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Probabilistic Reasoning

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  • 1. Probabilistic Reasoning Tameem Ahmad Student M.Tech. Z.H.C.E.T. A.M.U.,Aligarh Copyright, 1996 © Dale Carnegie & Associates, Inc.
  • 2. References: NPTEL National Programme on Technology Enhanced Learning (NPTEL) is a Government of India sponsored collaborative educational programme. By developing curriculum-based video and web courses the programme aims to enhance the quality of engineering education in India. It is being jointly carried out by 7 IITS and IISc Bangalore, and is funded by the Ministry of Human Resources Development of the Governament of India. Computer Based Numerical & Statistical Techniques by M.Goyal Laxmi Publications, Ltd., 01-Jan-2008 2
  • 3. Probabilistic ReasoningWhat???• Capturing uncertain knowledge• Probabilistic inference 3
  • 4. Knowledge representation bird(tweety). fly(X) :- bird(X). ?- fly(tweety).Yes 4
  • 5. Knowledge representation (Cont.) But the real world is not so simple like this… There are many other factors… A way to handle knowledge representation in real problems is to extend logic by using certainty factors. IF condition with certainty x THEN fact with certainty f(x) Replacesmoking -> lung cancer orlotsofconditions, smoking -> lung cancerWithP(lung cancer | smoking) = 0.6 5
  • 6. probabilistic model A probabilistic model describes the world in terms of a set S of possible states - the sample space. We don’t know the true state of the world, so we (somehow) come up with a probability distribution over S which gives the probability of any state being the true one. The world usually described by a set of variables or attributes. 6
  • 7. Cont… let the random variable Sum (representing outcome of two die throws) be defined thus:Sum(die1, die2) = die1 +die2P(Sum = 2) = 1/36,P(Sum = 3) = 2/36, . . . ,P(Sum = 12) = 1/36 7
  • 8. Cont… Visit to Asia? A Tuberculosis? T Either tub. or lung cancer? E Lung cancer? L Smoking? S Bronchitis? B Dyspnoea? D Positive X-ray? X 8
  • 9. Cont…Sample SpaceS = {(A = f, T = f,E = f,L = f, S = f,B = f,D = f,X = f),(A = f, T = f,E = f,L = f, S = f,B = f,D = f,X = t), . . .……………….(A = t, T = t,E = t,L = t, S = t,B = t,D = t,X = t)} 9
  • 10. Cont… Marginal Probability Distribution For example, P(A = t,D = f) = P(A = t, T = f,E = f,L = f, S = f,B = f,D = f,X = f) + P(A = t, T = f,E = f,L = f, S = f,B = f,D = f,X = t) + P(A = t, T = f,E = f,L = f, S = f,B = t,D = f,X = f) + P(A = t, T = f,E = f,L = f, S = f,B = t,D = f,X = t) ... P(A = t, T = t,E = t,L = t, S = t,B = t,D = f,X = t) This has 64 summands! 10
  • 11. Cont… Conditional Probablity 11
  • 12. Knowledge engineering foruncertain reasoningDecide what to talk aboutDecide on a vocabulary of random variablesEncode general knowledge about thedependenceEncode a description of the specific probleminstancePose queries to the inference procedure and getanswers 12
  • 13. Probabilistic Inference Rules Two rules in probability theory are important for inferencing, namely, the product rule and the Bayes rule. 13
  • 14. Cont… Suppose one has been tested positive for a disease; what is the probability that you actually have the disease? It depends on the accuracy and sensitivity of the test, and on the background (prior) probability of the disease. Let P(Test=+ve | Disease=true) = 0.95 (95%), so the true negative rate, P(Test=-ve | Disease=true), is 0.05 (5%). Let P(Test=+ve | Disease=false) = 0.05, so the false positive rate is also 5%. Suppose the disease is rare: P(Disease=true) = 0.01 (1%). 14
  • 15. Cont…Applying Bayesian 15