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 sigma s, and then we just say

that sigma s is of Gaussian 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 the temperature soon. So we have T and T prime.

Now, T prime now depends, the, the temperature soon depends on the current

temperature, as well as on the outside temperature.

Because of some equalization of temperatures from the inside to the

outside. So, what model we, might we have, for p

prime as a function of its 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 P is equal to some weighted combination,

sorry, P 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 stochasticitian noise in

the process, we're going to say P prime isn't exactly equal to this, but rather

is a Gaussian around this mean with some standard deviation, sigma T, to be

distinguished from the standard deviation sigma S, which was the sensor, variance.

Let's moke, make, let's make life even more interesting.

Let's imagine that there's a door in the room.

The door can be opened or closed, so it's a discreet variable.

It takes two values. And clearly the extent of the diffusion,

is going to depend on whether the door is open,

and we would expect different parameters to the system in the case of, the two

values of a discreet variable. And so if we write the model now, we're

going to have that the temperature time. It's the temperature soon T prime, is

going to be a Gaussian that whose parameters, alpha and sigma, depend on

the value of the door variable. So if D equals zero, we're going to have

parameters alpha zero and sigma zero T. And if D equals one, we have a different

set of parameters that reflect the different diffusion process.

So, just to give all these things names, this model that we had over here was

called a linear Gaussian. And we'll define that more thoroughly in

the next slide, and this model is called a conditional, linear Gaussian.

Because it's a 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, whose

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

Gaussian. It's a linear function of the parents x,

i, and importantly, whose variance doesn't depend at all on the parents.

The variance is fixed. That's the definition of a linear

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 for simplicity, but you can have more than one.

And this is just a linear Gaussian whose parameters depends on the value of a.