How can we calculate likelihood of sensor values wrt to actual real world?

Question

I am trying to understand Bayes Filter. In its update step we have P(z_t|x_t) as observation model.

Slide 37: Bullet 5 of this lecture states "Likelihood of measurement is given by “probabilistically comparing” the actual with the expected measurement."

For example from the following data:

Observation: [0, 1, 0, 0, 0, 0, 1, 0, 0, 0] -> This what the robot sense

Actual World: [0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0]

And actual world we can say [0, 1] resembles to [white, black] tile and sensor recognize them with [White: 0.7, Black: 0.9] probability. I am not able to understand here how can we calculate this likelihood for each step ?

Answer 1

The likelihood is with respect to your predicted state not the world state (You will never know the world state)

For example, if you predict you will be at [0,0] but the measurement tells you [.5,.5] you can calculate the likelihood of that measurement given your predicted state. In that scenario, you can use a normal distribution or some more complex function to calculate the likelihood.

In your case of a white black tile vector you can possibly treat each entry as a binomial distribution then calculate the joint probability.