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Currently I'm working on a decision making system and I have the following elements:

  • Complex bayesian network
  • Decision CPD (Conditional probability distribution)
  • Utility factor (Which maps weights to certain probability assignments) (Only the parents of U)

I'm calculating my utility value for the complete network without observing any evidence and that works fine.

However now I have some evidence observed and what that does is set the probability for every illegal assignment to 0. Legal assignments still have the same probability. When I then run my utility calculation function it changes the decision CPD (due to the fact that illegal options are removed (multiplied by 0)) but it still sees the probability of the evidence occurring with prior probability.

My question is, do I need to normalize the probabilities after observing the evidence and not taking into account that the evidence occurs with a certain probability. This heavily impacts the outcome of the utility function and therefore the outcome of my decision.

ROMANIA_engineer
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Marcel
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1 Answers1

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The sum of the probabilities of all possible outcomes must always sum to 1. So if some outcomes are eliminated and hence assigned zero probability, then YES, you must re-normalise so that the sum of all the remaining probabilities adds up to 1.

Stochastically
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  • Thanks for your answer, sometimes in complex situations you tend to forget the rules that makes your life easy! But is it possible to take into account the probability of the evidence occurring. Can I then after normalisation multiply it by the chance of the evidence occurring? – Marcel May 26 '13 at 10:23
  • What you're doing isn't 100% clear to me, but here's my best guess. If the probability of the evidence occurring is exactly equal to the sum of the probability of the remaining outcomes, then normalising just means dividing all probabilities by sum of the remaining probabiliites. However, if the probability of the evidence is different from the sum of the remaining probabilities, I think it's a harder problem because the evidence must affect the remaining probabilities in different ways. – Stochastically May 26 '13 at 10:30
  • It is the latter problem (I think), I don't always know how the bayesian network is structured and how evidence affects the probability distributions. So it could be the first case but also the last case which makes it way more complicated. But for now normalisation will do the job. In the future I'll take a look at more complex situations. – Marcel May 26 '13 at 10:40
  • It is neither case (although it's closer to the second). Simply normalising will yield a probability distribution but **not** in general the correct one. This is simple conditional probability. If the original event is A, and the evidence is event B, then the new probability for the occurrence of A is given simply by P(A|B) where P is the original probability distribution. As mentioned, this will be different for different events. However it's got nothing to do with the probability of the evidence occurring being equal to anything. – davin May 26 '13 at 16:29
  • A simple example will suffice: Consider a bag with 3 balls, one white and round, one white and square, and one yellow and square. We choose one ball at random (i.e. with uniform distribution) and are interested in its colour. If I gather evidence that the chosen ball is **not** round, then I'm clearly left with equal chance to get either colour. But just normalising (since nothing is multiplied by 0 because I can still get both colours) would maintain the uneven distribution that white has *double* chance of being chosen, which is clearly not the case. – davin May 26 '13 at 16:36