So we're going to go over and
build Bayes filter framework and
so given four things.
First is I'm going to need some prior
belief about where I'm starting okay, so
that's p of x.
Where did that come from again?
Remember that drive up to Delphi and
asking the Oracle and all that stuff.
So you forget your luggage so
you went back to Delphi.
You got another prior and it tells
you how things are starting, but
again, maybe with a large uncertainty.
So here we have our
dynamical system model.
And so it's the probability
of some new x given my belief
about the previous x plus this input.
By the way,
you see here it says u sub t minus 1?
The, the, the world is sometimes
schizophrenic about whether
you think of the input as happening
just after the last measurement,
so it was u at t minus 1 or
u happen now and
I want to estimate
what the state is now.
So, whether it's ut or
ut minus 1, it's whatever
u you're going to add onto xt minus 1 in
order to make your prediction about xt.
All right?
So, we're given these two things.
We're given a prior, an action model.
We're also given a sensor model.
And this is going to
be really important,
a little bit important this lesson,
very important next lesson.
And the sensor model is our likelihood,
okay?
And it is not,
I'll tell you what it isn't, first.
It is not, given some sensor reading,
reading, where do I think the object is?
You might think that's what it is, but
then you would be thinking incorrectly.
No, a likelihood model is
if the object were really someplace,
what's the likelihood of my measurement?
So normally what you'll see is and we're
going to see some later like a little
Gaussian blob drawn over the place
where the measurement was.
That does not mean, okay,
that given this measurement,
this is my distribution
of where things are.
No, no, no.
What it means is,
this is my likelihood, so
it's highest if x was actually at this
point, and then it falls off slowly.
All right.
That's the likelihood.
And then finally,
we're given the stream of observations.
The z, so z1 through zt minus 1,
or through zt.
And also, we know the actions, okay?
So we know the action data u,
u1 through u t minus 1, okay?
So those are the four
things that we have.
All right?
What do we want?
Well what we want is
the estimate of x at time t.
Okay?
Same thing we've always wanted
since we were little children.
And in the Bayesian world,
this is called the belief for
the posterior of this state.
Right?
And so sometimes it's even
written like this.
Bel for belief of xt.
What's my belief about xt?
Given everything that I've observed so
far, knowing all their prohibitions and
given my current measurement.
