Imprecise Probability

Standard probabilism requires that an agent's credence in every proposition be a single real number. Imprecise probability denies this. On the imprecise view, an agent's doxastic state is represented not by one probability function but by a set of them — a credal set — and a proposition receives an interval of values rather than a point.

The motivation is the poverty of a single number as a representation of evidence. Consider two propositions: that a thoroughly tested fair coin lands heads, and that a coin of entirely unknown provenance, possibly double-headed, lands heads. Standard Bayesianism assigns $0.5$ to both. But the epistemic situations are utterly different: the first credence is stable and evidentially grounded, the second is a shrug. Any evidence at all about the second coin should move it sharply, while a great deal would be needed to move the first. A single number cannot record that difference, and the difference is real.

The framework

An agent's state is a set $\mathcal{P}$ of probability functions, all of them coherent. The lower and upper probabilities are

$$\underline{P}(A)=\underset{P\in \mathcal{P}}{\inf }P(A),\overline{P}(A)=\underset{P\in \mathcal{P}}{\sup }P(A)$$

The width $\overline{P}(A)-\underline{P}(A)$ measures how much the evidence fails to determine. Total ignorance about the second coin is represented by $\mathcal{P}$ containing every distribution over $\{$heads, tails$\}$, giving $[0,1]$; the tested coin gives the sharp interval $[0.5,0.5]$. Precise probability is the special case where $\mathcal{P}$ is a singleton.

Walley's theory of lower previsions provides the general framework, developing the idea in terms of buying and selling prices for gambles rather than in terms of sets of measures. The behavioural interpretation is natural: $\underline{P}(A)$ is the highest price the agent will pay for a bet on $A$, and $\overline{P}(A)$ the lowest price at which they will sell it. The gap between them is a region of indeterminacy in which the agent declines to trade in either direction — which is exactly what a real agent facing genuine ignorance does.

Note that this reveals a hidden assumption in the Dutch-book setup. The standard argument assumes the agent will take either side of a bet at their credence, which builds in precision from the start. An agent with a buying price below their selling price is not thereby exploitable: no book can be made against them, provided the interval is coherent. So imprecise agents are not irrational by the Dutch-book standard, and the argument for precision has to come from elsewhere.

Arguments for imprecision

Evidence is often imprecise. Bayesian orthodoxy requires a precise prior even in complete ignorance, and the choice is then arbitrary — the indifference paradoxes show that "the" ignorance prior does not exist. Imprecise probability's response is that if the evidence does not determine a number, the representation should not contain one. The arbitrariness in the choice of prior is not a technical embarrassment to be managed but a signal that the model is over-specified.

Ellsberg's paradox. An urn contains 30 red balls and 60 that are black or yellow in unknown proportion. Most people prefer betting on red over black, and simultaneously prefer betting on "black or yellow" over "red or yellow". No single probability function can rationalise both preferences: the first implies $P(\text{red})>P(\text{black})$ and the second implies the reverse. The pattern is robust and survives explanation of the inconsistency to subjects.

The standard Bayesian must classify these preferences as irrational. The imprecise theorist says they are a sensible response to ambiguity — the difference between known and unknown chances — which precise probability cannot represent. Whether ambiguity aversion is rational or merely widespread is disputed, but the case that it is intelligible is strong.

Group and scientific contexts. A scientific community that has not converged on a prior is well represented by the set of priors its members hold, and results robust across that set are exactly the ones warranting collective assertion. This is close to what sensitivity analysis does in practice.

Problems

Dilation

The most serious technical difficulty. Ordinarily, learning narrows one's opinions. With imprecise credences, learning can widen the interval — and can do so whatever is learned.

The standard example: a fair coin is tossed, and there is a proposition $H$ about the outcome with $\underline{P}(H)=\overline{P}(H)=0.5$. Introduce a second event $E$ about which the agent is entirely ignorant, but which is known to be correlated with $H$ in an unknown way. Conditioning on $E$, or on $\neg E$, can send the interval for $H$ to $[0,1]$ in both cases. The agent knows in advance that whichever way the evidence falls, their opinion about $H$ will become less determinate than it is now.

This violates a strong intuition about learning and is difficult to reconcile with reflection: the agent's current precise credence is not the expectation of their future imprecise one in any straightforward sense. Defenders argue that dilation is the correct response — the agent has discovered that their previous precision was unwarranted, since it rested on ignoring a relevant correlation — and that the intuition it offends is a residue of thinking precision is always available.

Decision-making

Precise credence plus utility yields a determinate expected utility and hence a recommendation. A credal set yields a set of expected utilities, and there is no agreed rule for choosing among the acts they rank differently. The candidates all have defects:

• $\Gamma$-maximin — maximise the minimum expected utility over the credal set. Systematically pessimistic, and gives ambiguity aversion by fiat rather than deriving it.
• E-admissibility (Levi) — an act is permissible if it maximises expected utility relative to some member of the set. Often leaves many acts permissible.
• Maximality — reject acts dominated relative to every member. Similarly permissive.

The upshot is decision paralysis: with a wide credal set, most acts are permissible and the theory gives little guidance. There is also a diachronic worry that sequences of individually permissible choices can be jointly exploitable, though the details are contested.

Is imprecision ever mandatory?

A precise Bayesian can grant everything above as a modelling convenience — a way of summarising an agent's uncertainty about which prior to adopt — while denying that any rational agent's state is genuinely imprecise. The imprecise theorist owes an argument that precision is not merely unavailable in practice but incorrect in principle, and the accuracy-based arguments cut the other way: precise credences generally do better on scoring rules, and it is not clear how to score an interval at all.

Assessment

Imprecise probability is best seen not as a rival interpretation of probability but as a liberalisation of the Bayesian framework, retaining coherence while dropping precision. Its appeal is proportional to how seriously one takes the problem of the priors: if the choice of prior is genuinely arbitrary in cases of deep uncertainty, then a framework that refuses to make the choice is more honest. If priors can be constrained — by invariance, maximum entropy, or known chances — the motivation weakens considerably.

Its practical uses are real: robust Bayesian analysis, sensitivity analysis, and risk assessment under deep uncertainty all effectively work with sets of priors, whatever their official epistemology.

Where this sits

This completes the treatment of rational credence. The folder's arc is that probabilism is well supported by two independent arguments, conditionalization less securely by weaker ones, and that both leave the priors open — with imprecise probability the most direct response to that gap, at the cost of decision-theoretic guidance.

The alternative response is to look outward rather than inward: to hold that priors are constrained not by any internal principle of rationality but by the world, through knowledge of objective chance. That requires an account of what chance is and why credence should defer to it, which is the Principal Principle and the subject of the next folder.