In this video, we're going to look at how we can use the basic concepts of probability inside an estimation
problem, so we're going to represent the current state to be estimated as a random variable vector
with a probability distribution.
The distribution main is going to be the best estimate of the current vector.
So the solution to the minimum main squared error, while the distribution covariance is the estimate
of the uncertainty around the current state.
And this is, for example, the covariance of the estimation error.
We can also represent the measurements or observations as a random variable vector as well, with a
given probability density function.
The distribution main is then the measurement value, while the distribution covariance is the estimation
of the uncertainty of the measurement.
We can also use the dynamic system to estimate changes in the state domain and the uncertainty.
So the covariance between the measurements or updates with time.
So, for example, we could try to estimate the position of a car and we can represent the estimated
position for the car.
So this is the car position here.
This is the estimated position.
We can represent the position estimate as a normal distribution.
So we have a main one.
So the current main four timestep one is this estimate and we have a given uncertainty of the measurement.
So this is our sigma squared one.
So the uncertainty at this point of time and we can represent it as a normal or Gaussian distribution.
So we know that the car moves with time so we can use the dynamic system to predict the estimate forward.
So basically using the dynamic system, we can predict the estimate forward.
So we move this position estimate through the dynamic system to get this estimate over here.
So now we have a new position estimate for time, too.
It has a new main, has a new variance, and in this case, the variance here.
So Sigma squared two is going to be greater than our original position.
Estimate Sigma squared one, because as time goes, we get less certain with the position of the vehicle.
So now you can see that the estimate of position in here, which is the true position, is here.
So there's a bit more error now because we've had to predict forward and we haven't constrained the
system with any extra information, however.
Now, let's say we get a bit of additional information so we get a sense of measurement, so we get
a sense of mission such as from GPS.
So this is going to measure the position of the car.
So the measurement is going to be basically the main position of the measurement.
So this is going to be our Munim and the measurement itself is going to have some level of uncertainty
to it.
And again, we model this as a normal distribution.
So this is going to have our sigma squared m sorry, the uncertainty inside the measurement.
Now we can take our sense of measurement and our predicted state and we can combine them together to
get a better estimate.
So once we do that, once we use the measurement in, we're going to get this new estimate.
So this is our updated position estimate for the car and see now if the main is well and truly near
the position of the car and it's going to have a new main and it's going to have a new uncertainty.
So new sigma squared.
Now, what we can see is that once we used in the measurement, we can see the new uncertainty for the
updated position for time.
Step two is going to be smaller than the original position four times up to without the additional information.
So this is a quick example of how we can use probability in our estimation problem.
We can represent the things that we want to estimate as a probabilistic distribution, and then we can
use rules of probability to fuse in different estimates and then use Bayes Theorem to update our estimates
to get a better estimate using all the available information.
So this is one path away of how we can use Bayesian estimation.
So mixing the concepts of probability and estimation theory together to form a powerful framework for
estimation.