Bayesian Belief Updating

About Bayesian Epistemology

Bayesian epistemology treats rational belief as degrees of confidence that should be updated according to Bayes' theorem when new evidence arrives. Your credences (degrees of belief) must satisfy the probability axioms to avoid Dutch book vulnerabilities—betting arrangements that guarantee you'll lose money.

Bayesian Updating

When you observe evidence E, you update your belief in hypothesis H using Bayes' rule:

P(H | E) = P(E | H) × P(H) / P(E)

• Prior P(H): Your initial credence before seeing evidence
• Likelihood P(E | H): How well the hypothesis predicts the evidence
• Posterior P(H | E): Your updated belief after seeing evidence
• Bayes factor: P(E | H) / P(E | ¬H) measures evidence strength independent of priors

Jeffrey Conditioning

Sometimes evidence is uncertain—you're not completely sure what you observed. Jeffrey conditioning handles this by updating on a probability distribution over possible observations:

P_new(H) = P(H | E) × P_new(E) + P(H | ¬E) × P_new(¬E)

This maintains coherence when your evidence itself has a degree of uncertainty, such as unreliable testimony or ambiguous perceptual data.

Philosophical Implications

• Convergence: Rational agents with different priors will converge toward agreement given enough shared evidence
• Extremism: Very strong priors require very strong evidence to shift; this can lead to rational polarization
• Problem of priors: Where do initial credences come from? Maximum entropy? Indifference principles?
• Logical omniscience: Real agents have computational limits, but Bayesian updating assumes all logical consequences are recognized