Reasoning under uncertainty: probability, Bayes and Bayesian networks
Reasoning under uncertainty uses probability to represent and update beliefs. Bayes' theorem provides a rule for updating probabilities based on evidence. Bayesian networks combine probability with graph structures to model dependencies among variables.
Uncertainty and reasoning in AI
Reasoning in artificial intelligence uses available information to generate predictions, make inferences, and draw conclusions. A typical reasoning system comprises a knowledge base that stores structured knowledge and an inference engine that applies logic and reasoning methods, as described by IBM's overview of reasoning in AI. IBM's overview of reasoning in AI
Among the many reasoning strategies, probabilistic reasoning explicitly models uncertainty with probabilities. Other approaches include deductive, inductive, abductive, and fuzzy reasoning. These methods combine when an application requires dealing with incomplete or noisy data.
The UC Berkeley CS188 textbook introduces the logical and probabilistic foundations of intelligent systems, covering problem solving, reasoning, decision making, and learning. UC Berkeley's CS188 textbook
Probability and Bayes' theorem
Probability quantifies the likelihood of events. A conditional probability measures the probability of a hypothesis given observed data. The Stanford Encyclopedia of Philosophy defines conditional probability as the ratio of the joint probability to the probability of the data, provided the latter is positive. Stanford Encyclopedia of Philosophy on Bayes' theorem
Bayes' theorem relates the conditional probability of a hypothesis given data to the likelihood of the data under that hypothesis, multiplied by the ratio of the prior probabilities. This rule is central to updating beliefs when new evidence arrives.
The theorem simplifies calculation because inverse probabilities often are easier to estimate than direct ones. It also clarifies how hypotheses are confirmed by data that they render probable.
Bayesian epistemology and degrees of belief
Bayesian epistemology treats belief as a matter of degree, called credence. Rational agents assign numerical probabilities to possibilities and update them with evidence. The Stanford Encyclopedia of Philosophy describes the norms of probabilism and conditionalization that govern these degrees. Stanford Encyclopedia of Philosophy on Bayesian epistemology
Probabilism requires that a set of credences obey the laws of probability. Conditionalization specifies how credences should change when new evidence is received, typically by setting the new credence equal to the old conditional probability on the evidence.
This framework provides a normative account of learning and is widely applied in statistics and AI, where models maintain probability distributions over hypotheses.
Bayesian networks and causal models
Bayesian networks represent dependencies among variables as a directed acyclic graph, where edges encode conditional independence relationships. The Stanford Encyclopedia of Philosophy's entry on causal models explains how these graphs support inferences about causal structure and interventions. Stanford Encyclopedia of Philosophy on causal models
A causal model includes variables, structural equations, and a probability distribution. It can predict effects of interventions and evaluate counterfactual claims. The Markov condition links the graph to probability, allowing factorization of the joint distribution.
Research at the UCLA Cognitive Systems Laboratory continues to develop methods for evidential reasoning and causal inference, including learning and representing causal networks from observations. UCLA Cognitive Systems Laboratory
Alternatives to probability: fuzzy logic
Fuzzy logic models reasoning with vague statements by allowing truth values between 0 and 1. The Stanford Encyclopedia of Philosophy describes fuzzy logic as a family of many-valued logics where connectives operate truth-functionally over degrees of truth. Stanford Encyclopedia of Philosophy on fuzzy logic
In contrast to probability, which quantifies uncertainty about whether a proposition is true or false, fuzzy logic represents partial truth. This distinction matters in control systems and classification tasks where imprecision is intrinsic.
| Approach | Handling of uncertainty | Key mechanism | Example use |
|---|---|---|---|
| Probabilistic | Quantifies uncertainty with probabilities | Probability distributions and Bayesian updating | Diagnostic systems |
| Deductive | Assumes certainty if premises are true | If-then rules applied to a knowledge base | Expert systems |
| Abductive | Infers the most likely explanation for observations | Selects best explanation from available data | Medical diagnosis |
| Fuzzy | Uses degrees of truth for vagueness | Membership functions and fuzzy connectives | Control systems |
Frequently asked questions
What is Bayesian reasoning?
Bayesian reasoning uses probability to update beliefs as new evidence arrives, based on Bayes' theorem.
How does a Bayesian network differ from a simple probability model?
A Bayesian network uses a graph to represent conditional dependencies among many variables, making inference tractable.
Why is fuzzy logic not a substitute for probability?
Fuzzy logic handles partial truth, while probability handles uncertainty about truth; they address different aspects of imprecision.
What are the main norms in Bayesian epistemology?
Probabilism requires coherence, and conditionalization specifies how degrees of belief should change with evidence.