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Probability Theory III [2 P]

Uncertainty arises in the wumpus world because the agent's sensors give only partial, local information about the world. In this example the agent faces a situation in which each of the three reachable squares-$ [1,3]$ , $ [2,2]$ , and $ [3,1]$ - in a $ 4 \times 4$ grid world might contain a pit, because the squares $ [1,2]$ and $ [2,1]$ were breezy (no breeze was detected on square $ [1,1]$ ). Pure logical inference can conclude nothing about which square is most likely to be safe, so a logical agent might be forced to choose randomly. A probabilistic agent can do much better than the logical agent.

The goal is to calculate the probability that each of the three squares contains a pit. (For the purposes of this example, we will ignore the wumpus and the gold.) The relevant properties of the wumpus world are that (1) a pit causes breezes in all neighboring squares, and (2) the square $ [1,1]$ does not contain a pit. The given set of random variables are:

  1. We use one Boolean variable $ P_{i,j}$ for each square, which is true iff square $ [i, j]$ actually contains a pit.
  2. We also have Boolean variables $ B_{ij}$ that are true iff square, $ [i, j]$ is breezy; we include these variables only for the observed squares-in this case, $ [1,1]$ , $ [1,2$ ], and $ [2,1]$ .

In our analysis of the wumpus world in the excercise hour, we used the fact that each square contains a pit with probability 0.2, independently of the contents of the other squares. Suppose here instead that exactly $ N/5$ pits are scattered uniformly at random among the $ N=15$ squares other than $ [1,1]$ .

  1. Are the variables $ P_{ij}$ and $ P_{kl}$ still independent?
  2. What is the joint distribution $ P(P_{11}, ... , P_{44})$ now?
  3. Redo the calculation for the probabilities of pits in $ [1,3]$ and $ [2,2]$ .


next up previous
Next: Linear models for regression Up: NNA_Exercises_2012 Previous: Probability Theory II [2
Haeusler Stefan 2013-01-16