I am reading Probabilistic Robotics and I don't know how to solve the exercise problem number 4 at the end of the second chapter. There are no solutions to this text.
The exercise states:
In this exercise we will apply Bayes rule to Gaussians. Suppose we are a mobile robot who lives on a long straight road. Our location x will simply be the position along this road. Now suppose that initially, we believe to be at location xinit = 1000m, but we happen to know that this estimate is uncertain. Based on this uncertainty, we model our initial belief by a Gaussian with variance σ2init = 900m2.
To find out more about our location, we query a GPS receiver. The GPS tells us our location is zGPS = 1100m. This GPS receiver is known to have an error variance of σ2init = 100m2.
(a) Write the probability density functions of the prior p(x) and the measurement p(z|x).
(b) Using Bayes Rule, what is the posterior p(x|z)? Can you prove it to be Gaussian?
P.S.: This chapter talks about Recursive State Estimation using Bayes Filter Algorithm and also this exercise hints that it is plain manipulation of Quadratics..
What I have so far is that:
- p(x) is a normal distribution (gaussian) with mean=1000m and σ2 = 900m2.
- I'm thinking that p(z|x) is a normal distribution with mean=x and σ2 = 100m2.
Any help will be greatly appreciated..Thank you in advance...
You are right so far. $p(x) \sim \mathcal{N}(1000m, 900m^2)$. And $p(z|x) \sim \mathcal{N}(x, 100m^2)$
About (b): Bayes' rule is pretty easy and gives you: $p(x|z) = \frac{p(z|x) p(x)}{p(z)} $
You can prove this to be a Gaussian by writing the Gaussian formula down. First look at the exponent and prove that they stay quadratic and what the new mean and variance has become. Then, show that the normaliser is that of a Guassian and that the variance matches the one from the exponent.