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.
