1 An Old Quarrel in a New Arena
The Bayesian-frequentist divide is the oldest argument in statistics, but causal inference gives it an unusually sharp edge. The reason is structural: causal estimands are defined through potential outcomes Yᵢ(1) and Yᵢ(0), of which we observe at most one per unit. The other is missing not unknown in the usual sampling sense, but counterfactual. How we ought to reason about that missing half divides the field. The Bayesian sees a missing-data problem to be solved by a model and a prior; the frequentist sees a sampling problem to be solved by the assignment mechanism and a sampling distribution. With the recent surge of Bayesian machine-learning methods for causal effects Bayesian DiD, Bayesian regression discontinuity, Bayesian double machine learning the quarrel is no longer philosophical garnish. It shapes which estimators applied researchers reach for.
2 The Bayesian Case
Causal inference is missing-data imputation. Rubin [1978] gave the Bayesian programme its founding statement: treat the unobserved potential outcomes as missing data, posit a joint model for (Yᵢ(0),Yᵢ(1)), place a prior on its parameters, and the posterior predictive distribution of the missing outcomes induces a posterior over any causal estimand. The average treatment effect, the effect on the treated, quantile effects, even individual-level effects all fall out of one coherent calculation:
There is no need to derive a separate sampling distribution for each new estimand: the posterior delivers full uncertainty for all of them simultaneously.
Honest small-sample uncertainty. Frequentist causal inference leans on asymptotics- √n normality, large-sample confidence intervals. In the small or few-cluster samples common in field experiments and synthetic-control studies, those approximations can be poor. The Bayesian posterior is exact in the sense that it requires no large-n appeal; it propagates uncertainty about nuisance parameters automatically, rather than plugging in estimates and hoping the standard error captures the cost.
Priors as a feature, not a bug. Causal questions rarely arrive context-free. A prior lets the analyst encode that pre-trends are probably smooth, that effects are probably bounded, or that a discontinuity is probably modest regularising estimation in a transparent, debatable way. Bayesian regression discontinuity with Gaussian-process priors, for instance, yields flexible curves at the cutoff with calibrated credible intervals, and recent Bayesian double machine learning shows that informative priors can improve finite-sample efficiency over plug-in frequentist DML while remaining asymptotically equivalent to it.
Decisions, not just estimates. Policy is a decision problem, and Bayesian decision theory speaks its language directly: maximise expected utility under the posterior. "What is the probability this programme has a positive net benefit?" is a question the posterior answers and a confidence interval does not.
3 The Frequentist Case
Randomisation licenses inference without a model. The frequentist's trump card is design-based inference. In a randomised experiment the assignment mechanism is known, and Fisher's randomisation test and Neyman's repeated-sampling variance deliver valid inference using only that knowledge no outcome model, no prior. The inference is exact in finite samples and robust to any misspecification of how outcomes are generated, because it conditions on nothing but the randomisation the experimenter controlled. To a frequentist this is the whole point of running an experiment: to buy inference that does not depend on getting a model right.
Coverage guarantees and objectivity. A 95% confidence interval has a property the analyst can stand behind regardless of belief: across repeated samples it covers the truth 95% of the time. A Bayesian credible interval has correct frequentist coverage only if the prior happens to be "right," and a poorly chosen prior can produce intervals that systematically miss. For high-stakes, adversarial settings drug approval, regulatory cost-benefit the demand for procedures whose error rates hold irrespective of the analyst's beliefs is not dogmatism but accountability.
The Robins-Ritov problem: the assignment mechanism is informative. The most penetrating technical critique of pure Bayesianism in causal inference comes from Robins and Ritov [1997]. In high-dimensional observational settings, the propensity score (the assignment mechanism) is, by the likelihood principle, ancillary: a coherent Bayesian who models the outcome should ignore it. Yet frequentist semiparametric theory shows that consistent, efficient estimation of the treatment effect requires using the propensity score this is exactly why doubly robust and double machine-learning estimators work. A Bayesian who obeys the likelihood principle can therefore fail to be consistent in regimes where a design-aware frequentist succeeds. The known assignment mechanism carries information that the Bayesian framework, taken literally, throws away.
4 Where the Disagreement Really Lies
The clash is not about arithmetic given the same model and a flat prior, Bayesian and frequentist answers often nearly coincide. It is about what must be assumed and what is guaranteed. The frequentist minimises modelling assumptions by leaning on the design, and in return gets guarantees that hold come what may, at the cost of awkwardness in small samples and a restricted menu of estimands. The Bayesian embraces a full probability model, and in return gets coherent, all-estimands, small-sample uncertainty and a natural decision calculus, at the cost of sensitivity to priors and the Robins Ritov failure when the model ignores an informative design.
A second fault line is the treatment of nuisance functions. Modern causal inference is dominated by the need to estimate high-dimensional nuisances (propensity scores, outcome regressions) flexibly. Frequentist DML handles this with sample-splitting and Neyman-orthogonal moments that make the estimator first-order insensitive to nuisance error. The Bayesian analogue must confront "regularisation-induced confounding" priors that shrink the nuisance can bias the causal target and recent Bayesian DML work is precisely an attempt to engineer priors and reparameterisations that avoid it while keeping Bayesian coherence.
5 What Would Resolve It-and the Emerging Synthesis
Several developments are quietly dissolving the dichotomy.
- Bernstein-von Mises results. Where a Bernstein-von Mises theorem holds, the posterior is asymptotically a frequentist sampling distribution centred at an efficient estimate, so credible and confidence intervals coincide for large n. Establishing (or refuting) such theorems for specific causal estimators tells us exactly when the choice is innocuous.
- Design-aware priors. The constructive response to Robins-Ritov is to build the propensity score into the Bayesian model through balancing priors or by modelling the design explicitly restoring consistency without abandoning coherence. The success of these constructions is an empirical, checkable matter.
- Calibrated Bayes. A practical truce evaluates Bayesian procedures by their frequentist coverage: keep the posterior's coherence and decision-theoretic convenience, but report and stress-test the calibration of credible intervals across plausible data-generating processes.
The defensible position is ecumenical. In a clean randomised experiment, the design-based frequentist analysis should be the backbone it is the reason the experiment was run with a Bayesian model offered as a transparent, prior-disclosed complement for small samples and richer estimands. In messy high-dimensional observational work, neither pure stance suffices: the lesson of doubly robust and orthogonal estimation is that the assignment mechanism must be used, and the live research question is whether one reaches efficiency through frequentist orthogonalisation or through design-aware Bayesian modelling. The old quarrel, in causal inference, is converging on a shared requirement respect the design-that both traditions, at their best, already honour.
References
- Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., and Robins, J. (2018). Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal, 21(1), C1-C68.
- Imbens, G. W., and Rubin, D. B. (2015). Causal Inference for Statistics, Social, and Biomedical Sciences. Cambridge University Press.
- Robins, J. M., and Ritov, Y. (1997). Toward a curse of dimensionality appropriate (CODA) asymptotic theory for semi-parametric models. Statistics in Medicine, 16(3), 285-319.
- Rubin, D. B. (1978). Bayesian inference for causal effects: the role of randomization. The Annals of Statistics, 6(1), 34-58.