Uncertainty

Abstract

Uncertainty, which MeSH classifies under decision making, is the condition in which an agent lacks reliable knowledge of a present state or a future outcome. Cognitive psychology distinguishes risk, where outcome probabilities are known, from ambiguity, where the probabilities themselves are unknown, and treats the felt difference between them as a core fact about human choice. The mind represents uncertainty as graded probability rather than binary doubt, weighs it through nonlinear functions that overweight small chances, and estimates it in dedicated neural circuits that separate familiar noise from genuine change. Learning adapts to it, accelerating belief updates when the world grows volatile, while individual differences in how aversively uncertainty is experienced help explain vulnerability to anxiety. This article surveys its definition, its measurement, its neural basis, and its clinical significance.

Keywords: uncertainty, ambiguity, risk, probability weighting, volatility

Every consequential choice is made without full knowledge of its outcome. Uncertainty is the general name for that shortfall of knowledge, and it arrives in kinds the mind treats differently. A gambler who knows a fair wheel pays even money on red faces risk: the odds are given. An investor weighing an untested market faces something more unsettling, because the odds are not merely unfavourable but unknown, a condition the economist Frank Knight named uncertainty to set it apart from measurable risk. The distinction is psychologically real. People will pay to avoid unknown odds even when the known odds are no better (Ellsberg, 1961), a preference no account that collapses every unknown into a single number can explain. Uncertainty is therefore not one quantity but a small family of them, and how the mind estimates, weighs, and reduces each member is among the central problems of the field.

What Uncertainty Is

Uncertainty is best defined by what it is not: certainty is a probability of one or zero, and everything between is uncertain to a degree. The mind does not represent that degree as mere doubt but as a graded expectation, closer to a probability than to a yes-or-no. Formalising the degree was the achievement of information theory. Shannon defined the entropy of a distribution as the average surprise of its outcomes, and showed that this single quantity measures how much is unknown before an observation and how much is learned after one (Shannon, 1948). For a binary event with probability p, entropy is highest at p = 0.5, where the outcome is least predictable, and falls to zero as p approaches either extreme. Entropy gives uncertainty a currency: bits, the number of yes-or-no questions needed on average to resolve it.

Entropy alone, however, does not capture the distinction Knight drew. A fair coin and a coin of wholly unknown bias can share the same first-order entropy of one bit, yet they are not equally uncertain, because in the second case the probability itself is in doubt. The modeling literature separates these as risk, uncertainty about the outcome given a known probability, and ambiguity, uncertainty about the probability (Camerer & Weber, 1992). Ambiguity is, in effect, uncertainty of a higher order: a spread over the possible probabilities rather than over the outcomes. That higher-order spread is what a purely outcome-based measure like entropy misses, and it is what most sharply distinguishes human choice from the predictions of classical expected-utility theory.

Risk and Ambiguity

The cleanest demonstration that risk and ambiguity are psychologically distinct is the paradox Ellsberg posed against the axioms Savage had laid down for rational choice. An urn holds ninety balls: thirty are red, and the remaining sixty are black and yellow in an unknown proportion. Offered a bet that pays on red versus one that pays on black, most people take red, the known one-in-three chance. Offered a bet on red or yellow versus black or yellow, the same people switch to black-or-yellow, the known two-in-three chance. No single assignment of probabilities to black and yellow can rationalise both choices at once (Ellsberg, 1961). The pattern is not an error of arithmetic; it is a systematic preference for known odds over unknown ones, now called ambiguity aversion.

Ambiguity aversion is robust, and it is a preference, not a confusion: people display it while understanding the bets perfectly. Modeling it requires letting the decision weight on an event depend not only on its judged probability but on how confidently that probability is held (Camerer & Weber, 1992). The most influential such account is the two-stage model, in which a person first judges the probability of an uncertain event and then transforms that judgment through the same nonlinear weighting function that governs choice under risk (Tversky & Fox, 1995). Ambiguity aversion, in this view, is not a separate faculty but the compound of an ordinary probability judgment and the ordinary curvature of the weighting function, with source-dependent sensitivity added on top.

Judgment Under Uncertainty

Where the probabilities of events are not given, people must estimate them, and the estimates are produced by a small set of heuristics rather than by the calculus of probability. Representativeness judges likelihood by resemblance to a prototype, availability by the ease with which instances come to mind, and anchoring by adjustment from a salient starting value (Tversky & Kahneman, 1974). Each is efficient and usually serviceable, and each produces a characteristic bias: base rates are neglected when a description is representative, frequencies are overestimated when instances are vivid, and estimates cling to arbitrary anchors. These are not failures of motivation but signatures of the machinery that generates probability judgments in the first place.

The heuristics matter for uncertainty specifically because they determine the input to every later stage of choice. A weighting function, however well calibrated, operates on the probability the person actually believes, and that belief is a heuristic construction. When the two-stage model measures how judged probabilities are weighed, it inherits whatever distortions representativeness and availability introduced upstream (Tversky & Fox, 1995). Uncertainty is thus doubly transformed: first in the estimation of the odds, then in their weighting, and the observable choice reflects the composition of both.

Uncertainty in the Brain

If the mind estimates and weighs uncertainty, some neural system must compute it, and the evidence is that several do. When people choose between gambles whose odds are known and gambles whose odds are ambiguous, the level of ambiguity tracks activity in the amygdala and the orbitofrontal cortex, while the striatum follows the expected reward; lesions to the orbitofrontal region blunt the normal sensitivity to ambiguity (Hsu et al., 2005). Uncertainty, in other words, is not a nuisance the valuation system tolerates but a variable it explicitly encodes, with partly separable circuits for how much is at stake and how confidently the odds are known.

At the level of perception, uncertainty is represented even more directly. A population of noisy neurons, on the probabilistic-population-coding account, does not encode a single estimated value but an entire probability distribution over values, with the width of the distribution standing for the uncertainty of the estimate; combining two such populations implements Bayes' rule automatically, weighting each cue by its reliability (Ma et al., 2006). This gives uncertainty a concrete neural currency at the earliest stages of processing. Higher in the system, distinct signals index distinct varieties of uncertainty, so that expected noise, volatility, and ambiguity are dissociable rather than folded into one quantity (Bach & Dolan, 2012). The brain appears to keep its uncertainties itemised, because acting well requires knowing not just that one is unsure but of what kind.

Learning Under Uncertainty

Uncertainty is not static; it is the quantity learning exists to reduce, and a good learner tunes its own rate of updating to how uncertain it currently is. The decisive demonstration tracked people predicting outcomes in an environment whose reward probabilities were sometimes stable and sometimes rapidly changing. When the environment became volatile, participants raised their learning rate, giving recent outcomes more weight, exactly as a Bayesian observer should; activity in the anterior cingulate cortex tracked the estimated volatility that governed the adjustment (Behrens et al., 2007). The learning rate is thus not a fixed trait but a controlled variable, set by a second-order estimate of how fast the world is moving.

This requires the brain to distinguish two reasons a prediction might fail. Expected uncertainty is the known, irreducible noise of a stable environment, and a surprise it produces should be largely ignored. Unexpected uncertainty is a surprise too large to be noise, a signal that the underlying situation has changed and old learning should be discarded (Soltani & Izquierdo, 2019). Telling them apart is hard, and misattributing a change to noise, or noise to a change, is a generic failure mode of adaptive systems. The most general framing casts the whole enterprise as uncertainty minimisation: perception and action alike serve to reduce the discrepancy between predictions and inputs, a quantity that bounds the organism's long-run surprise (Friston, 2010). On this view uncertainty is not one problem the brain solves among others but the problem around which its architecture is organised.

Intolerance of Uncertainty

How aversively uncertainty is experienced varies across people, and the variation is clinically consequential. Anticipating a possible harm whose timing and likelihood are unknown recruits a coordinated set of processes — inflated estimates of threat, heightened physiological arousal, and avoidance — that is adaptive in moderation and pathological in excess (Grupe & Nitschke, 2013). The disposition to find uncertainty itself distressing, independent of the feared outcome, is measured as intolerance of uncertainty, and it behaves as a transdiagnostic construct: elevated across generalised anxiety, social anxiety, obsessive-compulsive disorder, and depression rather than specific to any one (Carleton, 2016). The proposal that intolerance of uncertainty may be a fundamental fear, as basic as the fear of pain, reframes a spread of disorders as downstream expressions of a single decision-theoretic vulnerability, and it has made uncertainty a target for treatment in its own right rather than a byproduct of whatever is feared.

Worked Example

Consider two urns, each holding one hundred balls that are either red or black. The known urn is announced to hold fifty of each. The unknown urn holds red and black in a proportion never disclosed. A bet pays one hundred dollars if a ball of a colour the chooser names in advance is drawn.

Under expected value the two bets are identical. For the known urn the probability of the named colour is exactly 0.5, so the expected payoff is 0.5 × $100 = $50. For the unknown urn, a chooser with no reason to favour red over black should, by the principle of insufficient reason, also treat the named colour as equiprobable, again giving an expected payoff of $50. The first-order entropy is identical too: at p = 0.5 each bet carries −0.5 log₂ 0.5 − 0.5 log₂ 0.5 = 1 bit of outcome uncertainty. On every quantity classical theory recognises, the bets are the same.

They differ only in a quantity that classical theory ignores: the second-order spread over the probability. The known urn fixes the probability at 0.5 with no spread at all. The unknown urn, modelled with a uniform prior over its composition, has an expected probability of 0.5 but a variance of 1/12 ≈ 0.0833, a standard deviation of about 0.289 — an entire distribution of possible odds where the known urn has a point. Ambiguity aversion is the willingness to pay to avoid that spread. A chooser who would sacrifice up to ten dollars to switch from the unknown urn to the known one reveals a ten-dollar ambiguity premium, a preference invisible to expected value and to first-order entropy alike (Ellsberg, 1961). Sampling collapses the spread: drawing one red ball from the unknown urn updates the uniform prior to an expected probability of 2/3 ≈ 0.667, so that a few observations turn ambiguity back into something close to risk.

Discussion

The recurring lesson across these literatures is that uncertainty is plural. Treating it as a single scalar — a probability, or its entropy — captures the mind's behaviour under risk and fails everywhere else, because people respond not only to how likely an outcome is but to how firmly that likelihood is known, where the estimate came from, and whether a surprise signals noise or change. The neural evidence reinforces the point: the brain does not maintain one uncertainty register but several, dissociating ambiguity from volatility from irreducible noise (Bach & Dolan, 2012), because behaving adaptively requires different responses to each. Overweighting a genuine change as noise leaves a learner stubbornly out of date; overweighting noise as change leaves it thrashing. Table 1 sets the principal varieties side by side, with the response each demands.

That plurality connects the decision-theoretic and clinical faces of the topic. The same second-order uncertainty that makes an ambiguous bet aversive, when chronic and dispositional, becomes intolerance of uncertainty and a marker of anxiety risk (Carleton, 2016). A construct first isolated in gambling experiments turns out to index vulnerability to disorder, which is why uncertainty has become a bridge between the study of normal choice and the study of psychopathology. The unifying theoretical claim — that the nervous system exists to minimise surprise (Friston, 2010) — is contested in its strong form, but its weaker reading is now common ground: estimating and reducing uncertainty is not a peripheral task but close to the core of what cognition is for.

Current Directions

Recent work has sharpened the distinction between the kinds of uncertainty a learner must track. A leading model separates stochasticity, the irreducible noise in outcomes, from volatility, the rate at which the underlying situation changes, and shows that estimating the two jointly is necessary because they pull the learning rate in opposite directions: volatility should raise it, stochasticity should lower it, and confusing the two produces the maladaptive updating seen in anxiety and depression (Piray & Daw, 2021). The framework recovers a range of otherwise puzzling learning effects as consequences of misestimating which kind of uncertainty is present, and it has given computational psychiatry a precise vocabulary for what goes wrong.

A second front extends uncertainty into the social domain, where the quantities to be inferred are other minds. Reducing uncertainty about what another person believes, wants, or will do engages the same estimation-and-updating machinery as physical uncertainty, but the target is a hidden mental state rather than a hidden urn, and the volatility of other agents is often higher and strategically produced (FeldmanHall & Shenhav, 2019). Casting social cognition as uncertainty reduction links it to the neural and computational accounts developed for perception and reward, and suggests that the machinery for coping with an unpredictable world was recruited, largely intact, for the harder problem of coping with unpredictable people.

Common Misconceptions

Uncertainty and risk are the same thing.

Risk is a special case in which the probabilities are known; uncertainty in the strict sense refers to situations where they are not. The two are chosen between differently, and people will pay to avoid unknown odds even when the known odds are identical (Ellsberg, 1961). Collapsing the distinction discards the very fact that makes human choice under uncertainty distinctive.

People weigh probabilities as they are stated.

Judged probabilities are constructed by heuristics such as representativeness and availability, so they diverge systematically from the stated odds before any weighting occurs (Tversky & Kahneman, 1974). What later choice acts on is the believed probability, not the announced one.

More surprise always means the environment has changed.

A large prediction error can mean either that the world has changed or merely that it is noisy, and treating every surprise as change makes a learner unstable. Adaptive systems must estimate expected and unexpected uncertainty separately and attribute surprise accordingly (Soltani & Izquierdo, 2019).

Glossary

Ambiguity aversion.

The systematic preference for known probabilities over unknown ones, even when the known odds confer no advantage.

Ambiguity, second-order.

Uncertainty about the probabilities themselves, as when the composition of an urn is undisclosed; a higher-order uncertainty distinct from risk.

Bayesian inference.

The updating of a prior probability distribution into a posterior in proportion to the likelihood of the observed evidence.

Ellsberg paradox.

A choice pattern in which preferences among bets on an urn cannot be reconciled with any single assignment of probabilities, revealing ambiguity aversion.

Entropy.

The average surprise of a probability distribution, measured in bits; a formal measure of uncertainty that is greatest at even odds.

Expected uncertainty.

The known, irreducible noise of a stable environment; surprises it produces should be largely ignored rather than learned from.

Expected value.

The probability-weighted average of an outcome's possible payoffs; the classical benchmark that ambiguity aversion violates.

Intolerance of uncertainty.

A dispositional tendency to find uncertainty itself distressing; a transdiagnostic marker elevated across the anxiety disorders and depression.

Knightian uncertainty.

Immeasurable uncertainty, in which probabilities cannot be assigned, as opposed to measurable risk; named for the economist Frank Knight.

Precision.

The inverse of variance; in predictive accounts, the weight assigned to a signal is set by its precision, so uncertain signals count for less.

Prediction error.

The discrepancy between a predicted and an observed outcome; the quantity learning uses to update beliefs, scaled by the learning rate.

Probability weighting.

The nonlinear transformation by which stated probabilities become decision weights, overweighting small chances and underweighting large ones.

Risk.

Uncertainty about an outcome when its probability is known, as in a fair wheel or an announced fifty-fifty urn.

Uncertainty.

The condition in which reliable knowledge of a present state or future outcome is unavailable; the general class of which risk and ambiguity are members.

Unexpected uncertainty.

A surprise too large to be noise, signalling that the underlying situation has changed and that prior learning should be discarded.

Volatility.

The rate at which the state of the environment changes; higher volatility warrants a higher learning rate, weighting recent evidence more.

Key Researchers

Timothy E. J. Behrens (b. 1976). Professor of Computational Neuroscience at the University of Oxford; with colleagues he showed that people track environmental volatility and raise their learning rate when the world becomes less predictable. Wikipedia - Faculty Page - ORCID

Colin F. Camerer (b. 1959). Robert Kirby Professor of Behavioral Economics at the California Institute of Technology; he helped separate risk, uncertainty, and ambiguity in formal models and co-authored the neuroimaging of ambiguity aversion. Wikipedia - Faculty Page - ORCID

R. Nicholas Carleton (b. 1974). Professor of Psychology at the University of Regina; he advanced intolerance of uncertainty as a transdiagnostic construct and proposed it may be a fundamental fear. Faculty Page - ORCID

Daniel Ellsberg (1931-2023). Economist and RAND analyst; his two-urn paradox demonstrated that ambiguity aversion violates the Savage axioms of rational choice. Wikipedia

Craig R. Fox (b. 1966). Professor at the UCLA Anderson School of Management; with Tversky he built the two-stage model of decision under uncertainty. Faculty Page - ORCID

Karl J. Friston (b. 1959). Professor of Neuroscience at University College London; he formulated the free-energy principle, casting perception and action as the minimisation of uncertainty. Wikipedia - Faculty Page - ORCID

Ming Hsu (b. 1978). Professor at the University of California, Berkeley; he led the study dissociating the neural response to ambiguity from the response to risk. Faculty Page - ORCID

Daniel Kahneman (1934-2024). Eugene Higgins Professor of Psychology, Emeritus, at Princeton University; with Tversky he documented the heuristics by which people judge probability under uncertainty. Nobel laureate in Economic Sciences, 2002. Wikipedia

Frank H. Knight (1885-1972). Economist at the University of Chicago; he drew the founding distinction between measurable risk and immeasurable uncertainty. Wikipedia

Wei Ji Ma (b. 1978). Professor of Neural Science and Psychology at New York University; he showed that populations of noisy neurons can represent probability distributions and implement Bayesian inference. Wikipedia - Faculty Page - ORCID

Leonard J. Savage (1917-1971). Mathematician and statistician at Yale University; he built the axiomatic foundation of subjective expected utility that Ellsberg's paradox was designed to test. Wikipedia

Amos Tversky (1937-1996). Davis-Brack Professor of Behavioral Sciences at Stanford University; he co-founded the heuristics-and-biases program and, with Fox, the two-stage model of judgment under uncertainty. Wikipedia

Frequently Asked Questions

What is the difference between risk and uncertainty?
Risk refers to situations in which outcome probabilities are known, such as a fair coin, whereas uncertainty in the strict sense refers to situations in which the probabilities are themselves unknown, and people treat the two differently (Ellsberg, 1961).

How is uncertainty measured?
Shannon entropy measures the uncertainty of a probability distribution as the average surprise of its outcomes, expressed in bits, and reaches its maximum when outcomes are equally likely (Shannon, 1948).

What is ambiguity aversion?
Ambiguity aversion is the tendency to prefer bets with known probabilities over bets with unknown probabilities even when the known odds offer no advantage, and it is a preference rather than a misunderstanding (Ellsberg, 1961).

Do people judge probabilities accurately?
People estimate probabilities using heuristics such as representativeness and availability, which are efficient but produce systematic biases, so judged probabilities can diverge from the true odds (Tversky & Kahneman, 1974).

Does the brain have a dedicated system for uncertainty?
Ambiguity in a choice tracks activity in the amygdala and orbitofrontal cortex while expected reward tracks the striatum, indicating that uncertainty is explicitly encoded rather than merely tolerated (Hsu et al., 2005).

Why should a learning rate change with the environment?
When an environment becomes volatile, recent outcomes are more informative than old ones, so raising the learning rate is the statistically correct response, which people approximate (Behrens et al., 2007).

What is the difference between expected and unexpected uncertainty?
Expected uncertainty is the known noise of a stable environment, which should be discounted, whereas unexpected uncertainty is a surprise large enough to signal that the environment itself has changed (Soltani & Izquierdo, 2019).

How does uncertainty relate to anxiety?
A dispositional difficulty tolerating uncertainty, called intolerance of uncertainty, is elevated across the anxiety disorders and depression and may act as a transdiagnostic vulnerability (Carleton, 2016).

References