A Brief Introduction to Uncertainty and Probabilistic Reasoning

• Problems of interest
  • (Logical reasoning:
    if a patient has a toothache and there is a large black hole in the aching tooth, then the patient has a cavity)
  • Probabilistic reasoning:
    if a patient has a toothache, then there is a probability of .9 that he has a cavity
    if a patient has a cavity, then there is a probability of .6 that he has a toothache
  • Causes and effects (symptoms)
• Joint probability distribution: probability for each variable assignment (atomic event)
• Probabilities of conjunctions and disjunctions

  

  

• Statistical independence

  

  

• Prior and conditional probability, conditional independence

  

  

  For independent variables,

  

  Conditional independence of S1 and S2, given D,

  

  
  (In the figures above, if the blue region is D and the two larger regions S1 and S2, S1 and S2 are conditionally independent, given D, in the figure on the left but not conditionally independent in the figure on the right.)

• Where probabilities come from: experiments vs. beliefs, inconsistencies in beliefs
• Bayes' rule

  

• Avoiding estimation of prior probability of effects (symptoms)

  

• Combining evidence with Bayes' rule: Bayesian updating

  

  For "symptoms" which are unconditionally independent and conditionally independent given the "disease",