Posterior probability after robot movement

Question

Suppose robot moves in cells and there are 5 cells. Their dist. is as follows: | 1/9 | 1/3 | 1/3 | 1/9 | 1/9 |

The robot moves one cells towards right. And the world is cyclic. When it moves to the most right cell, it return back to the most left one.

And the posterior probability after one cell movement is as follows: | 1/9 | 1/9 | 1/3 | 1/3 | 1/9 |

The following diagram is a good illustration.

Can any guy tell me why the posterior probability shifts to the right one cell? Thanks in advance!

Answer 1

Think about the probability of the robot being in any cell A at Time t in terms of its probability of being or not being in cell A-1 at Time t-1:

Break event up into mutually exclusive joint events:

--> P(Robot loc @ T = A ) = P(Robot loc @ T=A, Robot loc @ T-1 = A-1) + P(Robot loc @ T=A, Robot loc @ T-1 <> A-1)

Use conditional probability to break those joint events up into independent events:

--> P(Robot loc @ T= A ) = P(Robot loc @ T=A | Robot loc @ T-1 = A-1) . P(Robot loc @ T-1 = A-1) + P(Robot loc @ T=A | Robot loc @ T-1 <> A-1) . P(Robot loc @ T-1 <> A-1)

And that allows us to use the fact that the robot is moving to the right (the event that the robot has moved to the right has prob 1, any other possibility has prob 0).

--> P(Robot loc @ T= A ) = 1 . P(Robot loc @ T-1 = A-1) + 0 . P(Robot loc @ T-1 <> A-1)

Simplify, and get the answer you wanted.

--> P(Robot loc @ T= A ) = P(Robot loc @ T-1 = A-1)

Answer 2

After reading a bunch of posts, I got an intuitive understanding, which I think I can convey without having to go into the math.

Main confusion is because we first wrongly assume that the probabilities represent the probabilities of the doors being at positions x1,x2,..., xn.

It is in fact the probability of the robot being present at locations x1, x2, ..., xn.

Now, due to some sensor, the robot knows it's in front of a door. It does not know which door it is in front of. So, given that we know that the robot has sensed that it is in front of the door, we can say that there is a higher probability that the robot is at locations where the doors are present. This results in a posterior distribution.

Now, if we know that the robot moved, say , one unit to the right, then we can deduce that the posterior distribution also shifts by one unit to the right.

Hope this helps!