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So there are many classes of models that,
that allow us to represent in a single

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concise representation, a template over,
rich models that incorporate multiple

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copies of the same variable. And also
allows to represent multiple models

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within, as a byproduct of the single
representation. But, one of the most

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commonly used among those, is for
reasoning about temporal models, where we

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have a system that evolves over time. So.
So let's look at what we have, [inaudible]

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represent a distribution over temporal
trajectories. So the first thing we wanna

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do when representing a distribution over
continuous time, is, in most cases, not

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always, is to try and forget that time's
actually continuous. Because continuous

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quantities are harder to deal with. So
we're going to discretize time. And

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specifically. Eyes closed. And
specifically, we're going to do that by

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picking a particular time granularity
delta. Which is the time granularity; it's

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which we're going to measure time. Now, in
many cases, this is something that we

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already, is given to us by the granularity
of our sensor. So, in many cases, for

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example, when we have, a video or a robot,
there is a certain time granularity at

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which we obtain measurements. And so
that's usually the granularity that we'll

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pick. But in other cases, we might want to
have a different, a different granularity.

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So there is a choice here. So here's our
time granularity, for example. And now we

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have a set of template random variables, X
of P. And X of P denotes the value of a

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particular variable X. A variable X being
a template variable. X of P being a

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particular instantiation of that variable
at time point P delta. So that we have

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multiple copies, one for each time point.
Now here's some notation that we're going

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to end up using later on. So [inaudible]
introduce it. Where X of T denotes the

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variable X at time T. Xt: T prime denotes
the set. [inaudible] between P and P

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prime. So a discrete in this case because
we've discredited time so a finite set of

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random variables that spans these two time
points inclusive. Now our goal is that we

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would like to be able to have a, a concise
representation that allows us to represent

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this probability distribution over the
traject, over a trajectory of the system

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of any duration. So we want to start at a
particular time point, usually this is

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going to be zero, and then how, what is
the probability distribution over

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trajectories of arbitrary length. Well how
do we represent what is, a, first of all,

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an infinite family of probability
distributions because you could look at

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trajectories of, of duration two, five,
ten, a million, select an infinite family

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of distributions. And each of these is a
distribution over of an unbounded number

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of random variables, because if you have a
distribution over a trajectory length a

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million, you have to represent a million
dimensional distribution. So how do we.

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Compactify that, how do we make that a
much more concise representation? So,

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there's different pieces to this. The
first of those is what's typically called

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the Markov assumption. And the Markov
assumption, goes, is effectively a type of

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conditional independence assumption. It's
the same building block that we use to

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[inaudible] general purpose graphical
models we're going to use here in the

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context of time course data. So here,
we're saying that the probability of the

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variable X. Sorry the set of variable
expanding the time from zero all the way

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to capital T so I haven't made any
assumptions yet in the statement so I am

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just writing this down, we expressing it
in term of the chain rule for probability,

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there's no chain rule for daisy and
network yet here, this is just a chain

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rule for probabilities and the chain rule
for probabilities in this context

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basically says that, that probability is
equal to the probability of X, at time

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zero. Times the probability of each
consecutive time points, t + one, given.

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So, this is the state at t + one. Given
the state. At all previous claim points,

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zero up to T. So this is not in any way an
assumption, this is just a way of

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re-expressing this probability
distribution in the way that time flows

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forward. But it's not an assumption. You
can represent any probability distribution

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over these random variables in this way.
But now we're going to add this

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assumption. And this is an assumption.
This is an independence assumption. And

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this independence assumption tells me that
X of P+1, that is the state of time P+1,

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the next step. So this is the next step.
Is independent of the past. Given the

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present. So this is a forgetting
assumption. Once you know the current

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states. You don't care anymore about your
past. [sound] If you do that, we can now

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go back to this chain rule over here, and
simplify it. Because whereas, before, we

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conditioned on X up to zero, from time
zero to time T. Now everything up to T-1

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is conditionally independent. Given, so
all this is conditionally independent of

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P+1, given X of T. Which means that I've
allowed myself to move X of P here as the

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only thing that I'm conditioned on,
conditioning on, in order to determine the

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probability of distribution of X, P+1. So
to what extent is this assumption

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warranted? So is this true? And let's take
as an example; x equals the location, or

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pos. Of a robot, or an object that's
moving. Is it the case that the location

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of the robot at T+1, so L of T+1+1, is
independent of, say, L of P-1, just to

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simplify our lives, given L of P. Is this
a reasonable assumption? Well, in most

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cases, probably not. And the reason is
that it completely ignores the issue of

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velocity. Which direction are you moving
and how fast? And so this is a classical

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example of where the Markov assumption,
for this particular model, is probably too

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strong of an assumption. So what do we do
fix it? The one, one way to fix it is to

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enrich the state description. So estimate
the mark-up assumption to better

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approximation. Just like any independent
estimation the mark-up assumption is

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always going to be an approximation. But
the question is, how good of an

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approximation? And if we add for example
ZUT. Which is the velocity at time t and

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maybe the [inaudible] at time t, maybe the
robot's intent, where its goal is, I mean,

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all sorts of additional stuff into the
state, then at that point. The Markov

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assumption becomes much more warranted.
Okay? And so that's one way of proving the

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Markov assumption true. An alternative
strategy which we are not going to talk

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about right now is to move away from the
Markov assumption by adding dependencies.

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They go further back in time. Back in
time. And that's called a semi mark up

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model. And we're not going to talk about
that right now. The second big assumption,

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that we are going to have to make, in
order to simplify the model, is intended

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to deal with the question of, well fine,
so we've reduced the model to encoding A

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probability of X of P+1, given X of P. But
it's still an unbounded number of

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conditional probabilities. Now at least
each of them is compact, but there is

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still a probabilistic models for every P.
And this is where we're going to end up

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with a, with a template based model. We're
going to stipulate that there is a

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probabilistic model, P of X prime, given
X. X prime denotes the next time point,

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and X denotes the current time point. And
we're going to see that, that model is

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replicated. For every single time point.
That is, when you're moving from time zero

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to time one, you use the small one. You
move from time one to time two, you also

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use the small. And, and that assumption,
for obvious reasons, is called time

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invariance, because it assumes that the
dynamics of the system, not the actual

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position of the robot, but rather the
dynamics that move it from state to state

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or the dynamics of the system, don't
depend on the current time point t. And

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once again, this is an assumption and it's
an assumption that's warranted in certain

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cases and not in others. So, let's imagine
that this represents now the traffic. On

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some road. Well, does that traffic. Do the
dynamics of that traffic depend on; on say

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the current time point of the system? On
most roads, the answer is probably yes. It

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might depend on the time of day. On the
day of the week. And on whether there is a

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big football match and all sorts of thing
that might affect the dynamics of traffic.

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The point being that just like in the
previous example we can correct

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inaccuracies in our assumption by
enriching the model. So once again we can

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enrich the model by including. These
variables in it and once we have that then

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again the model becomes a much better
reflection of reality. So now, how do we

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represent this probabilistic model in, in
the context of a graphical model like we

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had before? So let's now assume that our
[inaudible] is composed of a set of random

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variables. And so we're [inaudible], so we
have we have a little baby traffic system,

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where we have the weather at the current
time point. The location of, say, a

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vehicle, the velocity of the vehicle. We
also have the sensor, whose observation we

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

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

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probabilistic model of, this, next state.
So W prime, V prime, L prime, F prime, and

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O prime, Given, the previous state. So
given W V L F. Why is O not here on the

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right hand side? It's not here on the
right hand side, because it doesn't affect

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any of the next state variables. So it
would be kinda hanging down here, if we

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included it. But since it doesn't affect
anything, we don't choose to, to represent

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it. So this model represents a conditional
distribution. Now we have a little network

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fragment, so this is a network fragment.
And it doesn't represent a joint

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distribution. It represents additional
distribution. The condition distribution

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of the C plus one, gives them C. But, wa,
but. In order to represent that we still

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use the same tools that we had in the
context of vary, of graphical models. And

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so we can write that as the same kind of
chain rule that we used before. So this

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would be the probability of W prime, given
W, based on the schedule over here. The

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probability of the prime, the velocity, so
this says that the weather, the first one

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says that the weather [inaudible] depends
on the weather time T. The second one that

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the velocity of time T plus one depends on
the weather time T and the velocity of

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time T which indicates a certain
persistence in the velocity as well as the

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fact that if it's raining you might slip
sideways so the velocity might change.

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Also if you're careful you might
[inaudible] down if it's raining and so

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again there might be an effect of the
weather and the velocity. The probability

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of the location at time t plus one given
the location at time t and the velocity at

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time t. The probability of the sensor
failure time t plus one give them a

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failure and the previous time and the
weather. Which indicates that once the

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sensor is failed it?s probably more likely
to stay failed but if many rain can make

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the sensor behave badly. And then finally,
the probability of the observation of time

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t plus one gives them the location of time
t plus one and the failure time t plus

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one. So there are several important things
to note about this diagram that are worth

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

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So here we have a dependency that goes
from T to T+1. And here we have a

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dependency that is within T plus one
alone. What, what induces us to make a d-,

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a modeling choice like this go one way
versus the other, the assumption here is

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that this is a fairly [inaudible]. The
[inaudible] dependency. So that a, the

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observation is relatively instantaneous
compared to our time granularity. And so

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we, we don't want that to go across time,
but rather, we want it to be within a time

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slice, because it's a better reflection
for, which variable is it that actually

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influences the observation? Is it the
current location or the previous location?

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So, these kinds of edges, let's just give
them names. These are called intra time

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slice. Edges. That means they're called,
inter, or between times five. And the

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model can include a combination of both of
these. Another kind of, anoth-, particular

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type of inter-time slice edge that's worth
highlighting specifically, are edges that

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go from a variable at one time point to
the value of that variable at the next

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time point. These are often called
persistence edges, because they indicate

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the, the tendency of a variable to persist
in state from one time point to another.

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Finally, let's just go back and look at
the parameterization that we have in. This

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model. So what CPDs did we actually
need to include in this model? So we can

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see that we have CPDs for the
variables of the right hand side, the

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prime variables. But there is no CPDs for
the variables, that are unprimed, the

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

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represent the distribution o-, over W, V,
L and F. It doesn't try and do that. It

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tries to represent the probability of the
next time slice, given the previous one.

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

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variables in it. The ones that represent
the next time point. So that represents

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the transition dynamics if we want to
represent the probability distribution over

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the entire system we also need to provide
a distribution over the initial state and

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this is just the standard generic Bayesian
network which represents the probability

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over the state at time zero using some
appropriate chain rule. So, nothing very

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fancy here. With those two pieces we can
now represent probability distributions

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over arbitrarily long trajectories. So we
represent this by taking for time slice zero

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and copying the time zero bayesian network.
Which represents the probability

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distribution over the time zero variables,
and now we have a bunch of copies, that

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represent, the probability distribution at
time one given time zero. And here we

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have another copy of exactly the same set
of parameters that represents time two

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00:17:37,493 --> 00:17:43,325
given time one. And if we continue copying
this indefinitely, and each copy gives us

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the probability distributions of the next
time slice given the one that we just had.

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And so we can construct arbitrary long
bayesian networks. So to make this definition

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slightly more formal we define the notion
of a two time slide Bayesian network also

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known as a 2TBN. And the 2TBN over a
set of template variables X1 up to XN, is

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specifying as a Bayesian network fragment
along exactly the same lines that we used

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in the example. The nodes have two copies
the next time state variables X1 prime up to XN prime

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prime up to XN prime, and some subset of
X1 up to XN, which are the variables, the

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00:18:28,272 --> 00:18:44,386
time T variables. That affect directly.
The state at T plus one. Okay. And,

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because we want this to represent a
conditional probability distribution. Only

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the time T plus one nodes
have parents and a CPD. Because we don't

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00:18:54,105 --> 00:18:59,210
really want to model the distribution over
the variables of time T. And the 2TBN

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defines a conditional distribution,
using the chain rule. Using something that

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00:19:07,304 --> 00:19:12,588
looks exactly like the chain rule. So the
probability of x prime given x is the

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00:19:12,588 --> 00:19:18,073
product of each variable in time t plus
one so only the prime variables given its

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00:19:18,073 --> 00:19:23,624
parents. Which may or be, which may be in
time t plus one or time t or a combination

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00:19:23,624 --> 00:19:33,563
of both. A dynamic Bayesian network is now
basically defined by a two TBN. Which we

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just defined and a Basian network over time zero. So this is the dynamics. And this

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is the initial state. And we can use that
to define arbitrary probability

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distributions for sorry probability
distributions over arbitrarily long

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trajectories, using what's called the
unrolled network or also called the ground

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network. And this is exactly in the as in
the example that I showed, dependency

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model for time zero is copied from the
base net for time zero, and this is the

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transition is copied. From the bayes net that denotes
the conditional probability for

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transitions. So before we conclude this
lecture let's look at an example of a

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dynamic Bayesian network that is a more
realistic one than the simple examples

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that we've shown before. This is a network
that was actually designed for tracking

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vehicles in a traffic situation. And so we
can see that there are multiple variables

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here that represent both the position and
velocity. Of the vehicle. Both in an

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absolute sense. For example, this is X dot
and Y dot are the velocities. As well as

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various more semantic notions of location
like whether you?re in the lane.

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[inaudible] There are contextual
variables, such as left clear and right

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clear. The engine status for example. As
well as what the driver is currently

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doing. For example the forward action and
the lateral action. We can see that there

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are persistence edges. That denote the
persistence of. There is [inaudible] of

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the state from time t to time t plus one.
As well as a variety of this intermediate

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variables over here, that allow us to
represent the probability distribution in

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even more compact way by incorporating
variables that do not persist to at least

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a simple five model that not persist. And
finally. We see that there are a large

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number of sensor observations, such as
turn signal. Whether it be cars clear on

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the right and on the left, or appears to
be clear on the right and the left, and so

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on. So this is a much more realistic model
of how traffic evolves than the simplified

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one that we saw before. To summarize,
dynamic Bayesian networks provide us with

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a language for encoding structure
distributions over time. And by making the

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assumptions of a Markovian evolution, as
well as time and variants, we can use a

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single compact network to allow us to
encode arbitrarily long transitions, over

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arbitrarily long time sequences. [sound].
But these assumptions that we made, the

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Markovian assumption, and the time
invariance assumption, are not

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immediately, correct, and might require
that we redesign the model, so as to,

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enforce, so as to make these assumptions a
better approximation to, the true

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underlying distribution. By, for example,
adding variables as we showed. [sound]