Prior sampling
Let’s use stochastic sampling instead!
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Prior sampling
How would you estimate P pc, ␣s, r, wq?
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Take a step back
How would you estimate P pdie shows 7q?
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Stochastic simulation
How would you estimate P pdie shows 7q?
Simple: you take n random samples from the network
Let Xi be the binary r.v. that is 1 if the event sampled in the ith run is 7 Simply output
řn Xi Pˆ p 7 q “ i “ 1
n Law of large numbers says that limnÑ8 Pˆ “ P .
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Stochastic simulation
Back to our Bayesian network with cloudy, sprinkler, rain, and wet grass.
How would you estimate P pc, ␣s, r, wq?
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Stochastic simulation
Back to our Bayesian network with cloudy, sprinkler, rain, and wet grass.
How would you estimate P pc, ␣s, r, wq?
Simple you just take say n random samples from the network
Let Xi be the binary r.v. that is 1 if the event sampled in the ith run is c, ␣s, r, w Simply output
Xi
n Law of large numbers says that limnÑ8 Pˆ “ P .
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řn Pˆpc, ␣s, r, wq “ i“1
Prior sampling. Let’s generate a random sample!
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Prior sampling.
List of randomly generated numbers: 0.4, 0.2, 0.71, 0.2 for this run. We have 0.4 small than PpCq, so Cloudy “ true
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Prior sampling
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Prior sampling
List of randomly generated numbers: 0.4, 0.2, 0.71, 0.2 for this run We have 0.2 ě PpS | Cq and hence, Sprinkler “ false.
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Prior sampling
List of randomly generated numbers: 0.4, 0.2, 0.71, 0.2 for this run. We have 0.71 ă PpR | Cq and hence, Rain “ true
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Prior sampling
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Prior sampling
List of randomly generated numbers: 0.4, 0.2, 0.71, 0.2 for this run. We have 0.2 ă PpW | S,Rq and hence, WetGrass “ true
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Prior sampling
So, this time we get the event
rCloudy “ true,Sprinkler “ false,Rain “ true,WetGrass “ trues Will write this:
rtrue, f alse, true, trues
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Prior sampling
If we repeat the process many times, we can count the number of times rtrue, f alse, true, trues is the result.
The proportion of this to the total number of runs is:
Ppc,␣s,r,wq The more runs, the more accurate the probability.
Similarly for other joint probabilities.
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Prior sampling
function PRIOR-SAMPLE(bn) returns an event sampled from bn inputs: bn, a belief network specifying joint distribution
PpX1,…,Xnq
x Ð an event with n elements fori “ 1tondo
xi Ð a random sample from PpXi | parentspXiqq given the values of ParentspXiq in x
return x
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Prior sampling limitation
How would you get the following marginal distribution?
PpX |eq
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