3203. Introduction to Artificial Intelligence
Bayesian Reasoning
1. Probabilistic reasoning
To reason with uncertain knowledge, one popular idea is to use probability theory to extending the truth value of a proposition {0, 1} to [0, 1], with binary logic as its special case. Though no conclusion is absolutely true, the one with the highest probability is preferred.
To extend the basic Boolean connectives to probabilty functions:
- negation: P(¬A) = 1 − P(A)
- conjunction: P(A∧B) = P(A) * P(B) if A and B are independent of each other
- disjunction: P(A∨B) = P(A) + P(B) if A and B never happen at the same time
From the definition of conditional probability of B given A, P(B|A) = P(B∧A) / P(A),
Bayes' Theorem is derived: P(H|E) = P(E|H) * P(H) / P(E).
To apply these mathematical results to practical problems, the notion of "probability" needs to be interpreted. There have been several influential interpretations:
- Classical: If an experiment has n equally likely outcomes, each outcome has probability 1/n.
- Frequentist: The probability of an event is the limiting relative frequency with which it occurs in a large number of repeated trials.
- Subjective: Probability represents a degree of belief. Different individuals may assign different probabilities, provided their beliefs are internally consistent.
- Logical/Evidential/Bayesian: The probability of an event measures the degree to which the available evidence supports that event.
Under the last interpretation, Bayes' theorem can be used to update a system's beliefs in light of new evidence, transforming a prior probability into a posterior probability.
2. Bayesian Networks
In AI, one influential approach of probabilistic reasoning is Bayesian Networks are also known as Belief Networks. Basic ideas are:
- Interpret probability as degree of belief given available evidence,
- Update probability with new evidence according to Bayes's rule,
- Use assumptions on independence to reduce computational cost.
Bayesian Networks are directed acyclic graphs in which the nodes represent variables of interest and the links represent informational dependencies among the variables. The strength of dependency is represented by conditional probabilities.
3. An Example
Example 9.13. An diagnostic assistant