The Bayes linear approach, developed primarily by Michael Goldstein and colleagues at Durham University, replaces the full probabilistic machinery of Bayesian inference with a simpler structure based on expectation as the primitive. Rather than specifying a complete prior distribution p(θ) and computing the posterior p(θ | D) via Bayes' theorem, the Bayes linear approach requires only the prior expectation E[θ], the prior variance Var(θ), and the covariance Cov(θ, D) between the quantity of interest and the data. The adjusted (updated) expectation is then computed by a linear projection.
This is not an approximation to full Bayes — it is a different foundational framework, rooted in de Finetti's theory of previsions and Bruno de Finetti's argument that expectation, not probability, is the fundamental concept for expressing uncertainty.
Adjusted Variance Var_D[θ] = Var[θ] − Cov(θ, D) · Var(D)⁻¹ · Cov(D, θ)
Where E_D[θ] → Updated expectation of θ given observation D
Var_D[θ] → Updated variance of θ given observation D
Foundations in Expectation
In the Bayes linear framework, the fundamental operation is expectation rather than probability. An agent specifies their beliefs about uncertain quantities by declaring expectations, variances, and covariances. These specifications are treated as primitive — they need not arise from any underlying probability distribution. This is a weaker requirement than full probabilistic specification, and the Bayes linear approach argues this is a strength: we can often say "I expect this quantity to be about 5, with uncertainty of about 2" without committing to a specific distributional form.
The update rule is the minimum mean-squared-error linear predictor — the best linear unbiased adjustment of the prior expectation given the data. When the joint distribution of (θ, D) is multivariate normal, the Bayes linear adjustment coincides exactly with the full Bayesian posterior. For non-normal distributions, it provides the best linear approximation.
Full Bayesian analysis requires specifying a complete joint probability distribution over all unknowns. For complex systems — climate models with thousands of parameters, engineering simulators with dozens of inputs, or large-scale industrial processes — this is often impractical. The analyst may have reliable beliefs about means and correlations but no justified basis for choosing, say, a log-normal over a gamma distribution. The Bayes linear approach allows honest uncertainty analysis without unjustified distributional assumptions. It trades the exactness of full Bayes for robustness to distributional misspecification.
Bayes Linear Sufficiency and Resolution
The framework introduces its own analogues of key Bayesian concepts. Bayes linear sufficiency defines when a subset of observations captures all the information in the full data set (in the mean-squared-error sense). The resolution transform decomposes the variance of a quantity into a resolved component (explained by the data) and a remaining component (unresolved uncertainty):
= Resolved variance + Remaining variance
Size of adjustment (resolution ratio) ρ²(θ, D) = 1 − Var_D[θ] / Var[θ]
The resolution ratio ρ² ranges from 0 (data tell nothing about θ) to 1 (data determine θ exactly). It is the Bayes linear analogue of R² in regression and provides a natural measure of how informative the data are.
Applications
Bayes linear methods have been widely applied in computer model emulation (building fast statistical surrogates for expensive simulators), uncertainty quantification for engineering systems, and reservoir modeling in petroleum engineering. The methodology is particularly suited to problems where the simulator is expensive to run, the parameter space is high-dimensional, and the analyst has expert knowledge expressible as means and covariances but not full distributions.
The Bayes linear framework has also been extended to handle exchangeable sequences, hierarchical structures (Bayes linear Bayes), and temporal updating (Bayes linear kinematics). The software tools MUCM (Managing Uncertainty in Complex Models) and BayesLinear (R package) implement these methods.
Historical Development
Bruno de Finetti introduces the concept of prevision (expectation as a primitive) and argues that probability can be founded on expectation rather than the other way around.
Michael Goldstein begins developing the Bayes linear framework at University College London and later at Durham University, building on de Finetti's foundational ideas.
Goldstein and Wooff publish Bayes Linear Statistics: Theory and Methods, the definitive textbook, providing a comprehensive treatment of the theory and its applications.
"We do not need the full apparatus of probability to reason carefully under uncertainty. Expectation alone, with its natural geometry of inner products and projections, provides a rigorous and practical foundation." — Michael Goldstein, Bayes Linear Statistics: Theory and Methods (2007)
Worked Example: Predicting Crop Yield from Rainfall
An agronomist uses Bayes linear methods to predict crop yield (Y) from rainfall (X), using only means, variances, and covariances — no distributional assumptions required.
(100, 2.1), (200, 4.3), (300, 5.8), (400, 8.1), (500, 10.2),
(600, 11.8), (700, 14.1), (800, 16.0), (900, 18.3), (1000, 20.1),
(1100, 22.0), (1200, 23.8), (1300, 26.1), (1400, 28.2), (1500, 30.0)
Step 1: Prior Specifications E[X] = 800, Var(X) = 186,667
E[Y] = 16.05, Var(Y) = 72.16
Cov(X, Y) = 3,610
Step 2: Bayes Linear Adjustment Adjusted slope = Cov(Y, X)/Var(X) = 3610/186667 = 0.01934
Adjusted intercept = E[Y] − slope × E[X] = 16.05 − 0.01934 × 800 = 0.58
Adjusted variance = Var(Y) − Cov²/Var(X) = 72.16 − 69.82 = 2.34
Step 3: Prediction E_D[Y | X = 600] = 0.58 + 0.01934 × 600 = 12.18 tonnes/ha
Variance resolved: (72.16 − 2.34)/72.16 = 96.8%
The Bayes linear approach resolves 96.8% of the prior uncertainty about yield using only the first two moments. The adjusted expectation gives the same point estimates as ordinary least squares, but Bayes linear methods make no normality assumption — they work with any underlying distribution. The residual variance of 2.34 represents irreducible uncertainty about yield beyond what rainfall explains.