Imprecise Probability

Origin. Peter Walley, Statistical Reasoning with Imprecise Probabilities (Chapman and Hall, 1991), which unified the field around coherent lower previsions and a behavioural interpretation. The ancestry is long: Boole (1854) computed bounds rather than values where the data underdetermine a probability; Keynes (1921) argued some probabilities are not comparable at all; Smith (1961) and Williams (1975) developed betting-rate accounts with a gap between buying and selling prices; Dempster (1967) and Shafer (1976) gave a special case; Isaac Levi, The Enterprise of Knowledge (1980), argued for convex sets of credal states as the representation of rational belief.

Models. One probability measure encodes a determinate opinion about everything, which is more than evidence usually supports. Replace it with a set of measures — a credal set — and read the lower probability as the greatest lower bound over the set. Walley's interpretation is behavioural rather than epistemic: the lower prevision of a gamble is the highest price at which one is willing to buy it, the upper prevision the lowest price at which one is willing to sell, and the interval between them is the range over which one declines to trade in either direction. Precise probability is the degenerate case where the two prices meet, so the theory contains Bayesianism rather than competing with it.

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)]. Conjugacy: P*(A) = 1 − P_*(¬A).

Lower prevision: E_(X) = inf_{P∈K} E_P(X), for a gamble X. Supremum acceptable buying price for X. Superadditive: E_(X + Y) ≥ E_(X) + E_(Y). Positively homogeneous, monotone, and E_(X + c) = E_(X) + c.

Coherence: Avoiding sure loss: no positive combination of accepted gambles is uniformly negative. Coherence: no accepted gamble's price can be raised by combining the others. Avoiding sure loss ⟺ K ≠ ∅. Coherence ⟺ E_* is the lower envelope of K. 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: optimal under at least one P ∈ K. Maximality: not strictly dominated in expectation by any rival under every P ∈ K. Interval dominance: compare [E_, E] intervals. E-admissible ⊆ maximal ⊆ interval-undominated. Γ-maximin picks one option and may pick a non-E-admissible one.

Natural extension: Extend assessments coherently. Smallest credal set compatible; equivalently the least committal coherent lower prevision dominating the assessment. Inference principle. Tight bounds. On a finite space it is a linear program.

Conditioning rules: Generalized Bayes rule: elementwise updating, undefined when P_*(B) = 0. Regular extension: condition on every P with P(B) > 0. The two disagree when the conditioning event has lower probability zero.

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

Symbols.

Metatheory. Walley's two consistency conditions have exact set-theoretic content. An assessment avoids sure loss iff the credal set it induces is nonempty, and it is coherent iff it coincides with the lower envelope of that set — the lower envelope theorem, which makes "a coherent lower prevision" and "the infimum over a set of linear previsions" two descriptions of one object. Natural extension is the least committal coherent lower prevision dominating an assessment, and on a finite space it is computed by a linear program, so inference is a matter of optimization rather than of new axioms. The hierarchy of models is strict: precise probabilities sit inside belief functions, belief functions inside 2-monotone capacities, and 2-monotone capacities inside coherent lower previsions, each inclusion proper, and the Choquet integral agrees with the lower envelope expectation exactly at the 2-monotone level and not above it. Seidenfeld and Wasserman (1993) identified dilation: there are partitions on which conditioning strictly widens the interval whichever cell is observed.

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

Limitations. Dilation is not a pathology but a theorem: for some partitions the posterior interval strictly contains the prior interval under every cell, so the agent knows in advance that the evidence will make her less determinate whatever it turns out to be, and no analogue of the martingale property rules this out. Conditioning is not unique — the generalized Bayes rule is undefined when the conditioning event has lower probability zero, regular extension gives an answer there, and the two disagree; nor does updating each measure elementwise generally commute with taking lower envelopes. Decision is not unique either: Γ-maximin, maximality and E-admissibility give different answers on the same credal set, Γ-maximin can select an option optimal under no measure in K, and sequential application is not dynamically consistent, so the framework yields a set of acts rather than a choice. Convexity of the credal set is a modelling convention, and the convex hull of two measures licenses commitments neither source made. Computation degrades sharply: de Campos and Cozman (2005) proved inference in credal networks NP-hard even on polytrees, where the precise case is linear.

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