What Incomplete Records Can Reveal About Joint Outcomes

Missing records still impose constraints

Questions about income persistence, treatment responses and overlapping deprivation concern outcomes occurring together. Yet researchers may observe only fragments of each person’s or household’s outcome vector. Whether a value is recorded may itself depend on that value or on other unobserved outcomes.

In his November 2026 preprint, Rami V. Tabri develops bounds that combine two inputs: known population distributions for individual outcomes, called marginals, and the distribution of incomplete records, including which outcomes were recorded. He places no restrictions on how observation depends on outcomes.

The distinction matters. A selectively observed group need not represent the population. Nevertheless, its recorded values must fit inside the known population distributions. That requirement can exclude otherwise possible relationships among outcomes.

From broad bounds to sharp endpoints

Classical Fréchet–Hoeffding bounds describe how large or small a joint probability can be when only the individual outcome probabilities are known. They allow every dependence structure consistent with those marginals.

Tabri asks how those bounds change when candidate distributions must also reproduce the incomplete records. His main target is the probability that every outcome falls at or below its specified threshold—for example, that three generations all belong to their respective bottom income quintiles.

The preprint establishes an exact reduction to binary indicators recording whether each threshold is met. Crucially, Tabri proves that every feasible allocation of these indicators can be extended to a full outcome distribution preserving both the original population marginals and the entire observed-record distributions. The calculation therefore does not merely approximate a continuous-outcome problem by discarding detail.

The resulting bounds are described as sharp: compatible full-data distributions attain both endpoints. No narrower interval can be justified from the stated information alone.

Two ways to allocate missing outcomes

For a record to qualify for the joint event, all its recorded outcomes must already meet their thresholds. Tabri calls the probability mass of such records eligible. Missing outcomes then determine whether an eligible record ultimately qualifies.

To obtain the lower bound, the method assigns threshold failures to missing coordinates, excluding as much eligible mass as possible. The known marginals limit how many failures are available. Tabri represents this calculation as a maximum-flow problem.

For the upper bound, every missing coordinate of a qualifying record must meet its threshold. Different observation patterns compete for the available marginal success probabilities. This produces a fractional-packing linear program, with one variable per occurring observation pattern and one resource constraint per outcome.

The paper also gives necessary and sufficient conditions for strict improvement over the classical bounds. Observing some outcomes does not automatically guarantee tighter bounds.

An income example without linked pairs

In a constructed three-generation example, each generation has a 20% population probability of bottom-quintile membership. A quarter of lineages have no recorded income; each remaining quarter has income recorded for just one generation. No pair of generations is observed together.

Among lineages observed for a particular generation, 50% belong to that generation’s bottom quintile. Combining these stipulated recording-pattern probabilities with the population marginals reduces the sharp upper bound on three-generation persistence from 20% to 11.25%. The lower bound remains zero.

Conditional on the first generation being in the bottom quintile, the upper bound on both subsequent generations also being there falls from 100% to 56.25%. These are illustrative population calculations, not empirical estimates of mobility.

A separate constructed split-questionnaire example considers 20 indicators observed through four five-item modules, with selective nonresponse. Although each criterion is met by 84% of the population, the sharp probability of meeting all 20 lies between zero and 20%, rather than zero and 84% using marginals alone.

What the results do not establish

The framework requires population marginals and observed records referring to the same target population. In potential-outcome applications, knowing those marginals is an additional input, not something selective treatment-group observations establish by themselves.

Sharpness is generally pointwise: a distribution attaining an endpoint at one threshold vector need not attain endpoints elsewhere. Computational savings also depend on how many observation patterns occur; that number can grow exponentially with the number of outcomes.

Finally, the preprint addresses population identification, not uncertainty from estimating its inputs with samples. Its contribution is a characterization of what the specified information permits, rather than a sampling-inference procedure or an imputation model.