It has a sensor that can detect the density of some rock.
Unfortunately the sensor is noisy. The number returned by the sensor is: x = 3∙d + N(3, 100), where
d is the actual density, and N(μ, σ²) specifies a Gaussian distribution with mean μ and variance σ².
You decide to use a Bayes’ filter to help overcome the noise of the sensor and to help you
detect which type of rock you are sensing.
Here are rough numerical approximations to the sensor model in tabular form.
......................................Sensor Output............................................
...........|..x < 2..|..2 < x < 8..|..8 < x < 14..|..14 < x < 20..|..x > 20 D....0..|...0.46...|......0.23......|.......0.18.........|........0.09..... ..|......0.04 E N....2..|...0.24...|......0.22......|.......0.23.........|........0.17..... ..|......0.14 S I....4..|....0.10...|......0.15......|.......0.22.........|........0.23.... ...|......0.30 T Y....6..|...0.03...|......0.07......|.......0.15.........|........0.22..... ..|......0.53
(a) Derive Bayes’ rule for two probabilities (starting with a Venn diagram).
P(A|B) =?
(b) Which type of Bayes’ filter is most appropriate for this problem: a Kalman
filter, a particle filter or a table based Bayes’ filter. Why?
For the rest of this question we will assume a table based filter (because it is easiest to ask
questions about, don’t assume it is the best for the question above simply because the following
questions use it):
(c) What is your state representation for your table based filter?
(d) What is your initial state assuming you have no idea what sort of rock you
are currently sensing?
Your sensor returns output of 3 when sensing a rock for the first time.
(e) What is the your new state?
(f) At this point, what is your best estimate as to what density of rock you are
sensing?
You sense the same rock again, and this time get receive output of −1.
(g) What is your new state?
(h) At this point, what is your best estimate as to what density of rock you are sensing?