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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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“One possibility is a computational formulation of the notion of ‘calibration’ from Bayesian analysis [10].”
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“If the predicted probability distribution is consistent with observed empirical frequencies of labels, the predictor is said to be calibrated [Dawid, 1982].”
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“Given a data distribution D of ( x, y ) ∈ X × { 0 , 1 } and let c be the predictive confidence, the predictor f : X → [0 , 1] is said to be calibrated [6], if: Notation.”
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“Model calibration is a crucial step in ensuring the accuracy and reliability of predictive models [18] .”
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“The quantitative study of event forecasting dates back at least three decades (Dawid, 1982; Murphy and Winkler, 1987).”
BackgroundEnsemble Bayesian forecasting system Part II: Experiments and properties
Herr, H. D., & Krzysztofowicz, R. (2019) · Journal of Hydrology · doi:10.1016/j.jhydrol.2019.02.057 · Cited by 6
A Constructivist View of the Statistical Quantification of Evidence
Hennig, C. M. (2009) · Cited by 6
Failure of calibration is typical
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“…the data sequences encode, for each of some sequence of days, whether or not it snows in a given locale on that day (1=snow; 0=no snow); each day, the forecasting system is asked to output a probability for snow on that day, given the pattern of snowy and snowless days in the extant data set.”
BackgroundFuzzy set theory, observer bias and probability theory
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Calibrating the prior distribution for a normal model with conjugate prior
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“Arguments in favour of this approach are discussed by Dawid [16] and Rubin [17].”
ResultDesiderata for a predictive theory of statistics
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“An alternative to the conventional model is provided by the Prequential approach, see Dawid (1984). The Prequential Principle comes in three forms, weak, strong and super-”
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BackgroundOn the Possibility of Testimonial Justice
Stewart, R. T., & Nielsen, M. (2020) · Australasian Journal of Philosophy · doi:10.1080/00048402.2019.1706183 · Cited by 5
“Calibration is still a plausible and widely-endorsed epistemic requirement (e.g., Dawid, 1982; van Fraassen, 1983; Shimony, 1988; Lange, 1999; Tetlock, 2005).6 On this way of viewing things, Theorem 1 shows that there is a conflict between requirements arising from considerations of justice and…”
BackgroundThe Right to Be an Exception to a Data-Driven Rule
Cen, S. H., & Raghavan, M. (2023) · doi:10.21428/2c646de5.a15f7255 · Cited by 12
SDE-HNN: Accurate and Well-Calibrated Forecasting Using Stochastic Differential Equations
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“In the previous works [7, 8, 32], the calibration error is the equivalent accumulation of different quantile (confidence) deviation, which may be inaccurate when the data distribution is uneven.”
High-confidence predictions under adversarial uncertainty
Drucker, A. (2013) · ACM Transactions on Computation Theory · doi:10.1145/2493252.2493257 · Cited by 5
“The second strand of related work is the so-called forecasting problem in decision theory (see [4] for an early, influential discussion).”
BackgroundEstimating Uncertainty Online Against an Adversary
Kuleshov, V., & Ermon, S. (2017) · Proceedings of the AAAI Conference on Artificial Intelligence · doi:10.1609/aaai.v31i1.10949 · Cited by 6
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“Restricting inference to one measure space requires that our procedures be ‘well-calibrated’ in the sense made precise by Dawid (1982).”
MethodUniversal Algorithm for Trading in Stock Market Based on the Method of Calibration
V'yugin, V. V. (2013) · Lecture notes in computer science · doi:10.1007/978-3-642-40935-6_5 · Cited by 5
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“Dawid, The well-calibrated bayesian, Journal of the American Statistical Association 77 (1982), no.”
MethodRejoinder: Calibration-Based Empirical Probability
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Self-citationEhud Lehrer, A. (2004) · Mathematics of Operations Research · doi:10.1287/moor.1030.0087 · Cited by 5
“Smorodinsky and Weiss (1987) gave an explicit construction for the case of ergodic Markov chains which was inspired by Champernowne (1933).”
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“On the more theoretical side, Dawid (1982) show that an internally consistent Bayesian can expect to become calibrated under feedback.”
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Gavrikov, A., Cerrone, V., Serafini, A., Brugnera, R., Garfagnini, A., Grassi, M., Jelmini, B., Lastrucci, L., Aiello, S., Andronico, G., et al. (2024) · Physics Letters B · doi:10.1016/j.physletb.2024.139141 · Cited by 4
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“Calibration is deemed desirable in different contexts: statistical inference (e.g. Box, 1980; Dawid, 1982; Rubin, 1984; Gneiting, Balabdaoui and Raftery, 2007; Draper and Krnjajic, 2010), predictions in a game-theoric setting (e.g. Foster and Vohra, 1998; Vovk and Shafer, 2005) and machine learning…”
On the Richness of Calibration
Höltgen, B. T. A., & Williamson, R. C. (2023) · ACM Conference on Fairness, Accountability, and Transparency (FAccT) · doi:10.1145/3593013.3594068 · Cited by 5
“9Dawid [13] mentions a related but slightly different trade-off between robustness and incisiveness w.r.t. the amount of information taken into account for groups on which to calibrate forecasts. b) For any predictor 𝑝 : X → [0, 1], |𝑐 (𝐼1) | ≤ 𝛼 and |𝑐 (𝐼2) | ≤ 𝛽 entails |𝑐 (𝐼1 ∪ 𝐼2) | ≤ 𝛼 · |𝐼1 |+𝛽 · |𝐼2 ||𝐼1 |+|𝐼2…”
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Source-free domain adaptation requires penalized diversity
Sevyeri, L. R., Sheth, I., Farahnak, F., See, A., Kahou, S. E., Fevens, T., & Havaei, M. (2025) · Machine Learning · doi:10.1007/s10994-025-06863-5 · Cited by 4
“assumption that training and test sets are drawn from the same distribution, calibration (Dawid 1982) is introduced to the traditional machine learning paradigm to elucidate the model uncertainty.”
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“Dawid 1982 tells us a well calibrated model implies for any given win probability percentage x, if we look at all instances where a win probability of x% was predicted, then approximately x% of those games would result in a victory for the offensive team.”
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“Bayesian models will also demonstrate better predictive performance than frequentist models by accounting for uncertainty in both the parameters of models and the choice of models themselves, they are better calibrated to reality (Dawid, 1982; Kaplan, 2021).”
Calibrating Bayesian decoders of neural spiking activity
Wei, G., Mansouri, Z. T., Wang, X., & Stevenson, I. H. (2024) · Journal of Neuroscience · doi:10.1523/jneurosci.2158-23.2024 · Cited by 3
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“⟨ 𝑝 𝑡 𝑡 ⟩ .”
Resiliency, robustness and rationality of probability judgements
Logue, J. (1997) · International Studies in the Philosophy of Science · doi:10.1080/02698599708573548 · Cited by 4
McCullough, C. L., Novobilski, A. J., & Fesmire, F. M. (2007) · International Conference on Machine Learning and Applications · doi:10.1109/icmla.2007.40 · Cited by 4
Cross-calibration of categorical variables: An evaluation of the genetic algorithm approach
M’Hallah, R., & Aboukhamseen, S. M. (2018) · Applied Soft Computing · doi:10.1016/j.asoc.2018.10.009 · Cited by 4
Testing whether a Learning Procedure is Calibrated
Cockayne, J., Graham, M. M., Oates, C. J., Sullivan, T. J., & Teymur, O. (2020) · ePrints Soton (University of Southampton) · doi:10.48550/arxiv.2012.12670 · Cited by 4
A Conversation with Eugenio Regazzini
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“Our curiosity to this problem was stimulated by a paper of Phil Dawid (Dawid, 1982).”
MethodMaking Good Probability Estimates for Regression
Carney, M., & Cunningham, P. (2006) · Lecture notes in computer science · doi:10.1007/11871842_55 · Cited by 4
“Calibration , the second goal, refers to the property that if a predicted density function suggests P percent probability of occurrence, the event truly ought to have probability P of occurring [12].”
BackgroundSuki, N., Poh, N., Senan, F. M., Zamani, N. A., & Darus, M. (2016) · IEEE International Conference on Biometrics: Theory, Applications and Systems (BTAS) · doi:10.1109/btas.2016.7791192 · Cited by 4
“In addition, although our work echoes those of existing literature [27, 13, 3, 21, 4], our study also shows that likelihood ratio has its own uncertainties, in terms of confidence intervals [32].”
BackgroundCalibration and Internal no-Regret with Partial MonitoringPreprint
Perchet, V. (2010) · arXiv (Cornell University) · doi:10.48550/arxiv.1006.1746 · Cited by 4
“Indeed, calibration has been introduced by Dawid [8] for repeated games of predictions: at each stage, Nature chooses an outcome s in a finite set S and Predictor forecasts it by announcing, stage by stage, a probability over S .”
BackgroundMethodInfluentialAn Application of Operational-Subjective Statistical Methods to Rational Expectations
Blattenberger, G., & Lad, F. (1988) · Journal of Business and Economic Statistics · doi:10.1080/07350015.1988.10509690 · Cited by 4
Bayesian priors for tree calibration: Evaluating two new approaches based on fossil intervalsPreprint
Norris, R. W., Strope, C. L., McCandlish, D. M., & Stoltzfus, A. (2015) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/014340 · Cited by 5
“We did not test the precision of the Obs method because there were too few instances of the actual value falling within the 95 % HPD. Table 4.”
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“We can now apply our law of large numbers for submartingale differences (De Cooman et al., 2016, Theorem 7) to get to the following result, which generalises Philip Dawid’s well-known consistency result for Bayesian Forecasters (Dawid, 1982, Theorem 1), to deal with imprecise assessments: Theorem 29 (The well-calibrat…”
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Gelman, A., & Hennig, C. M. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1508.05453 · Cited by 4
“There is also some work on (partial) intersubjective agreement on prior specifications, e.g. Dawid (1982a), providing a rationale for consensus (virtue V2(c)).”
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Huang, H., Sharma, S., Loquercio, A., Angelopoulos, A. N., Goldberg, K., & Malik, J. (2024) · Proceedings - IEEE International Conference on Robotics and Automation/Proceedings · doi:10.1109/icra57147.2024.10610223 · Cited by 8
“Such adversarial settings have been popular since the early literature on calibration [23] and were introduced into the realm of conformal prediction by [17].”
BackgroundImproving Uncertainty-Error Correspondence in Deep Bayesian Medical Image Segmentation
Mody, P., Chaves-de-Plaza, N. F., Rao, C., Astrenidou, E., de Ridder, M., Hoekstra, N., Hildebrandt, K., & Staring, M. (2024) · The Journal of Machine Learning for Biomedical Imaging · doi:10.59275/j.melba.2024-5gc8 · Cited by 3
“…to this as uncertainty calibration (Kumar et al., 2019; Krishnan and Tickoo, 2020; Zhang et al., 2020; Camarasa et al., 2021; Gruber and Buettner, 2022), but we find this term incorrect as historically, calibration is referred to in context of probabilities of a particular event (Dawid, 1982).”
BackgroundDeterministic Convergence and Strong Regularity
Nielsen, M. (2018) · The British Journal for the Philosophy of Science · doi:10.1093/bjps/axy038 · Cited by 3
Gerevini, A., Maroldi, R., Olivato, M., Putelli, L., & Serina, I. (2023) · IEEE Access · doi:10.1109/access.2023.3296260 · Cited by 3
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“A closely related concept to uncertainty estimation is a model calibration [25, 93].”
Background“Calibeating”: Beating forecasters at their own game
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“The calibeating in this case thus yields K ′ = 0 and The procedure of this second result is stochastic , as it must be in order to guarantee calibration (cf. Dawid 1982, Oakes 1985, and Foster and Vohra 1998).”
Calibration of Deep Learning Classification Models in fNIRS
Cao, Z., & Luo, Z. (2024) · IEEE International Conference on Automation Science and Engineering (CASE) · doi:10.1109/case59546.2024.10711714 · Cited by 3
“One mathematical formulation of the reliability of confidence is calibration [16], which is important for classification models in various fields, ensuring probability estimates align closely with correctness likelihood.”
BackgroundCalibration verification for stochastic agent-based disease spread models
Horii, M., Gould, A., Yun, Z., Ray, J., Safta, C., & Zohdi, T. I. (2024) · PLoS ONE · doi:10.1371/journal.pone.0315429 · Cited by 3
Masterton, G. (2013) · Synthese · doi:10.1007/s11229-013-0384-z · Cited by 3
“This confirms the theorem by Dawid (1982) that, in the presence of feedback, a coherent Bayesian invariably expects themselves to be well calibrated.”
Bayesian Inference Is Unaffected by Selection: Fact or Fiction?
Harville, D. A. (2020) · The American Statistician · doi:10.1080/00031305.2020.1858963 · Cited by 3
“somewhat akin to unbiasedness) and performs well in other respects in a framework like that considered by Dawid (1982). In any case, it could be argued that the evaluation of a procedure’s performance in repeated application should be based in whole or in part on its performance when applied to a sequence of yvalues (…”
BackgroundInfluentialProbabilistic Iterative Methods for Linear Systems
Cockayne, J., Ipsen, I. C. F., Oates, C. J., & Reid, T. W. (2020) · ePrints Soton (University of Southampton) · doi:10.48550/arxiv.2012.12615 · Cited by 3
“(1), with contributions in Hennig et al. [2015], Bartels and Hennig [2016], Bartels et al.”
BackgroundAgricultural Intensification, Environmental Ethics and Sustainability: Some Ethical Observations
Dower, N. (2008) · The international library of environmental, agricultural and food ethics/The International library of environmental, agricultural and food ethics · doi:10.1007/978-1-4020-8722-6_11 · Cited by 3
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A cautionary note on the discrete uniform prior for the binomial N: reply
Link, W. A. (2014) · Ecology · doi:10.1890/14-0857.1 · Cited by 3
Observations on the Concept of Risk and Arab Culture
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Bayesian Classification and Regression with High Dimensional FeaturesPreprint
Li, L. (2007) · arXiv (Cornell University) · doi:10.48550/arxiv.0709.2936 · Cited by 3
“Bias in this respect is referred to as a lack of “calibration” — that is, the predictive probabilities do not represent the actual chances of events (Dawid 1982).”
BackgroundMethodInfluentialIs Information Uncertainty Diversifiable? A Short Bayesian AnswerPreprint
Johnstone, D. J. (2015) · SSRN Electronic Journal · doi:10.2139/ssrn.2626624 · Cited by 3
Revisiting Explicit Regularization in Neural Networks for Well-Calibrated Predictive UncertaintyPreprint
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“The calibrated behavior refers to the ability to match its predictive probability of an event to the long-term frequency of the event occurrence [1], and the uncertainty representation ability refers to the ability to represent uncertainty about its predictions.”
BackgroundRubin, D. B. (1983) · Journal of the American Statistical Association · doi:10.1080/01621459.1983.10477023 · Cited by 3
Stolz, U., Gavryushkina, A. “., Vaughan, T. G., Stadler, T., & Allen, B. J. (2025) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2025.04.17.649084 · Cited by 3
“This study is inspired by the well-calibrated validation approach [16], which considers a Bayesian model well-calibrated if the highest posterior density (HPD) region at x % credibility level contains the true value with a frequency approximately equal to x %.”
Cognitive and Representational Cues for Assigning Weight to Numerical Information in Decision-Making
Rinne, L. F. (2010) · ProQuest Demo Repository · Cited by 3
H-Calibration: Rethinking Classifier Recalibration With Probabilistic Error-Bounded Objective
Huang, W., Cao, G., Xia, J., Chen, J., Wang, H., & Zhang, J. (2025) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2025.3582796 · Cited by 6
“Calibration ensures that machine learning models provide meaningful and interpretable predicted probabilities, consistent with realized outcomes, making it a key mathematical formulation for model reliability [12], [17], [30], [31], [31], [32], [33], [34], [35].”
On universal algorithms for adaptive forecasting
V'yugin, V. V. (2011) · Problems of Information Transmission · doi:10.1134/s0032946011020074 · Cited by 2
“In [2] a method for evaluating forecasts was proposed, the calibration method.”
MethodWhy Average When You Can Stack? Better Methods for Generating Accurate Group Credences
Kinney, D. B. (2022) · Philosophy of Science · doi:10.1017/psa.2022.3 · Cited by 2
“, wN} is used to calculate each group credence Cr∗(F ) and that there is an event F ∈ F that has non-zero probability according to each individual’s credence function (Dietrich, 2019). As mentioned in the introduction, my goal in this paper is not to adjudicate between which of these two means provides the better mech…”
BackgroundMethodology of Causal Modelling
(2008) · Springer eBooks · doi:10.1007/978-1-4020-8817-9_3 · Cited by 2
Bayesian historical borrowing with longitudinal large-scale assessments
Kaplan, D. E., Chen, J., Lyu, W., & Yavuz, S. (2023) · Large-scale Assessments in Education · doi:10.1186/s40536-022-00140-w · Cited by 3
“Scoring rules provide a measure of the accuracy of probabilistic forecasts, and a prediction can be said to be “well-calibrated” if the assigned probability of the outcome matches the actual proportion of times that the outcome occurred (Dawid, 1982).”
BackgroundPushing the Limits of Fairness in Algorithmic Decision-Making
Shah, N. (2023) · International Joint Conference on Artificial Intelligence · doi:10.24963/ijcai.2023/806 · Cited by 2
On the Performance of Horseshoe Priors for Inducing Sparsity in Structural Equation Models
Harra, K., & Kaplan, D. (2023) · Structural Equation Modeling: A Multidisciplinary Journal · doi:10.1080/10705511.2023.2280895 · Cited by 2
“In other words, a key characteristic of statistics is to develop accurate predictive models, and all other things being equal, a given model is to be preferred over other competing models if it provides better predictions of what actually occurred (Dawid, 1982, 1984, see also; Kaplan, 2023).”
Probabilistic neural networks for improved analyses with phenomenological R -matrix
Kim, C.-H., Chae, K. Y., Smith, M. S., Bardayan, D. W., Brune, C. R., deBoer, R. J., Lu, D., & Odell, D. (2024) · Physical Review C · doi:10.1103/physrevc.110.054609 · Cited by 2
Forecasting Performance of an Estimated Open Economy DSGE Model for the Euro Area
Malin Adolfson, A., Lindé, J., & Villani, M. (2005) · Cited by 2
Howson, C. (2007) · Synthese · doi:10.1007/s11229-006-9135-8 · Cited by 2
Prediction of the insulin sensitivity index using Bayesian networks
Bøttcher, S. G., & Dethlefsen, C. (2004) · Cited by 2
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
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