
to the original table. And instead of counting up the
probability we've got a statue, as we just did, we're
going to calculate the probability that there
is an art show. I guess that would be that one
and that one, not that one, but that one. So the probability there's an
art show is one chance in 10. Or we can do the same
thing with a hack. In that case, we get that one
off, that one on, that one off, that one on, that
one off, that one on, that one off. So the probability of a hack
on any given time period is about 50-50. So I've cooked up this little
demo so it does the "ands" of all these things. It could do "ors," too, with
a little more work. But these are just the "ands" of
these various combinations. Then you can ask more
complicated questions, like for example, you could say, what
is the probability of a hack given that there's
a statue? And that would be limiting the
calculations to those rows in which the statue
thing is true. And then what I get is 0.781. Now, what would happen to the
probability that it's a hack if I know that there's
an art show? Will that number
go up or down? Well, let's try it. Ah, it went down. So that's sort of because the
existence of the art show sort of explains why the statue
might be there. Now, just for fun, I'm going
to switch to another situation, very similar. And the situation here is that
a neighbor's dog often barks. It might be because
of a burglar. It might be because
of a raccoon. Sometimes, there's a burglar
and a raccoon. Sometimes, the damn
dog just barks. So let's do some calculations
there and calculate the probability that a raccoon
is true, similar to what we did last time. Looks like on any
given night-- it's kind of a wooded are--
there's a high probability of a raccoon showing up. And then we can ask, well, what
is the probability of the dog barking given that
a raccoon shows up? Well, in that case, we want to
just limit the number of rows to those where a raccoon-- or where the dog is barking. Looks like the probability of
the dog barking, knowing nothing else, is about
[? 3/7. ?] But now we want to know the
probability of the raccoon-- that's these guys here
need to get checked. These are off. So that's the probability
of a raccoon. Did I get that right? Oh, that's probability
of a burglar. Sorry, that was too hard. So let me go back
and calculate-- I want to get the probability
of a raccoon. That's true, false, true, false,
true, false, true. So the probability of
a raccoon, as I said before is 0.5. Now, what happens to that
probability if I know the dog is barking? Well, all I need to do is limit
my rows to those where the dog is barking,
those bottom four. And I'll click that there, and
you'll notice all these tallies up above the midpoint
have gone to zero, because we're only considering
those cases where the dog is barking. In that case, the probability
that there's a raccoon-- just the number of
tallies over the total number of tallies-- gee, I guess it's 225 plus
50 divided by 370. That turns out to be 0.743. So about 75% of the time, the
dog barking is accounted for-- well, the probability of a
raccoon under those conditions is pretty high. And now, once again, I'm going
to ask, well, what is the probability of a raccoon, given
that the dog is barking and there's a burglar? Any guess what will
happen there? We did this once before
with the statue. Probability first went up when
we saw the statue and then went down when we saw
another explanation. Here's this one here. Wow, look at that. It went back to its original
condition, its a priori probability. So somehow, the existence of
the burglar and the dog barking means that the
probability of a raccoon is just what it was before
we started this game. So those are kind of interesting
questions, and there's a lot we can do when we
have this table by way of those kinds of calculations. And in fact, the whole miracle
of probabilistic inference is right in front of us. It's the table. So why don't we go home? Well, because there's a little
problem with this table-- with these two tables that
I've shown you by way of illustration. And the problem is that there
are a lot of rows. And I had a hard time making
up those numbers. I didn't have the patience to
wait and make observations. That would take too long. So I had to kind of
make some guesses. And I could kind of manage
it with eight rows-- those up there. I could put in some tallies. It wasn't that big of a deal. So I got myself all those
eight numbers up there like that. And similarly, for the art show
calculations, produced eight numbers. But what if I added something
else to the mix? What if I added the
day of the week or what I had for breakfast? Each of those things would
double the number of rows of their binary variables. So if I have to consider 10
influences all working together, then I'd have
2 to the 10th. I'd have 1,000 numbers
to deal with. And that would be hard. But if I had a joint probability
table, then I can do these kinds of miracles. But Dave, if I could have this
little projector now, please. I just want to emphasize that
although we're talking about probabilistic inference, and
it's a very powerful tool, it's not the only tool
we need in our bag. Trouble with most ideas in
artificial intelligence is that their hardcore proponents
think that they're the only thing to do. And probabilistic inference
has a role to play in developing a theory of
human intelligence. And it certainly has a practical
value, but it's not the only thing. And to illustrate that point,
I'd like to imagine for a few moments that MIT were founded in
1861 BC instead of 1861 AD. And if that were so, then it
might be the case that there would be a research program
on what floats. And this, of course, would be
a problem in experimental physics, and we could imagine
that those people back there in that early MIT would, being
experimentally minded, try some things. Oh, I didn't know that's
what happened. It looks like chalk floats. Here's a rock. No, it didn't float. Here's some money. Doesn't float. Here's a pencil. No, it doesn't float. Here's a pen. Here's a piece of tin foil
I got from Kendra. That floats. That's a metal. The other stuff's metal, too. Now I'm really getting
confused. Here's a little wad of paper. Here's a cell ph--
no, actually, I've tried that before. They don't float. And they also don't work
afterward, either. I don't need to do any of that
in the MIT of 1861 AD and beyond, because I know
that Archimedes worked this all out. And all I have to do is measure
the volume of the stuff, divide that by the
weight, and if that ratio is big enough, then the
thing will float. But back in the old days, I
would have to try a lot of stuff and make a big table,
taking into account such factors as how hard it is, how
big it is, how heavy it is, whether it's alive or not. Most things that are
alive float. Some don't. Fish don't, for instance. So it would be foolhardy
to do that. That's sort of a probabilistic
inference. On the other hand, there are
lots of things where I don't know all the stuff I need to
know in order to make the calculation. I know all the stuff I need to
know in order to decide if something floats, but not all
the stuff I need to know in order, for example, to decide
if the child of a Republican is likely to be a Republican. There are a lot of subtle
influences there, and it is the case that the children of
Republicans and the children of Democrats are more likely to
share the political party of their parents. But I don't have any direct
way of calculating whether that will be true or not. All I can do in that case is
what I've done over here, is do some measurements, get some
frequencies, take some snapshots of the way the world
is and incorporate that into a set of probabilities that can
help me determine if any given parent is a Republican, given
that I've observed the voting behavior their children. So probability has a place,
but it's not the only tool we need. And that is an important
preamble to all the stuff we're going to do today. Now, we're really through,
because this joint probability table is all that there is to
it, except for the fact we can't either record all those
numbers, and it becomes quickly a pain to
guess at them. There are two ways to think
about all this. We can think about these
probabilities as probabilities that come out of looking
at some data. That's a frequentist view
of the probabilities. Or we could say, well, we can't
do those measurements. So I can just make them up. That's sort of the subjective
view of where these probabilities come from. And in some cases, some people
like to talk about natural propensities, like in
quantum mechanics. But for our purposes, we either
make them up, or we do some tallying. Trouble is, we can't deal
with this kind of table. So as a consequence of not being
able to deal with this kind of table, a gigantic
industry has emerged for dealing with probabilities
without the need to work up this full table. And that's where we're
going to go for the rest of the hour. And here's the path we're
going to take. We're going to talk about some
basic overview of basic probability. Then we're going to move
ourselves step by step toward the so-called belief networks,
which make it possible to make this a practical tool. So let us begin. The first thing is basic
probability. Let us say basic. And basic probability-- all probability flows from
a small number of axioms. We have the probability of some
event a has got to be greater than 0 and
less than 1. That's axiom number one. In a binary world, things have
a probability of being true. Some have a probability
of being false. But the true event doesn't have
any possibility of being anything other than true, so
the probability of true is equal to 1, and the probability
of false-- the false event, the
false condition-- has no possibility of being
true, so that's 0. Then the third of the axioms
of probability is that the probability of a plus the
probability of b minus the probability of a and
b is equal to the probability of a or b. Yeah, that makes sense, right? I guess it would make more sense
if I didn't switch my notation in midstream-- a and b. So those are the axioms that
mathematicians love to start up that way, and they can derive
everything there is to derive from that. But I never can deal with
stuff that way. I have to draw a picture and
think of this stuff in a more intuitionist type of way. So that's the formal approach
to dealing with probability, and it's mirrored by intuitions
that have to do with discussions of spaces,
like so, in which we have circles, or areas, representing
a and b. And to keep my notation
consistent, I'll make those lowercase. So you can think of those as
spaces of all possible worlds in which these things
might occur. Or you can think of them
as sample spaces. But in any event, you associate
with the probability of a the size of this area here
relative to the total area in the rectangle-- the universe. So the probability of a is the
size of this circle divided by the size of this rectangle
in this picture. So now all these axioms
make sense. The probability that a is
certain is just when that fills up the whole thing, and
there's no other place for a sample to be, that means
it has to be a. So that probability goes
all the way up to 1. On the other hand, if the size
of a is just an infinitesimal dot, then the chances of landing
in that world is 0. That's the bound on
the other end. So this-- axiom number one-- makes
sense in terms of that picture over there. Likewise, axiom number two. What about axiom number three? Does that make sense in terms
of all this stuff? And the answer is, sure, because
we can just look at those areas with a little
bit of colored chalk. And so the probability of a
is just this area here. The probability of b
is this area here. And if we want to know the
probability that we're in either a or b, then we just have
to add up those areas. But when we add up those areas,
this intersection part is added in twice. So we've got to subtract that
off in order to make this thing make a rational equation,
so that makes sense. And axiom three makes
sense, just as axioms one and two did. So that's all there is
to basic probability. And now you could do all sorts
of algebra on that, and it's elegant, because it's like
circuit theory or electromagnetism, because
from a very small number of axioms-- in this case three-- you can build an elegant
mathematical system. And that's what probability
subjects do. But we're not going to go there,
because we're sort of focused on getting down to a
point where we can deal with that joint probability
table that we currently can't deal with. So we're not going to go into
a whole lot of algebra with these things. Just what we need in order to
go through that network. So the next thing we need to
deal with is conditional probability. And whereas those are axioms,
this is a definition. We say that the probability of
a given b is equal to, by definition, the probability
of a and b. I'm using that common notation
to mean [INAUDIBLE] as is conventional
in the field. And then we're going to divide
that by the probability of B. You can take that as a
definition, and then it's just a little bit of mysterious
algebra. Or you could do like we did up
there and take an intuitionist approach and ask what that
stuff means in terms of a circle diagram and some
sort of space. And let's see, what
does that mean? It means that we're trying to
restrict the probability of a to those circumstances where
b is known to be so. And we're going to say that-- we've got this part here,
and then we've got the intersection of a with b. And so it does make sense as a
definition, because it says that if you've got b, then the
probability that you're going to get a is the size of
that intersection-- the pink and orange stuff-- divided by the whole of b. So it's as if we restricted the
universe of consideration to just that part of the
original universe as covered by b. So that makes sense
as a definition. And we can rewrite that, of
course, as P of a and b is equal to the probability
of a given b times the probability of b. That's all basic stuff. Now, we do want to do a little
bit of algebra here, because I want to consider not just two
cases, but what if we divide this space up into
three parts? Then we'll say that the
probability of a, b, and c is equal to what? Well, there are lots of ways
to think about that. But one way to think about it is
that we are restricting the universe to that part
of the world where b and c are both true. So let's say that y is
equal to b and c-- the intersection of b and c,
where a and b are both true. Then we can use this formula
over here to say that probability of a, b, and c is
equal to the probability of a and y, which is equal to the
probability of a given y times the probability of y. And then we can expand that back
out and say that P of a given b and c is equal
to the probability-- sorry, times the probability
of y, but y is equal to the probability of b
and c, like so. Ah, but wait-- we can run this idea over that
one, too, and we can say that this whole works is equal to the
probability of a given b and c times the probability
of b given c times the probability of c. And now, when we stand back and
let that sing to us, we can see that some magic is
beginning to happen here, because we've taken this
probability of all things being so, and we've broken up
into a product of three probabilities. The first two are conditional
probabilities, so they're really all conditional
probabilities. The last one's conditional
on nothing. But look what happens as we
go from left to right. a is dependent on two things. b is only dependent on one thing
and nothing to the left. c is dependent on nothing
and nothing to the left. So you can sense a
generalization coming. So let's write it down. So let's go from here over
to here and say that the probability of a whole
bunch of things-- x1 through x10-- is equal to some product
of probabilities. We'll let the index
i run from n to 1. Probability of x to the last one
in the series, conditioned on all the other ones-- sorry, that's probability
of i, i minus 1 down to x1 like so. And for the first one in this
product, i will be equal to n. For the second one, i will
be equal to n minus 1. But you'll notice that as I
go from n toward 1, these conditionals get smaller-- the number of things on
condition get smaller, and none of these things
are on the left. They're only stuff that
I have on the right. So what I mean to say is all of
these things have an index that's smaller than
this index. None of the ones that have a
higher index are appearing in that conditional. So it's a way of taking a
probability of the end of a whole bunch of things and
writing it as a product of conditional probabilities. So we're making good progress. We've done one. We've done two. And now we've done three,
because this is the chain rule. And we're about halfway through
our diagram, halfway to the point where we can
do something fun. But we still have a couple more
concepts to deal with, and the next concept is the
concept of conditional probability. So that's all this
stuff up here-- oops. All this stuff here is the
definition of conditional probability. And now I want to go to the
definition of independence. So that's another definitional
deal. But it's another definitional
deal that makes some sense with a diagram as well. So the definition
goes like this. We say that P of a given b
is equal to P of a if a independent of b. So that says that the
probability of a doesn't depend on what's going
on with b. It's the same either way. So it's independent. b doesn't matter. So what does that look like
if we try to do an intuitionist diagram? Well, let's see. Here's a. Here's b. Now, the probability
of a given b-- well, let's see. That must be this part here
divided by this part here. So the ratio of those areas is
the probability of a given b. So that's the probability of
this way divided by the probability of both ways. So what's the probability of
a in terms of these areas? Well, probability of a in terms
of these areas is the probability-- let's see, have I
got this right? I've got this upside down. The probability of a given b
is the probability of the stuff in the intersection-- so that's both ways-- divided by the probability
of the stuff in b, which is going this way. And let's see, the probability
of a not conditioned on anything except being in this
universe is all these hash marks, like so, divided
by the universe. So when we say that something's
independent, it means that those two ratios
are the same. That's all it means in the
intuitionist's point of view. So it says that this little
area here divided by this whole area is the same as this
whole area for a divided by the size of the universe. So that's what independence
means. Now, that's quite
a lot of work. But we're not done with
independence, because we've got conditional independence
to deal with. And that, too, can be viewed
as a definition. And what we're going to say is
that the probability of a given b and z is equal to the
probability of a given z. What's that mean? That means that if you know
that we're dealing with z, then the probability of
a doesn't depend on b. b doesn't matter anymore
once you're restricted to being in z. So you can look at
that this way. Here's a, and here's
b, and here is z. So what we're saying is that
we're restricting the world to being in this part of the
universe where z is. So the probability of a given b
and z is this piece in here. a given b and z is
that part there. And the probability of a given
z is this part here divided by all of z. So we're saying that the ratio
of this little piece here to this part, which I'll mark that
way, ratio of this to this is the same as the
ratio of that to that. So that's conditional
independence. So you can infer from these
things, with a little bit of algebra, that P of a and b given
z is equal to P of a given z times P of b in z. Boy, that's been quite a
journey, but we got all the way through one, two, three,
four, and five. And now the next thing is belief
nets, and I'm going to ask you to forget everything
I've said for a minute or two. And we'll come back to it. I want to talk about the dog
and the burglar and the raccoon again. And now, forgetting about
probability, I can say, look, the dog barks if a
raccoon shows up. The dog barks if a
burglar shows up. A burglar doesn't show up
because the dog is barking. A raccoon doesn't show up
because the dog is barking. So the causality flows from the
burglar and the raccoon to the barking. So we can make a diagram
of that. And our diagram will
look like this. Here is the burglar, and
here is the raccoon. And these have causal relations
to the dog barking. So that's an interesting idea,
because now I can say that-- well, I can't say anything yet,
because I want to add a little more complexity to it. I'm going to add two
more variables. You might call the police,
depending on how vigorous the dog is barking, I guess. And the raccoon has a propensity
to knocking over the trash can. So now, I've got
five variables. How big a joint probability
table am I going to need to keep my tallies straight? Well, it'll be 2 to the 5th. That's 32. But what I'm going to say is
that this diagram is a statement, that every node in it
depends on its parents and nothing else that's
not a descendant. Now, I need to say that about
50 times, because you've got to say it right. Every node there is independent
of every non-descendant other
then its parents. No, that's not quite right. Given its parents, every node
is independent of all other non-descendants. Well, what does that mean? Here's the deal with
calling the police. Here's its one and
only parent. So given this parent, the
probability that they were going to call the police doesn't
depend on anything like B, R, or T. It's because
all of the causality is flowing through this
dog barking. I'm not going to call the
police in a way that's dependent on anything else other
than whether the dog is barking or not. Because this guy has this as
a parent, and these are not descendants of calling the
police, so this is independent of B, R, and T. So let's go walk through
the others. Here's the dog. The dog's parents are burger appearing and raccoon appearing. So the probability that the dog
appears is independent of that trash can over there,
because that's not a descendant. It is dependent on
these parents. How about the trash can? It depends only on
the raccoon. It doesn't depend on any other
non-descendant, so therefore, it doesn't depend on D,
B, or P. How about B? It has no parents. So it depends on nothing else,
because everything else is either a non-descendant, because
B does not dependent on R and T, because they're
not descendants. It's interesting that B might
depend on D and P, because those are descendants. So it's important to understand
that there's this business of independence given
the parents of all other non-descendants. And you'll see why that funny,
strange language is important in a minute. But now, let's see-- I want to make a model of what's
going to happen here. So let me see what kind of
probabilities I'm going to have to figure out. This guy doesn't depend
on anything upstream. So we could just say that
all we need there is the probability that a burglar
is going to appear. Let's say it's a fairly
high-crime neighborhood-- 1 chance in 10-- 1 day in 10, a burglar
appears. The raccoon doesn't depend on
anything other than its own propensity, so its probability, we'll say, is 0.5. Raccoons love the place, so it
shows up about 1 day in 2. So what about the dog barking? That depends on whether there's
a burglar, and the other parent is whether
there's a raccoon. So we need to keep track of the
probability that the dog will bark for all four
combinations. So this will be the burglar, and
this will be the raccoon. This will be false, false,
true, true-- oops-- false, false,
true, false, false, true, true, true. So let's say it's a wonderful
dog, and it always barks if there's a burglar. So that would say that the
probability here is 1.0, and the probability here is 1.0. And if there's neither a burglar
nor a raccoon, the dog still likes to bark
just for fun. So we'll say that's a
chance of 1 in 10. And then in case there's a
burglar, let's say this. There's no burglar, but
there is a raccoon-- he's tired of the raccoons, so
he only barks half the time. Do these numbers, by the way,
have to add up to 1? They clearly don't. These numbers don't
add up to one. What adds up to 1 is this
is the probability that the dog barks. And then the other phantom
probability is out here. And these have to add up to 1. So that would be 0.9, that would
be 0.0, that would be 0.5, and this would be 0.0. So because those are just 1
minus the numbers in these columns, I don't bother
to write them down. Well, we still have a couple
more things to do. The probability that we'll call
the police depends only on the dog. So we'll have a column for the
dog, and then we'll have a probability of calling
the police. There's a probability for that
being false and a probability for that being true. So if the dog doesn't bark,
there's really hardly any chance we'll call the police. So make that 0, 0, 1. If the dog is barking, if he
barks vigorously enough, maybe 1 chance in 10. Here, we have the trash can--
the final thing we have to think about. There's the trash can;
rather, the raccoon. And here's the trash
can probability. Depends on the raccoon being
either present or not present. If the raccoon is not present,
the probability the trash can is knocked over by, say,
the wind is 1 in 1,000. If the raccoon is there, oh man,
that guy always likes to go in there, so that's 0.8. So now I'm done specifying
this model. And the question is, how many
numbers did I have to specify? Well, let's see. I have to specify that one, that
one, that one, that one, that one, that one-- that's
6, 7, 8, 9, 10. So I had to specify
10 numbers. If I just try to build myself
a joint probability table straightaway, how many numbers
would I have to supply? Well, it's 2 to the n. So it's 2 to the
5th, that's 32. Considerable saving. By the way, how do you suppose
I made that table? Not by doing all
those numbers. By making this belief network
and then using the belief network to calculate
those numbers. And that's why this is a
miracle, because with these numbers, I can calculate those
numbers instead of making them up or making a whole lot of
tally-type measurements. So I'd like to make sure
that that's true. And I can use this stuff here
to calculate the full joint probability table. So here's how this works. I have the probability
of some combination-- let's say the police, the dog,
the burglar, the trash can, and the raccoon. All the combinations that are
possible there will give me an entry in the table-- one row. But let's see-- there's some miracle here. Oh, this chain rule. Let's use the chain rule. We can write that as a
probability that we call the police given d, b, t, and r. And then the next one in my
chain is probability of d given b, t, and r. Then the next one in the chain
is the probability of b given t and r. And the next one in my chain
is P of t given r. And the final one in
my chain is p of r. Now, we have some conditional
independence knowledge, too, don't we? We know that this probability
here depends only on d because there are no descendants. So therefore, we don't have to
think about that, and all the numbers we need here are
produced by this table. How about this one here? Probability that the dog barks
depends only on its parents, b and r, so it doesn't
depend on t. So b, in turn, depends on-- what does it depend on? It doesn't depend on anything. So we can scratch those. Probability of t given r, yeah,
there's a probability there, but we can get
that from the table. And finally, P or r. So that's why I went through
all that probability junk, because if we arrange things in
the expansion of this, from bottom to top, then we arrange
things so that none of these guys depends on a descendant
in this formula. And we have a limited number
of things that it depends on above it. So that's the way we can
calculate back the full joint probability table. And that brings us to the end
of the discussion today. But the thing we're going to
think about is, how much saving do we really
get out of this? In this particular case, we
only had to devise 10 numbers out of 32. What if we had 10 properties
or 100 properties? How much saving would
we get then? That's what we'll take
up next time, after the quiz on Wednesday.
