John Guiver replied on 04-07-2009 3:57 AM
I am wondering why are you using a 4x4. The web site I was looking at shows a 5x5 - I just want to make sure that we are on the same page...
Going with my variable definitions for now, the first bit of the model sets the prior for whether a cell has a pit or not. There can be any number of pits as drawn from a Bernoulli prior with a probability of 0.2. The Wumpus is different in that there
can only be one Wumpus, and therefore we encode its position drawing from a uniform Discrete prior over 25 (5x5) possible positions. However, we would like a posterior probability of a Wumpus being on a particualr square, so the second bit of the model below
encodes that as a Bernoulli distributed variable for each square.
for (int i=0; i < 5; i++)
{
for (int j=0; j < 5; j++)
{
hasPit[i, j] =
Variable.Bernoulli(0.2);
hasWumpus[i, j] = (wumpusPosition == i + (j * 5));
}
}
The next bit of model, as hinted at yesterday represents the neighbourhood relationships between breeze and pit, and between stench and Wumpus. So for example, the corner cases encoded as follows:
// Corner cases
hasBreeze[0, 0] = hasPit[0, 1] | hasPit[1, 0]; hasStench[0, 0] = hasWumpus[0, 1] | hasWumpus[1, 0];
hasBreeze[0, 4] = hasPit[0, 3] | hasPit[1, 4]; hasStench[0, 4] = hasWumpus[0, 3] | hasWumpus[1, 4];
hasBreeze[4, 0] = hasPit[4, 1] | hasPit[3, 0]; hasStench[4, 0] = hasWumpus[4, 1] | hasWumpus[3, 0];
hasBreeze[4, 4] = hasPit[4, 3] | hasPit[3, 4]; hasStench[4, 4] = hasWumpus[4, 3] | hasWumpus[3, 4];
The edge cases and the non-edge cases can be similarly represented, though with three and four neighbours respectively.
At this point the model is complete, and you need to do inference. As you step through Wumpus world, you will need to write down the observations. After each new observation, you can run inference again. For example:
// Square (0,0)
hasBreeze[0, 0].ObservedValue = true;
hasStench[0, 0].ObservedValue = false;
hasPit[0, 0].ObservedValue = false;
hasWumpus[0, 0].ObservedValue = false;
// Square (1,0)
hasBreeze[1, 0].ObservedValue = false;
hasStench[1, 0].ObservedValue = true;
hasPit[1, 0].ObservedValue = false;
hasWumpus[1, 0].ObservedValue = false;
You can get the inferred posterior distributions for values by creating an inference engine and calling the Infer method. For example, continuing the above game, you are currently at square (1,0) and are interested in the posterior probabilities for hasPit
and has hasWumpus at (2,0) and (1,1). The code for this is:
InferenceEngine ie = new
InferenceEngine();
Bernoulli postPit20 = ie.Infer<Bernoulli>(hasPit[2, 0]);
Bernoulli postPit11 = ie.Infer<Bernoulli>(hasPit[1, 1]);
Bernoulli postWumpus20 = ie.Infer<Bernoulli>(hasWumpus[2, 0]);
Bernoulli postWumpus11 = ie.Infer<Bernoulli>(hasWumpus[1, 1]);
At any step, you could choose to either get the probabilties for just the neighbouring squares, or for the whole grid. If you get the probabilties for the whole grid, there are a couple of caveats. One is that each inference you do as above will recompile the
model and that will be excessive for 25 inferences; there is a way to stop this happening (the InferAll method) but I won't get into that now. The second is that, with the current release, trying to infer an observed variable (i.e. a variable in a square
that you have visited) it will return an error - of course the outcomes for these square are known, so this isn't a problem, but your code will need to check for this.
Let me know how you get on.
John G.
