Causation in Law and Statistics (CBR project)

Overview

The UKRI Cross-Research Council Responsive Mode Pilot Scheme (CRCM) was established to support breakthrough and transformative interdisciplinary research which spans multiple disciplines. The CBR has obtained funding under the CRCM Scheme in collaboration with the Statistics Lab of the Department of Pure Mathematics and Mathematical Statistics (DPMMS) at Cambridge, to carry out an interdisciplinary project on causation in law and statistics.

Background

Causal inference is a rapidly developing, interdisciplinary field at the intersection of mathematical, biomedical and social sciences. Advances in the past 4 decades have deepened our understanding of probabilistic causation and elucidated the necessary assumptions that underlie causal inference from observational data. New designs and methods such as instrumental variables, g-methods for longitudinal data, difference-in-differences, synthetic controls and negative controls have profoundly changed how researchers analyse empirical data across several many scientific fields. Another major breakthrough is a much better understanding about causal identification, that is, when it is possible to test a causal hypothesis or estimate a causal effect using data. However, these advances have not yet made a major impact on how causation is conceptualised in law and their impact on legal research has been limited.

Understanding counterfactuals is a key issue in law. Counterfactual reasoning is routinely applied in legal decision making, in the form of the ‘but-for test’ of causation. This test assigns liability where a court concludes on the balance of probabilities that the harm sustained by the claimant would not have happened ‘but for’ the fault of the defendant. However, this ubiquitous practice is far from unproblematic. The but-for test has proved especially difficult to apply in cases of long-tail diseases which may emerge decades after an initial exposure to a harmful substance. This has led courts to modify the strict but-for test by allowing exceptions based on concepts which include material contribution to harm and material increase of risk; however, these concepts lack clarity and have introduced unsustainable distinctions into the law. There is a similar debate over causation in civil law systems, where the concepts of equivalence, abnormal cause, and unlawful cause are approximately analogous to ideas used in the common law to distinguish causes from conditions. When courts have directly engaged with statistical concepts, the results have not been satisfactory. For example, the UK Supreme Court in Sienkiewicz misread statistical evidence and misapplied the epidemiological concept of doubling the risk (McIvor, 2013).
Counterfactual causation is also an extremely challenging concept for statisticians. Pearl (2009) places cases of counterfactual causation at the highest or most complex level of his ‘ladder of causation’, above association (eg a regularity observed in a class of cases) and intervention (eg the result of an experiment). Recent advances in causal inference have focused predominantly on estimating the effects of causes (EoC) from experimental and observational data, which belongs to the intervention rung of Pearl’s ladder of causation. There is much less attention to the causes of effects (CoE), which is directly relevant to the problems which courts must address.

Aims and objectives

The objective is to develop an integrated theory of causality for law and statistics. The project will address the following specific questions:

1. what is the extent of correspondence or overlap between existing models of causality in law and statistics?
2. how far can the foundational legal notions of causation, in particular the but-for test, be modelled statistically?
3. can the use of legal examples and materials be helpful for developing statistical models of causation at the individual level, that is, ‘actual causation’?
4. how far do Bayesian models of causal inference in statistics, which emphasise the importance of adjusting causal priors in the light of new information, offer a basis for understanding how courts resolve causation problems?

Methods

1. Mapping stage: legal-statistical case studies

To make progress on an understanding of causation which is interdisciplinary as opposed to simply cross- or transdisciplinary, we will firstly engage in an exercise of mapping concepts across the 2 fields. To do this, we will analyse leading common law cases on causation from the viewpoint of statistical theories and models described in the ‘Background’ section above. The cases we will analyse include leading decisions in the Anglo-American common law, including those on material contribution (Bonnington), exposure to risk (McGhee), market-share liability (Sindell), concurrent and overlapping causes (Baker, Jobling), probabilistic cause (Fairchild, Barker, Sienkiewicz, Baily, Heneghan), and the generic/specific causation distinction (Holmes). We will also identify and analyse equivalent cases in civil law jurisdictions.

2. Translation stage: defining and solving new problems about counterfactual causation

The rich discussion on causation in law can motivate new statistical problems. We will explore new causal models and estimands that are inspired by the mapping of legal cases in stage (i). Those estimands will be inherently counterfactual, which makes them only partially identifiable and challenging to deal with. Consider a baseline treatment assignment policy d0, and a proposed change to policy, d1. The policy change can be evaluated counterfactually by the incurred individual benefit and harm that can be quantified, respectively, by the concepts of probability of benefit (PB) and probability of harm (PH), as defined above (see ‘Background’). Because these quantities are counterfactual, they are only partially identifiable and can only be bounded using empirical data. Our preliminary investigation of this problem shows that the smallest possible PB is simply the average treatment effect (ATE) for the policy change, and the largest possible PH also admits a simple form. These results can then be used to understand counterfactual optimality of a policy change. Specifically, consider the optimisation problem that maximises the ATE subject to an upper bound on the worst-case PH. Our preliminary results give a closed-form solution to this problem, and we find that there exists a universal index to prioritise the individuals. This means that we can define a universal ordering on the subgroups so that a subgroup with a higher index should always be prioritised (if their treatment is not optimal yet), regardless of the current policy. This provides a way to design policy changes that optimally balance the average benefit and potential individual harm, which has applications in law and other social systems. As part of the proposed project, we will consider extensions of this model (eg to ordinal treatments and outcomes) and other related counterfactual problems.

3. Application stage: statistical analysis of factors influencing judicial decision making

In the third stage, we will apply our newly developed causal models in empirical settings, making use of the growing availability of data on judicial decisions.

1. Coevolution of law and economy
   Despite the advances recently made in applying natural language processing (NLP) and machine learning (ML) to law, most studies do not go beyond a purely associational analysis of patterns in textual corpora. In a recent paper (Deakin and Shuku, 2025), we took a more expansive approach, constructing a measure of the evolution of judicial sentiment in English poor law cases between the 1690s and 1830s, which we then regressed against data on wages and prices. We found that judicial sentiment during this period was pro-cyclical, meaning that courts interpreted workers’ poor law claims more restrictively during periods of recession. In the proposed project, we will extend our analysis of the poor law dataset we created (Deakin et al, 2024), to deepen our exploration of causal interactions between economic and legal change, and extend the analysis to cover a dataset of 20th century British and Irish workmen’s compensation cases that we are currently constructing (Deakin et al, 2025). We will use causal modelling to identify the impact on legal decision making of business cycle effects and worker militancy. We will also study the reverse causal flow, that is, the counterfactual impact of policy changes on the economy. Both parts of the study will inform our discussions with judges and legal practitioners on the current state of causation tests in workplace injury and disease cases.
2. The impact of automated decision-making systems on judicial outcomes
   We will apply the counterfactual policy method described in point (ii) above to the first randomised controlled trial assessing the impact of algorithmic recommendations on judicial decisions (Imai et al, 2021). While the original study focused on estimating the conditional average treatment effect to decide whether or not to recommend AI to judges, our approach is designed to operate at the individual level. This can offer particular value in providing a fairer recommendation in cases involving individuals closer to the decision boundary of the algorithm. While there is a growing trend of human decisions being influenced by algorithms, regulatory frameworks remain limited. Most of these algorithms are developed for general purposes, rather than being tailored to high-stakes contexts such as legal or medical decision-making. We aim to initiate a dialogue with legal professionals on how these algorithms can be improved and how legal principles can inform the development of statistical models used in such settings.

Progress

The project began on 1 July 2026. An update on progress will be made in next year’s Annual Report.