Imprecise Probability Logic

Origin. Walley (1991), de Finetti, Levi. Sets of probabilities. Lower/upper bounds. Robust Bayesianism. Uncertainty about uncertainty.

Models. Single probability insufficient. Sets of probability measures. Lower and upper probabilities. Credal sets. Interval-valued beliefs.

Formalism.

Credal set: K = set of probability measures. All measures in K compatible with information. Imprecision: multiple compatible. Not ignorance: structured uncertainty.

Lower/upper probability: P_(A) = inf_{P∈K} P(A). P(A) = sup_{P∈K} P(A). Bounds on probability. Interval [P_(A), P(A)].

Coherence: Avoiding sure loss. Desirability conditions. Rational constraints. Behavioral interpretation.

Conditioning: K|B = {P(−|B) : P ∈ K, P(B) > 0}. Conditional credal set. Dilation possible. Imprecision may grow.

Decision making: Γ-maximin: maximize minimum expected utility. E-admissibility: undominated in expectation. Interval dominance. Multiple decision rules.

Natural extension: Extend assessments coherently. Smallest credal set compatible. Inference principle. Tight bounds.

Comparison with Dempster-Shafer: Both handle imprecision. Different interpretation. DS: mass functions. IP: coherent betting rates.

Symbols.

Symbol	Unicode	Meaning
K	—	credal set
P_*	—	lower probability
P*	—	upper probability
E_*	—	lower expectation

Metatheory. Sets of measures. Coherence. Decision theory. Robust inference.

Applies to. Decision under ambiguity. Risk assessment. Machine learning. Statistics.

Limitations. Computational cost. Many decision rules. Interpretation debates. Limited tools.