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That exploit additional forms of local
structures some kind of pro matrix form

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are extremely valuable in context of
general, in the general context of

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graphical models because they allow us to
provide a much sparse representation. But

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they are absolutely essential when we have
in networks that involves continuous

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variables because their tables are simply
not an option. So let's look at some

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examples of networks that involve
continous variables, and see what kind of

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representations we might want to
incorporate here. So now, let's imagine

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that we have a continous temperature
variable. Say the temperature in a room.

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And we have a sensor, a thermos-, a
thermometer that mentions, that measures

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the, the temperature. Now, thermometers
aren't perfect, and so, what we would

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expect, then, is the sensor is around the
right temperature, but not quite. And so.

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One way to capture that is by saying that
the sensor S is a normal distribution. And

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here's an all distribution. Around the
true temperature, T, with some standard

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deviation, Sigma S. So this defines for
every value of T a distribution over S. In

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a very compact parametric form that has
just really the parameter of sigma S, and

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if we just say that S is a Gausian around
the variable, around the value of the

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variable T. Now let's make the situation a
little bit more interesting. This is the

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temperature now. And this is a
[inaudible]. So we have P. And P. Prime.

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Now, P. Prime now depend, the, the
temperature soon depends on the current

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temperature, as well as the outside
temperature, because of some equalization

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of temperatures from the inside to the
outside. So, what model what, we, might we

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have for T prime as the function of it's
two parents? Temperature the current

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temperature and the outside temperature.
Well, so one model might be just some kind

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of diffusion model that says that T is
equal to some weighted combination. Sorry,

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T prime is a Gaussian around,
a mean that's defined as a combination of

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the current temperature and the outside
temperature. So you kind of combine the

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two, and because there is stochasticity in
the process, we're going to say that T

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prime isn't exactly equal to this, but
rather is a Gaussian around this mean.

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With some standard deviation stigmatic. To
be distinguished from the standard

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deviation stigma x which is the center of
deviance. Let's make life even more

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interesting. Let's imagine that there is a
door in the room. The door can be opened

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or closed so it's a discreet variable,
takes two values and clearly the extent of

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the diffusion is going to depend upon
whether the door is open and we would

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expect different parameters to this
system. In the case of two values of the

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discreet variable and so if we write the
model now we're going to have that the

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temperature time, the temperature soon T
prime, is going to be a Gaussian

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who's parameters, alpha and sigma depend
on the value of a door variable. So, if b

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equals zero. We're going to have
parameters alpha zero and sigma zero T.

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And if b equals one, we have
a different set of parameters that equal

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to different diffusion process. So just to
give all these things name this model that

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we had over here was called a linear
Gaussian. We will define that more formally in the

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next slide. And this model is called the
conditional linear Gaussian. Because

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the linear Gaussian whose parameters are
conditioned on the discrete variable

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Door. So to generalize these models
to a broader setting where, where we have

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a general variable Y and Y has parents X1
up to XK the linear Gaussian model has the

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following form. It says that Y is a
Gaussian, so that's what the N stands for,

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who's mean, is a linear function, and
that's why it's called a linear

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Gaussian. Is a linear function of the
parents X<u>i. And importantly whose</u>

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variance doesn't depend at all. On the
parents So the variance is fixed. That's

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the definition of a liner Gaussian CPD,
and obviously, it's restricted. It doesn't

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capture every situation, but it's a useful
model and a useful first approximation in

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many cases. Conditional linear Gaussian
introduces into the mix, the possibility

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of one or more discrete parents. In this
case, we just drew one more simplicity,

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but you could have more than one. And this
is just a linear Gaussian whose

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parameters depend on the value of A. So,
writing it down it looks exactly like

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this. We now have, for every one of the
parameters, we have the ability for the

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parameters to depend on A. And this case,
the variants, can depend on the continue

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on the discreet parent. But not on the
continuous ones. And this is the

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conditional in your Gaussian model. And
again, similarly it is a restricted model,

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and one can certainly generalize beyond
that, as we'll show in a moment. But it's

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a very useful model that's used in a large
number of applications. One example

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application that we've seen that involves
continuous variables is the task of robot

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localization. I'm not going to show this
video again now. We're going to see it

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again when we talk about temporal models.
But, just as a reminder, what we have here

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is a robot whose location is a continuous
quantity. So are the sensor observations

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that give a noisy version of how far away
the robot is from an obstacle looking in

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each one of the different directions. Okay
and so we have both the continuous state

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variable as well as the continuous
observation that the robot needs to deal

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with. So what kind of observation model
makes sense in this study. So here lets

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imagine that this line over here
represents the true distance. From the

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robot's current location, from a given
location to obstacle. So if the robot

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is looking in, if we're conditioning on
the robots current location and we're

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asking, if I look in this diresction how
far is it before I hit an obstacle. In

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this case that distance is, 200 and, I
don't know, twenty centimeters. And what

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this tells us is that a sonar is a
Gaussian. You can see this is the sonar.

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The red is the sonar. Is a Gaussian around,
the true distance. Now, the laser, which

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is a different sensing modality for the
same robot is also a Gaussian around that

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true distance. But because the laser is a
more accurate sensor the standard

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deviation is lower for the laser than for
the sonar. And that reflects the accuracy

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of these two different sensor modalities.
Now, this is an idealized version. But

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surprisingly corresponds in useful ways to
the real model. So let's actually look at

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some of the, Let's look first at the model
that was used in the system. And then, at,

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which is the red line. And then we can
look at the blue line. The red line

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actually involves three different
components. So this is the actual center

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model. Used by the robots and we can see
that it has three components. It has this

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peak Which is the Gaussian around the
true distance. The next most obvious

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phenomenon is this ridiculous peak over
here. This corresponds to a max range

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reading. This is what you get if there
isn't an obstacle in that direction within

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a reasonable distance for the laser or the
sonar to return any signal. And so that's

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why there's a very large peak, at, at,
beyond a certain distance. That's the

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entire rest of the probability mass. The
final most more subtle aspect of this

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probability distribution is that you see
that this is higher than that. So, the,

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there's more probability mass, for the
density, before the obstacle than after

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the obstacle. And why is that? Because
once you get to the obstacle. The beam

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returns. But before you get to the
obstacle there might be some other like,

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transient things, like a person walking in
front of the obstacle. And that's going to

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return the beam in a way that doesn't
represent the actual structure of the map.

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And so that's why we have a certain
probability. Of having the ob, of having

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the, the beam return sooner than the
obstacle, and that probability doesn't

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exist on the downside once the obstacle's
been reached. So the actual probability

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distribution is an aggregation of these
three signals. The sensor model around the

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obstacle in a given direction. This
uniform distribution, before

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the obstacle would [inaudible]
distribution return and the max range

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reading at the end. The red line is the
model that was used and the blue

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line is, the actual measured distances in
different settings to sort of show whether this

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really does, the model that was used
really represents reality. And the answer

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is that, it does to a surprising extent.
So this, this is an example of how

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continuous sensor, continuous
distributions are going to be used in a

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real world application. The next example,
of that, is actually the robot motion

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model. So here are the robots. And the
robot is heading in a given direction

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alpha. Heading sorry, heading in this
direction that's what it thinks it's going

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and the question is if it moves a certain
distance it thinks it's moving a certain

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distance in a given direction. What is the
Actual distribution over it's next

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location. And the answer is it's a little
bit tricky because robots actually have

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heading and there's certain uncertainty
alpha, over which is the difference

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between where they think they are going
and where they are actually going. And

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then there is also noise. On the distant
delta, that they think they moved, and,

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so, when you put all these together, the
actual cloud, on, the distribution, on

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where the robots going to be, falling,
[inaudible] move, is this weird, banana

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shaped distribution, which is centered
around. This area where the robot thinks

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it is. But is, but has a sort of banana
shape that's, where the banana shape is

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induced by the uncertainty about the
angular trajectory. And if you actually

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run this for a while, you can see that the
banana shape gets more and more and more

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diffuse. So here is the first banana
shape, and now the robot turns, and

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there's more uncertainty over the heading.
And so, you get. A larger and larger

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banana shaped distribution, on the robots
position, assuming there's no evidence to

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correct, the position based on say sonar
or laser readings.