Sensor fusion and nonlinear filtering - Lars Hammarstrand

About professor Lars Hammarstrand

Associate Professor at Chalmers University of Technology, Göteborg, Sweden

LinkedIn:
https://www.linkedin.com/in/lhammarstrand/



YouTube channel:
https://youtube.com/playlist?list=PLTD_k0sZVYFqjFDkJV8GE2EwfxNK59fJY

1.3 Distributions

https://www.youtube.com/watch?v=X7ePpI4qgTQ&list=PLTD_k0sZVYFqjFDkJV8GE2EwfxNK59fJY&index=5&t=126s

Conditional Distributions

Conditional Distributions are indispensable in Sensor Fusion, Filtering, and Bayesian estimation in general.

Conditional distribution - product rule

Let x and z be two random variables with the "joint probability" pdf p(x,z)

z is some constant value

The function
p (x | z)
read as "conditional density" of x, given z,
is defined:

p(x, z) = p(z | x) p(x)

where 

p(x)

 is read as a "modular distribution" of x

and if 

p(x) =/= 0

read as: possible values of x for which probability density is non-zero,

when can write

p( z | x ) =  p(x, z) / p(x)

read as: 

The conditional density of x given z 

is the ratio of the joint probability of x and z

divided by modular distribution of x. 

This can be re-written as:

p(z | x = x') = p(x', z) / p(x')

where:

x' is some constant

p(x') is also some constant

and that is proportional \alpha to the joint probability (x', z).

we fixed one dimension x' which is a function of z.

Interpretation: Conditional density p( z | x) describes the distribution of z given that x is known. 

Example:

Sara decided how many pieces of candy she can have every day

by tossing a coin and rolling the dice.

40% coin

60% dice

Coin: 

- heads 1 candy

- tails 0 candy

Pr {z = i | Sara tosses coin} =
{ 

    0.5 if i = 0,1 

    0  otherwise

Pr {z = i | Sara throws dice} = 

{

    1/6 if i = 1,2,..., 6

    0  otherwise

Marginal distributions

TBC