Probabilistic logic
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This document discusses probabilistic logic and inference. It defines conditional probability and the product rule. It then shows examples of computing probabilities of events occurring together or conditionally, such as the probability of a toothache or the probability of a cavity given a toothache. The document also defines Bayes' rule for calculating conditional probabilities and shows an example of how a doctor could use it to determine the probability that a patient has meningitis given that they have a stiff neck.
Probabilistic logic
- 1. Rushdi Shams, Dept of CSE, KUET, Bangladesh 1 Knowledge Representation Probabilistic Logic Artificial Intelligence Version 2.0
- 2. Rushdi Shams, Dept of CSE, KUET, Bangladesh 2 Conditional Probability Definition of conditional probability: P(a | b) = P(a b) / P(b) if P(b) > 0 Product rule gives an alternative formulation: P(a b) = P(a | b) P(b) = P(b | a) P(a)
- 3. Rushdi Shams, Dept of CSE, KUET, Bangladesh 3 Inference with Probability
- 4. Rushdi Shams, Dept of CSE, KUET, Bangladesh 4 Inference in Probability P(toothache) = 0.108 + 0.012 + 0.016 + 0.064 = 0.2
- 5. Rushdi Shams, Dept of CSE, KUET, Bangladesh 5 Inference in Probability P(cavity V toothache) = 0.108 + 0.012 + 0.072 + .008 + 0.016 + 0.064 = 0.28
- 6. Rushdi Shams, Dept of CSE, KUET, Bangladesh 6 Inference in Probability Can also compute conditional probabilities:
- 7. Rushdi Shams, Dept of CSE, KUET, Bangladesh 7 Inference in Probability Can also compute conditional probabilities:
- 8. Rushdi Shams, Dept of CSE, KUET, Bangladesh 8 Baye’s Rule Product rule gives an alternative formulation: P(a b) = P(a | b) P(b) = P(b | a) P(a) Joining them together, we can find- P(a | b) = P(b | a) P(a) P(b)
- 9. Rushdi Shams, Dept of CSE, KUET, Bangladesh 9 Application of Bayes’ Rule A doctor knows that the disease meningitis causes the patient to have a stiff neck is 50% Means probability of stiff neck given the probability of having meningitis P(s | m) = 0.5 He also knows that in every 50000 patients, 1 may have meningitis Means probability that a patient has meningitis P (m) = 1/50000 He also knows that in every 20 patients, 1 may have stiff neck Means probability that a patient has meningitis P (m) = 1/20 Then, from Bayes’ rule P(m | s) = P(s | m) P(m) P(s)
- 10. Rushdi Shams, Dept of CSE, KUET, Bangladesh 10 Application of Bayes’ Rule P(m | s) = P(s | m) P(m) P(s) = 0.5 X (1/50000) 1/20 = 0.0002 Means he can expect only 1 in 5000 patients with a stiff neck to have meningitis
- 11. Rushdi Shams, Dept of CSE, KUET, Bangladesh 11 References Artificial Intelligence: A Modern Approach (2nd Edition) by Russell and Norvig Chapter 13