What would have needed to change for the past to turn out differently?

Imagine two production lines controlled by the same stop signal. Both have stopped. If someone overrides the first line so that it keeps running, the second can remain stopped. But ask a different question—what background circumstances would have allowed the first line to keep running without an override—and the answer changes. In the paper’s simplified system, the common stop signal would have to change, allowing both lines to run.

Neither answer is inherently wrong. They describe different hypothetical situations: an external manipulation versus an alternative history under unchanged operating rules.

In an arXiv preprint submitted on November 5, 2026, Zhangyi Liu, Pouria Fatemi and Suvrit Sra develop a mathematical approach to this second kind of reasoning, called backtracking counterfactual inference. Their focus is a complication that a single alternative-history story can conceal: the observed facts may be consistent with several hidden histories.

Moving possibilities, not just one explanation

The researchers work with structural causal models. These specify how each observed variable depends on its causes and on background variables, often called exogenous noises. Here, “noise” means a background input to an equation, not necessarily measurement error.

The model’s equations can map different background inputs to the same observation. Squaring a number, for example, hides its sign. Observing the result therefore need not reveal which input occurred. Instead, inference produces a probability distribution over background conditions consistent with the evidence.

The authors propose moving this entire distribution to background conditions that would produce a requested alternative event. The original equations remain fixed.

Their tool is optimal transport, a way of comparing probability distributions through the cost of moving probability mass. Here the cost is squared distance in the space of background variables. The goal is to find feasible alternative histories requiring the smallest average displacement from histories compatible with the observations.

That is an explicit modeling choice, not a uniquely correct definition of how history could have differed. The chosen distance determines what counts as a small change.

Two versions of a nearby alternative history

The first formulation, OT-CF, allows any counterfactual background distribution supported on conditions satisfying the target event. Under the paper’s assumptions, an optimal solution can be obtained by moving each possible factual background state to a nearest feasible state. Finding those nearest states can still require numerical optimization.

The second formulation, IOT-CF, adds a restriction: the counterfactual background variables must remain mutually independent.

This distinction needs care. The models start with independent background noises, but conditioning on evidence can create dependence, particularly with partial observations. A dependent counterfactual distribution is therefore not automatically invalid. Independence is an additional requirement when the result is intended to describe a new population-level distribution of independently generated background inputs.

The restriction can change the predicted outcome. In one three-variable example, both formulations make the second variable equal two. OT-CF gives two possible values for the third variable, approximately 1.374 and 0.626, with equal probability. The independence-constrained solution instead makes that variable equal one. The same target event can thus support different answers elsewhere in the model.

What the experiments establish

The researchers test numerical solvers on constructed models with five, six and ten variables, including polynomial and nonsmooth equations. They measure background displacement, error in satisfying the requested event and violations of the intended independence structure.

The solvers expose practical compromises. One preserves independence by constructing separate marginal distributions but only approximately enforces the target. Another penalizes both target error and departures from independence. A hard-constrained solver preserves the product structure and achieves target satisfaction to numerical precision when successfully solved. The implemented OT-CF solver can report lower movement costs while leaving a larger target residual; those figures are not exact feasible solutions merely because their costs are lower.

The central limitations are substantial. The causal graph, equations and background prior are assumed known, rather than learned from data. Numerical methods use finite representations of the factual posterior, and the number of joint possibilities can grow exponentially. The experiments do not establish performance on large real-world systems.

The contribution is a clearer accounting of assumptions: which mechanisms stay fixed, which hidden conditions may change, how uncertainty is carried forward, and what “as little change as possible” actually means.