on whether there is a big football match, on all sorts of things that might affect

the dynamics of traffic. The point being that just like in the

previous example, we can correct inaccuracies in our assumption by

enriching the model. So, once again, we can enrich the model

by including these variables in it. And once we have that, then the, again,

the model becomes a much better reflection of reality.

So now, how do we represent this probabilistic model in in the context of

a graphical model like we had before? So, let's now assume that our stayed

description is composed of a set of random variables.

And so, we're interested, we have we have a little baby traffic system where we

have the weather at the current time point, the location of, say, a vehicle,

the velocity of the vehicle. We also have a sensor, who's observation

we get at each of those time points. And the sensor may or may not be failing

at the current time point. And what we've done here is we've encoded

the the probabilistic model of this next state.

So, W prime, V prime, L prime, F f prime, and O prime, given the previous states.

So, given W, V, L, and F. Why is O not here on the right-hand side?

It's not here on the right-hand side because it doesn't affect any of the next

state variables. So, it would be kind of hanging down here

if we included it. But that doesn't, it doesn't affect

anything, we don't choose to to represent it.

So, this model represents a conditional distribution.

Now, we have a little network fragment. So, this is a network fragment.

And it doesn't represent a joint distribution,

it represents a conditional distribution. The conditional distribution of the t + 1

given t. But what, but in order to represent that,

we still use the same tools that we have in the context of variance, of graphical

models. And so, we can write that as the same

kind of chain rule that we used before. So, this would be the probability of W

prime, given W, based on this edge over here,

times the probability of V prime, the velocity.

So, this, this says that the weather, the first one says, that the weather at time

t plus one depends on the weather at time t.

The second one that the velocity of time t plus one depends on the weather at time

t and the velocity at time t which indicates a certain persistence in the

velocity as well as the fact that, you know, if there, if it's raining you might

slip sideways so the velocity might change.

Also if you're careful, you might slow down if it's raining.

And so again, there might be an effect of the weather on the velocity.

The probability of the location at time t + one, given the location at time t and

the velocity time t. The probability of a sensor failure at

time t1. + 1, given the failure, and at, at the

previous time and the weather. Which indicates that, once the sensor has

failed, it's probably more likely to stay failed.

But maybe rain can make the sensor behave badly.

And then, finally, the probability of the observation of time t + 1 given the

location of time t1. + 1, and the failure of time t1.

+ 1. So, there's several important things to

note about this diagram that are worth highlighting.

First of all, we have dependencies both within and across time.

So here, we have a dependency that goes from t to t plus one.

And here, we have a dependency that is within t T plus one alone.

What's, what induces us to make a modeling chose like this go one way

versus the other? The assumption here that this is a fairly

wide [UNKNOWN] dependency so that a, the observation is relatively instantaneous

compared to our time granularity. And so, we, we don't want that to go

across time but rather we want it to be within a time slice because it's a better

reflection for which variable is it that actually influences the observation.

Is it the current location or the previous location?

So these kinds of edges, let's just give the names.

These are called intra-time slice edges and these are called inter or between

time slice. And the model can include a combination

of both of these. Another kind of

anothwe, a particular type of inter-time slice edge that's worth highlighting

specifically are edges that go from a variable at one time point to the value

of that variable at the next time point. These are often called persistence edges

because they indicate the the tendency of a variable to persist in

state from one time point to another. Finally, let's just go back and look at

the parameterization that we have in this model. So, what CPDs did we actually need

to include in this model? And we can see that we have CPDs for the

variables on the right-hand side, the prime variables.

But there's no CPDs for the variables that are unprimed,

the variables on the left. And this is because the model doesn't

actually try and represent the distribution, O over W, V, L, and F.

It doesn't try and do that. It tries to represent the probability of

the next time slice, given the previous one.

So, as we can see, this graphical model only has CPD's for a subset of the

variables in it. The ones that represent the next time

point. So, that represents the transition

dynamics. If we want to represent the probability

distribution over an entire system, we also need to provide a distribution over

the initial state. And this is just the standard generic

Bayesian network which represent the probability over the state at times zero

using some appropriate chain rule. So, nothing very fancy here.