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The Well-Calibrated Bayesian
Abstract
Suppose that a forecaster sequentially assigns probabilities to events. He is well calibrated if, for example, of those events to which he assigns a probability 30 percent, the long-run proportion that actually occurs turns out to be 30 percent. We prove a theorem to the effect that a coherent Bayesian expects to be well calibrated, and consider its destructive implications for the theory of coherence.
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“of uncertainty [6], [27], [28]; measured by proper scoring rules like log-loss or Brier scoring.”
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Iturrieta, P. C., Bayona, J. A., Werner, M. J., Schorlemmer, D., Taroni, M., Falcone, G., Cotton, F., Asim, K. M., Savran, W. H., & Marzocchi, W. (2024) · Seismological Research Letters · doi:10.1785/0220230247 · Cited by 13
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“A model can be called well-calibrated if the confidence scores of its predictions correspond to the actual prediction accuracy in the long run (Dawid, 1982).”
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“A well-calibrated classifier prediction is such that the predicted probability of an event is close to the proportion of the those events among a group of similar predictions [7].”
BackgroundSoll, J. B., Palley, A. B., Klayman, J., & Moore, D. A. (2019) · Management Science · doi:10.1287/mnsc.2019.00660 · Cited by 13
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Economic Darwinism: Who has the Best Probabilities?
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Inducing Leaders to Take Risky Decisions: Dismissal, Tenure, and Term LimitsPreprint
Aghion, P., & Jackson, M. O. (2014) · SSRN Electronic Journal · doi:10.2139/ssrn.2441421 · Cited by 13
“This is related to the literature on “calibration”and the literature that followed (e.g., Dawid (1982), Foster and Vohra (1998), Sandroni, Smorodinsky, and Vohra (2003), Dekel and Feinberg (2006)).”
Making Decisions Using Sets of Probabilities: Updating, Time Consistency, and CalibrationPreprint
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“As pointed out by Dawid (1982), an agent updating her beliefs and making decisions on the basis of these beliefs should also be concerned about being calibrated.”
BackgroundInfluentialAvilés-Rivero, A. I., Papadakis, N., Li, R., Sellars, P., Alsaleh, S. M., Tan, R. T., & Schönlieb, C. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1906.08635 · Cited by 13
“Our work is motivated by the problems of network calibration and the confirmation bias in pseudolabelling [26], [27], [28], [29], where one seeks that the probability of the predicted label reflects the ground truth correctness likelihood.”
BackgroundFair Machine Learning in Healthcare: A Survey
Feng, Q., Du, M., Zou, N., & Hu, X. (2025) · IEEE Transactions on Artificial Intelligence · doi:10.1109/tai.2024.3361836 · Cited by 12
“Calibration, for instance, aims to align the proportion of positive predictions with the actual rate of positive outcomes across various subgroups [42].”
Partial information framework: Model-based aggregation of estimates from diverse information sources
Satopää, V., Jensen, S. T., Pemantle, R., & Ungar, L. (2017) · Electronic Journal of Statistics · doi:10.1214/17-ejs1346 · Cited by 12
Coherence and calibration in expert probability judgement
Bolger, F., & Wright, G. (1993) · Omega · doi:10.1016/0305-0483(93)90005-6 · Cited by 12
Sandroni, A., & Rann Smorodinsky, B. (2003) · Games and Economic Behavior · doi:10.1016/s0899-8256(03)00152-0 · Cited by 12
“In a Nash equilibrium all players take best responses to correct b However, it may be impossible to infer from the observed data if a player is or i randomizing.”
BackgroundInfluentialGeneral Representation of Epistemically Optimal Procedures
Dietrich, F. K. (2006) · Social Choice and Welfare · doi:10.1007/s00355-006-0094-2 · Cited by 12
“10My condition of Calibration (C) is related to, but not identical with the concept of well-calibrated probability assignments in the literature (e.g. Dawid (1982)).”
MethodThe Value of a Probability Forecast from Portfolio Theory
Johnstone, D. J. (2007) · Theory and Decision · doi:10.1007/s11238-006-9023-1 · Cited by 12
Li, L., Neal, R. M., & Zhang, J. (2008) · Bayesian Analysis · doi:10.1214/08-ba307 · Cited by 12
“Bias in this respect is also referred to as a lack of “calibration” — that is, the predictive probabilities do not represent the actual chances of events (Dawid 1982).”
BackgroundConstructing efficient belief network structures with expert provided information
Sarkar, S., & Murthy, I. (1996) · IEEE Transactions on Knowledge and Data Engineering · doi:10.1109/69.485642 · Cited by 12
The complexity of forecast testing
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“Dawid [5] was the first to define calibration, with numerous algorithms later given by Foster and Vohra [8], Fudenberg and Levine [10], Hart and Mas-Colell [12] and more (see e.”
MethodMoment Multicalibration for Uncertainty EstimationPreprint
Jung, C., Lee, C., Pai, M. M., Roth, A., & Vohra, R. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2008.08037 · Cited by 12
“This fundamental problem with uncertainty estimation in non-parametric settings is also a problem for mean estimation: what does it mean that a point prediction μ(x) is an estimate of E[y|x] if we have no knowledge of the distribution on y conditional on x (because we have observed no samples from this distribution)?…”
BackgroundMethodInfluentialRöchner, P., Marques, H. O., Campello, R. J. G. B., & Zimek, A. (2024) · Data Mining and Knowledge Discovery · doi:10.1007/s10618-024-01056-5 · Cited by 10
“Following DeGroot and Fienberg (1982) for evaluating probability estimates in supervised classification, we require good outlier probabilities to be refined and calibrated (Dawid 1982), also called reliable (Murphy 1973).”
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Approachability, fast and slow
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A Unifying Theory of Distance from Calibration
Blasiok, J., Gopalan, P., Hu, L., & Nakkiran, P. (2022) · Symposium on the Theory of Computing · doi:10.1145/3564246.3585182 · Cited by 71
“The most basic requirement of this form is calibration [8].”
BackgroundA Philosopher’s View on Bayesian Evaluation of Informative Hypotheses
Romeijn, J., & van de Schoot, R. (2008) · Springer eBooks · doi:10.1007/978-0-387-09612-4_16 · Cited by 11
“In the words of Dawid [7], the Bayesian is “well-calibrated”: Inherent to the choice of set of hypotheses (i.”
BackgroundCalibrated Bayes, an inferential paradigm for official statistics in the era of big data
Little, R. J. A. (2015) · Statistical Journal of the IAOS · doi:10.3233/sji-150944 · Cited by 11
“This capitalizes on the strengths of both paradigms, and is the essence of Calibrated Bayes (CB) [5,12,14,42,51, 64].”
MethodSmooth calibration, leaky forecasts, finite recall, and Nash dynamics
Foster, D. P., & Sergiu Hart, A. (2018) · Games and Economic Behavior · doi:10.1016/j.geb.2017.12.022 · Cited by 11
“Thus, in particular, there cannot be deterministic calibrated procedures (because there are no leaky calibrated procedures); see Dawid (1985) and Oakes (1985). In the case of smooth calibration, the procedure that we construct is deterministic, and thus leaky smoothly calibrated. However, there are also randomized lea…”
BackgroundMethodInfluentialPrice probabilities: a class of Bayesian and non-Bayesian prediction rules
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“As Dawid (1982) eloquently said: “If a subjective distribution P attaches probability zero to a non-ignorable event, and if this event happens, then P must be treated with suspicion, and modified or replaced.””
BackgroundRandomness is inherently imprecise
de Cooman, G., & Bock, J. D. (2021) · International Journal of Approximate Reasoning · doi:10.1016/j.ijar.2021.06.018 · Cited by 11
Non-stochastic infinite and finite sequences
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Walker, S. G., & Gutiérrez‐Peña, E. (2007) · Test · doi:10.1007/s11749-006-0008-8 · Cited by 11
“Thus, a forecaster “is well calibrated if, for example, of those events to which he assigns a probability of 30 percent, the long run proportion that actually occurs turns out to be 30 percent” ( Dawid 1982 ).”
BackgroundLittle, R. J. A. (1988) · Computational Statistics & Data Analysis · doi:10.1016/0167-9473(88)90090-4 · Cited by 11
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Model complexity validation for PDF estimation using Gaussian mixtures
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“The test involves a calibration [12] of the data frequencies, predicted by the estimator.”
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“This requires that the uncertainty in a prediction properly represents the probability that the prediction is correct, for example the 95% confidence interval should contain the correct answer 95% of the time [14, 15].”
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Gupta, C., & Ramdas, A. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2107.08353 · Cited by 11
“7 was the ‘true probability of rain’ without making unverifiable assumptions [Dawid, 1982].”
BackgroundCombining probability forecasts
Ranian, R. (2016) · Cited by 11
Pei, H., Xiong, Y., Wang, P., Tao, J., Liu, J., Deng, H., Ma, J., & Guan, X. (2024) · International World Wide Web Conference (WWW) · doi:10.1145/3589334.3645398 · Cited by 10
“Here, the calibration measures the discrepancy between the model’s confidence score in its predictions and the actual correctness of these predictions [8].”
BackgroundTowards Unraveling Calibration Biases in Medical Image Analysis
Lara, M. A. R., Mosquera, C., Ferrante, E., & Echeveste, R. (2023) · Lecture notes in computer science · doi:10.1007/978-3-031-45249-9_13 · Cited by 10
Frequentist confidence intervals for orbits
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Diniz, M. A., Izbicki, R., Lopes, D., & Salasar, L. E. (2018) · Journal of the Operational Research Society · doi:10.1080/01605682.2018.1457485 · Cited by 12
“with P(j) = (P(j) 1 ;P (j) 2 ;P (j) 3 ), is dened by 1 n Xn j=1 I X j 6= arg max x2f1;2;3g P(j) x ; where X j is the outcome of the j-th match. The second concept we use is that of calibration (Dawid, 1982). Probability assertions are said to be well calibrated at the level of probability pif the observed proportion o…”
BackgroundDichotomization of ICU Length of Stay Based on Model Calibration
Verduijn, M., Peek, N. B., Voorbraak, F. P. J. M., de Jonge, E., & de Mol, B. A. J. M. (2005) · Lecture notes in computer science · doi:10.1007/11527770_10 · Cited by 10
“The concept of model calibration concentrates on the difference between true classconditional probabilities P (Yt = 1|x) and probabilities M(Yt = 1|x) estimated by model M [10].”
The Need for More Emphasis on Prediction: A “Nondenominational” Model-Based Approach
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“The one aspect of performance in repeated application that seems to have gained acceptance as a legitimate concern among at least some Bayesians is that of being “well-calibrated”—refer, for example, to Dawid (1982), Rubin (1984), and Little (2006).”
BackgroundOn Supervised Learning of Bayesian Network Parameters
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“In this work, we focus on two aspects of uncertainty, often required addressing in practical uses of neural networks: calibration [2] and out-of-distribution (OOD) detection.”
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“We also validate the correct statistical properties of our implementation of the likelihood using a well-calibrated simulation study [50].”
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“Finally, there is a related literature within the statistical community—classic examples are Dawid (1982) and Efron (1986). Hunt and Mostyn 87”
BackgroundStopping rules and the likelihood function
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Unal, R., Keating, C. B., Chytka, T., & Conway, B. A. (2005) · Engineering Management Journal · doi:10.1080/10429247.2005.11415285 · Cited by 9
“Dawid (1982) makes a case that a well-calibrated assessor will likely not be coherent, because any recalibration required to bring prediction more in line with reality leads to incoherence.”
BackgroundMethodTesting Multiple ForecastersPreprint
Feinberg, Y., & Stewart, C. (2007) · SSRN Electronic Journal · doi:10.2139/ssrn.961856 · Cited by 9
“Thus we modify the inequality in (3) to a strict inequality (on one side of the interval) when needed, to reflect that the empirical frequency must converge to the appropriate interval.”
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Self-citationClosed-Form Power and Sample Size Calculations for Bayes Factors
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“Under this distribution, the distribution of the Bayes factor with normal analysis prior (2) can be derived in closed-form (Weiss, 1997; De Santis, 2004).”
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On calibration error of randomized forecasting algorithms
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“The identification of such a factor is the calibration meant by the title of this section, common practice in Statistics, supported by a vast literature (Brown, 1993; Dawid, 1982; Hastie & Tibshirani, 1998).”
BackgroundSome aspects of probability forecasting
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“But the fact that Bayesian inference is in this sense dogmatic is at the origin of many other criticisms, including the criticism of Dawid (1982)”
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Comparing the Accuracy of Default Predictions in the Rating Industry: The Case of Moody's vs. S&P
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“Such forecasters are called ”well calibrated” (Dawid 1982).”
BackgroundDawes, R. M. (2015) · Elsevier eBooks · doi:10.1016/b978-0-08-097086-8.42162-8 · Cited by 8
Diagnostic systems by model selection: a case study
Lauritzen, S. L., Thiesson, B., & Spiegelhalter, D. J. (1994) · Lecture Notes in Statistics · doi:10.1007/978-1-4612-2660-4_15 · Cited by 15
Reconciling Individual Probability Forecasts✱
Roth, A., Tolbert, A. W., & Weinstein, S. A. (2023) · ACM Conference on Fairness, Accountability, and Transparency (FAccT) · doi:10.1145/3593013.3593980 · Cited by 7
“[11] proposed calibration as a desirable frequentist condition for evaluating probabilistic forecasts: roughly speaking that the outcome being forecast should have appeared with empirical frequency p conditional on the forecaster predicting probability p of the outcome, simultaneously for all predictions p .”
BackgroundKim, J., Cha, J., Kim, T., Lee, H., Yu, H.-Y., & Suh, M. (2024) · Engineering Applications of Artificial Intelligence · doi:10.1016/j.engappai.2024.108967 · Cited by 7
SciPhy: A Bayesian phylogenetic framework using sequential genetic lineage tracing data
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“We validated SciPhy using well-calibrated simulations 16 to ensure that it draws samples from the correct posterior distribution.”
Adapting Ensemble‐Calibration Techniques to Probabilistic Solar‐Wind Forecasting
Edward-Inatimi, N. O., Owens, M. J., Barnard, L., Turner, H., Marsh, M. S., Gonzi, S., Lang, M. S., & Riley, P. (2024) · Space Weather · doi:10.1029/2024sw004164 · Cited by 7
“…is used to infer biases towards over/under confident predictions and can also distinguish between forecasts which are well calibrated but differ in their decisiveness; a model’s abil-273 ity to assign more extreme probabilities to events with high confidence (Dawid, 1982; DeGroot & Fienberg, 1983).”
Evidential time-to-event prediction with calibrated uncertainty quantification
Huang, L., Xing, Y., Mishra, S., Denœux, T., & Feng, M. (2025) · International Journal of Approximate Reasoning · doi:10.1016/j.ijar.2025.109403 · Cited by 7
On the Limits of Language Generation: Trade-Offs between Hallucination and Mode-Collapse
Kalavasis, A., Mehrotra, A., & Velegkas, G. (2025) · ACM Symposium on Theory of Computing (STOC) · doi:10.1145/3717823.3718108 · Cited by 31
“…for this work: a language model is calibrated if, roughly speaking, the strings that the model assigns probability mass 𝑝 , appear in a 𝑝 fraction of the true distribution [31]. next element in the order (since they allow both continuations) and, hence, the target sequence cannot be identified.”
Robust Bayesian inference for Big Data: Combining sensor-based records with traditional survey data
Rafei, A., Flannagan, C. A. C., West, B. T., & Elliott, M. R. (2022) · The Annals of Applied Statistics · doi:10.1214/21-aoas1531 · Cited by 7
Economic Theory: Economics, Methods and Methodology
Gilboa, I., Postlewaite, A., Samuelson, L., & Schmeidler, D. (2023) · Revue économique · doi:10.3917/reco.736.0897 · Cited by 8
On the trade-off between ranking effectiveness and fairness
Melucci, M. (2023) · Expert Systems with Applications · doi:10.1016/j.eswa.2023.122709 · Cited by 7
Forecast Hedging and Calibration
Foster, D. P., & Sergiu Hart, A. (2021) · Journal of Political Economy · doi:10.1086/716559 · Cited by 7
“11 Dawid (1982) discusses the connection of calibration to posterior probabilities, whereas here we want to connect it to the priors.”
BackgroundStronger calibration lower bounds via sidestepping
Qiao, M., & Valiant, G. J. (2021) · ACM Symposium on Theory of Computing (STOC) · doi:10.1145/3406325.3451050 · Cited by 7
“Calibration was later studied by Dawid [2] from a Bayesian perspective.”
Benedetti, M. H., Berrocal, V. J., & Little, R. J. A. (2022) · The Annals of Applied Statistics · doi:10.1214/21-aoas1585 · Cited by 7
Imprecise probabilistic inference from sequential data.
Prat-Carrabin, A., & Woodford, M. (2024) · Psychological Review · doi:10.1037/rev0000469 · Cited by 8
Calibration: Respice, Adspice, Prospice
Foster, D. P., & Vohra, R. (2013) · Cambridge University Press eBooks · doi:10.1017/cbo9781139060011.014 · Cited by 7
“Dawid [5] offers the following intuitive definition of calibration:”
BackgroundFrequentist confidence intervals for orbits
Lucy, L. B. (2014) · Springer Link (Chiba Institute of Technology) · doi:10.1051/0004-6361/201423661/pdf · Cited by 7
Calibrated Seismic Verification of the Threshold Test Ban Treaty
Picard, R. R., & Bryson, M. C. (1992) · Journal of the American Statistical Association · doi:10.1080/01621459.1992.10475208 · Cited by 7
“The alternative-obtaining a "posterior" distribution other than by Bayes theorem-has its own shortcomings (Dawid 1982).”
Background
Metadata
- Authors
- Dawid, A. P.
- Affiliations
- University College London (GB)
- Corresponding author
- Dawid, A. P.
- ORCID
- Dawid, A. P.
- Venue
- Journal of the American Statistical Association
- Publisher
- Informa UK Limited
- Published
- 1982-09
- Volume
- 77
- Issue
- 379
- Pages
- 605-610
- ISSN
- 0162-1459 (ISSN-L)
- 1537-274X
- OpenAlex
- W2023943903
- Cited by
- 625
- References
- 24
- Citations by year
- 2026: 22 · 2025: 26 · 2024: 45 · 2023: 51 · 2022: 27 · 2021: 31 · 2020: 39 · 2019: 18 · 2018: 19 · 2017: 15 · 2016: 13 · 2015: 18 · 2014: 23 · 2013: 18 · 2012: 17
- FWCI
- 6.76
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of the American Statistical Association (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Bayesian Modeling and Causal Inference
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Social Sciences › Decision Sciences › Management Science and Operations Research › Forecasting Techniques and Applications
- Keywords
- well calibration, Bayesian forecasting, coherence, subjective probability, Dutch book argument
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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