That exploit additional forms of local structures some kind of pro matrix form are extremely valuable in context of general, in the general context of graphical models because they allow us to provide a much sparse representation. But they are absolutely essential when we have in networks that involves continuous variables because their tables are simply not an option. So let's look at some examples of networks that involve continous variables, and see what kind of representations we might want to incorporate here. So now, let's imagine that we have a continous temperature variable. Say the temperature in a room. And we have a sensor, a thermos-, a thermometer that mentions, that measures the, the temperature. Now, thermometers aren't perfect, and so, what we would expect, then, is the sensor is around the right temperature, but not quite. And so. One way to capture that is by saying that the sensor S is a normal distribution. And here's an all distribution. Around the true temperature, T, with some standard deviation, Sigma S. So this defines for every value of T a distribution over S. In a very compact parametric form that has just really the parameter of sigma S, and if we just say that S is a Gausian around the variable, around the value of the variable T. Now let's make the situation a little bit more interesting. This is the temperature now. And this is a [inaudible]. So we have P. And P. Prime. Now, P. Prime now depend, the, the temperature soon depends on the current temperature, as well as the outside temperature, because of some equalization of temperatures from the inside to the outside. So, what model what, we, might we have for T prime as the function of it's two parents? Temperature the current temperature and the outside temperature. Well, so one model might be just some kind of diffusion model that says that T is equal to some weighted combination. Sorry, T prime is a Gaussian around, a mean that's defined as a combination of the current temperature and the outside temperature. So you kind of combine the two, and because there is stochasticity in the process, we're going to say that T prime isn't exactly equal to this, but rather is a Gaussian around this mean. With some standard deviation stigmatic. To be distinguished from the standard deviation stigma x which is the center of deviance. Let's make life even more interesting. Let's imagine that there is a door in the room. The door can be opened or closed so it's a discreet variable, takes two values and clearly the extent of the diffusion is going to depend upon whether the door is open and we would expect different parameters to this system. In the case of two values of the discreet variable and so if we write the model now we're going to have that the temperature time, the temperature soon T prime, is going to be a Gaussian who's parameters, alpha and sigma depend on the value of a door variable. So, if b equals zero. We're going to have parameters alpha zero and sigma zero T. And if b equals one, we have a different set of parameters that equal to different diffusion process. So just to give all these things name this model that we had over here was called a linear Gaussian. We will define that more formally in the next slide. And this model is called the conditional linear Gaussian. Because the linear Gaussian whose parameters are conditioned on the discrete variable Door. So to generalize these models to a broader setting where, where we have a general variable Y and Y has parents X1 up to XK the linear Gaussian model has the following form. It says that Y is a Gaussian, so that's what the N stands for, who's mean, is a linear function, and that's why it's called a linear Gaussian. Is a linear function of the parents Xi. And importantly whose variance doesn't depend at all. On the parents So the variance is fixed. That's the definition of a liner Gaussian CPD, and obviously, it's restricted. It doesn't capture every situation, but it's a useful model and a useful first approximation in many cases. Conditional linear Gaussian introduces into the mix, the possibility of one or more discrete parents. In this case, we just drew one more simplicity, but you could have more than one. And this is just a linear Gaussian whose parameters depend on the value of A. So, writing it down it looks exactly like this. We now have, for every one of the parameters, we have the ability for the parameters to depend on A. And this case, the variants, can depend on the continue on the discreet parent. But not on the continuous ones. And this is the conditional in your Gaussian model. And again, similarly it is a restricted model, and one can certainly generalize beyond that, as we'll show in a moment. But it's a very useful model that's used in a large number of applications. One example application that we've seen that involves continuous variables is the task of robot localization. I'm not going to show this video again now. We're going to see it again when we talk about temporal models. But, just as a reminder, what we have here is a robot whose location is a continuous quantity. So are the sensor observations that give a noisy version of how far away the robot is from an obstacle looking in each one of the different directions. Okay and so we have both the continuous state variable as well as the continuous observation that the robot needs to deal with. So what kind of observation model makes sense in this study. So here lets imagine that this line over here represents the true distance. From the robot's current location, from a given location to obstacle. So if the robot is looking in, if we're conditioning on the robots current location and we're asking, if I look in this diresction how far is it before I hit an obstacle. In this case that distance is, 200 and, I don't know, twenty centimeters. And what this tells us is that a sonar is a Gaussian. You can see this is the sonar. The red is the sonar. Is a Gaussian around, the true distance. Now, the laser, which is a different sensing modality for the same robot is also a Gaussian around that true distance. But because the laser is a more accurate sensor the standard deviation is lower for the laser than for the sonar. And that reflects the accuracy of these two different sensor modalities. Now, this is an idealized version. But surprisingly corresponds in useful ways to the real model. So let's actually look at some of the, Let's look first at the model that was used in the system. And then, at, which is the red line. And then we can look at the blue line. The red line actually involves three different components. So this is the actual center model. Used by the robots and we can see that it has three components. It has this peak Which is the Gaussian around the true distance. The next most obvious phenomenon is this ridiculous peak over here. This corresponds to a max range reading. This is what you get if there isn't an obstacle in that direction within a reasonable distance for the laser or the sonar to return any signal. And so that's why there's a very large peak, at, at, beyond a certain distance. That's the entire rest of the probability mass. The final most more subtle aspect of this probability distribution is that you see that this is higher than that. So, the, there's more probability mass, for the density, before the obstacle than after the obstacle. And why is that? Because once you get to the obstacle. The beam returns. But before you get to the obstacle there might be some other like, transient things, like a person walking in front of the obstacle. And that's going to return the beam in a way that doesn't represent the actual structure of the map. And so that's why we have a certain probability. Of having the ob, of having the, the beam return sooner than the obstacle, and that probability doesn't exist on the downside once the obstacle's been reached. So the actual probability distribution is an aggregation of these three signals. The sensor model around the obstacle in a given direction. This uniform distribution, before the obstacle would [inaudible] distribution return and the max range reading at the end. The red line is the model that was used and the blue line is, the actual measured distances in different settings to sort of show whether this really does, the model that was used really represents reality. And the answer is that, it does to a surprising extent. So this, this is an example of how continuous sensor, continuous distributions are going to be used in a real world application. The next example, of that, is actually the robot motion model. So here are the robots. And the robot is heading in a given direction alpha. Heading sorry, heading in this direction that's what it thinks it's going and the question is if it moves a certain distance it thinks it's moving a certain distance in a given direction. What is the Actual distribution over it's next location. And the answer is it's a little bit tricky because robots actually have heading and there's certain uncertainty alpha, over which is the difference between where they think they are going and where they are actually going. And then there is also noise. On the distant delta, that they think they moved, and, so, when you put all these together, the actual cloud, on, the distribution, on where the robots going to be, falling, [inaudible] move, is this weird, banana shaped distribution, which is centered around. This area where the robot thinks it is. But is, but has a sort of banana shape that's, where the banana shape is induced by the uncertainty about the angular trajectory. And if you actually run this for a while, you can see that the banana shape gets more and more and more diffuse. So here is the first banana shape, and now the robot turns, and there's more uncertainty over the heading. And so, you get. A larger and larger banana shaped distribution, on the robots position, assuming there's no evidence to correct, the position based on say sonar or laser readings.