Fair Inference On Outcomes

Razieh Nabi; Ilya Shpitser

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Fair Inference On Outcomes

Razieh Nabi, Ilya Shpitser

arXiv:1705.10378v4May 29, 2017stat.ML

TL;DR

The paper asks how to formalize and reduce discrimination in statistical inference involving sensitive features. It defines discrimination through path-specific causal effects, learns constrained outcome models, and addresses identification and inference complications.

Problem
Statistical models do not inherently encode human intuitions about fairness, motivating a framework for assessing discrimination involving sensitive features and outcomes.
Method
The paper models discrimination as an effect along disallowed causal pathways and learns outcome distributions by maximizing likelihood subject to path-specific-effect constraints.
Results
The approach formalizes fair inference, addresses out-of-sample prediction, non-identification, and Bayesian computation, and is illustrated on real datasets.
Takeaways & Limitations
Fairness can require estimators independent of the selected outcome model, correctly specified supporting models, and path-specific constraints rather than simply dropping sensitive features.
Takeaways & Limitations
Dropping a sensitive feature does not necessarily remove discrimination and can be inappropriate when that feature is predictive of the outcome.

Abstract

from arXiv · show

In this paper, we consider the problem of fair statistical inference involving outcome variables. Examples include classification and regression problems, and estimating treatment effects in randomized trials or observational data. The issue of fairness arises in such problems where some covariates or treatments are "sensitive," in the sense of having potential of creating discrimination. In this paper, we argue that the presence of discrimination can be formalized in a sensible way as the presence of an effect of a sensitive covariate on the outcome along certain causal pathways, a view which generalizes (Pearl, 2009). A fair outcome model can then be learned by solving a constrained optimization problem. We discuss a number of complications that arise in classical statistical inference due to this view and provide workarounds based on recent work in causal and semi-parametric inference.

Introduction

The paper frames fair outcome inference as a response to discriminatory biases in statistical and machine-learning models. It defines discrimination through effects of sensitive features along disallowed causal pathways and develops constrained inference methods.

Statistical and machine-learning models do not automatically encode human intuitions about fairness and bias.Their deployment has raised concerns among policymakers, regulators, and advocates about harmful discriminatory biases.
Discrimination is modeled as an effect of a sensitive feature, such as race or gender, on an outcome along specified disallowed causal pathways.This formulation is intended to capture intuitive properties of discrimination and generalize prior formal and informal proposals.
The paper treats fair inference as a constrained optimization problem that restricts path-specific effects.It also examines implications for out-of-sample prediction, non-identification, and computational Bayesian methods.

Notation And Preliminaries

The paper establishes notation for variables, values, sets, datasets, features, and outcomes, then situates fairness within outcome-based inference tasks involving a sensitive feature.

Variables use uppercase letters, values use lowercase letters, and sets use bold letters.A variable’s state space is denoted X_V.
Datasets are represented as D = (Y, X), where Y is the outcome and X is the feature vector.The ith realization of feature X_j is x_i^j, while x_i denotes the entire feature vector for instance i.
The framework considers classification and regression with a sensitive feature S whose careless use in inference about Y may result in discrimination.Examples include gender in hiring and race in recidivism prediction, and the approach extends to other outcome-based inference tasks.

Causal Inference

The paper introduces causal and mediation concepts for separating outcome effects by treatment, mediator, and pathway. Identification depends on causal assumptions, and path-specific effects can be estimated when suitable functionals exist.

Causal Inference: Causal inference compares potential outcomes Y(a) to quantify average causal effects, rather than interpreting E[Y|A] as necessarily causal.Associations between treatment and outcome may be spurious or only partly causal.
Causal Inference: Consistency and conditional ignorability connect observed data to counterfactual outcomes and yield the adjustment formula under observed confounding control.Adjusting for C is intended to remove non-causal dependence between A and Y.
Causal Diagrams: Causal graphs represent variables as vertices, with missing directed edges indicating absent direct causal relationships and mediators lying on directed paths.Figure 1 includes single- and multiple-mediator structures and a graph where the natural direct effect is not identified.
Mediation Analysis: Mediation analysis decomposes an average causal effect into direct and mediator-mediated indirect effects using nested counterfactuals such as Y(a, M(a′)).The natural direct effect is E[Y(a, M(a′))] − E[Y(a′)], while the natural indirect effect is E[Y(a)] − E[Y(a, M(a′))].
Mediation Analysis: Sequential ignorability and related conditional independence assumptions are required to identify nested mediation counterfactuals.These assumptions concern treatment, mediator, outcome, and baseline factors jointly.
Path-Specific Effects: Path-specific effects quantify causal influence along selected pathways and can be represented by nested counterfactuals.Under stronger assumptions, the edge g-formula identifies certain path-specific counterfactual means, which may then be estimated by plug-in methods.

Approaches To Fair Inference

The paper contrasts associative fairness criteria with causal and path-specific approaches. It argues that causal pathways can distinguish discriminatory from permissible effects, while non-identification requires principled treatment rather than ad hoc repair.

Associative Approaches: Associative criteria such as selection-rate comparisons and equal false-positive rates can give unintuitive results when sensitive features are confounded.These approaches assess relationships such as p(Y|S), rather than causal effects.
Causal Fairness: In the hiring example, p(H(C = 1)) = 0.035 and p(H(C = 0)) = 0.125 despite similar observed rates p(H = 1|C = 1) = 0.0595 and p(H = 1|C = 0) ≈ 0.0515.The counterfactual comparison reveals a strong hiring preference against applicants with prior convictions.
Causal Fairness: Causal fairness is motivated by randomized counterfactual comparisons that isolate the effect of the sensitive feature or treatment.The example treats a hiring rule as fair when randomized assignment of convictions yields equal hiring probabilities.
Path-Specific Fairness: The paper connects employment-discrimination reasoning to the mediation counterfactual Y(G = 1, C(G = 0)).“Everything else had been the same” corresponds to holding job characteristics at their original-gender values.
Path-Specific Fairness: Discrimination may involve only selected causal paths, so the domain determines whether a pathway is disallowed.A gender-to-fitness-test-to-hiring path may be discriminatory for accounting but appropriate for construction.
Prior Work: The paper formulates removal of selected path-specific effects as constrained inference on statistical models and rejects ad hoc repair when the effect is not identifiable.Without additional assumptions, an unidentified discriminatory effect cannot be removed reliably from observed data.

Discriminatory Path-Specific Effects

The paper defines discrimination as an effect of a sensitive feature along disallowed causal pathways and learns a fair distribution by constrained optimization. It addresses prediction under distribution shift, estimator choice, outcome-model misspecification, and extensions to identifiable path-specific effects.

Fairness formulation: Discrimination is represented by constraining a path-specific effect (PSE) of the sensitive feature on the outcome within specified bounds.The framework transforms the observed distribution into a KL-close fair distribution whose PSE lies within (ε_l, ε_u).
Fair prediction: New instances are mapped into the fair world because they are drawn from the observed distribution rather than the constrained fair distribution.Without this mapping, discrimination guarantees established under p* do not directly apply to new draws from p.
Model specification: Choosing W trades predictive information from new instances against KL closeness to the observed model, and the paper leaves more principled selection methods for future work.Larger W supports more discriminative prediction, whereas smaller W yields a model that is at least as KL-close in the comparison described.
Finite-sample inference: For finite samples, the fair distribution is approximated by constrained maximum likelihood using an estimator of the PSE.The choice of W is guided by which model components the selected PSE estimator uses; examples include plug-in and inverse-probability-weighted estimators.
Path-specific effects: The framework generalizes to any setting with an identifiable PSE, using identification theory and plug-in maximum-likelihood estimators.The construction can constrain effects along multiple pathways rather than only a single direct-effect edge.
Robust estimation: When prediction requires a potentially misspecified outcome model, estimators that do not rely on that model, including IPW and a triply robust estimator, are used instead.The triply robust estimator remains consistent when any two of the outcome, mediator, and treatment models are correctly specified.
Bayesian computation: The paper adapts Bayesian conditional-density methods as computational tools but does not extend its fairness approach to a fully Bayesian procedure.Its fair-inference methods remain fundamentally frequentist, constraining fitted parameters within specified bounds.

Dealing With Non-Identification of the PSE

Non-identification prevents estimating some PSEs from observed data, so the paper proposes workarounds that retain the original causal model. These use additional measurements, identified broader effects, or bounds for non-identifiable effects.

Non-identification: The natural direct effect is not identified in some causal models, so it cannot be estimated as a function of the observed data distribution.The problem arises in the example models because unobserved confounders prevent identification.
Workarounds: If unobserved confounders can be measured or reliably proxied, the non-identified NDE becomes identifiable.This is one of the paper’s proposed approaches for recovering information about the target PSE.
Workarounds: If the target PSE is not identifiable, an identified PSE containing the target pathways and additional pathways can serve as an alternative constraint.In Fig. 1(b), the broader PSE including A → Y, A → M → Y, and A → M → W → Y is identified while the NDE is not.
Bounds and assumptions: Using a broader PSE to limit the target effect requires its magnitude to upper-bound the target PSE, which on the mean-difference scale can require all relevant causal paths to have the same sign.Without that relationship, constraining the broader effect need not constrain the discrimination effect of interest.
Bounds and assumptions: When neither confounder measurement nor a suitable identified broader PSE is available, bounds for non-identifiable PSEs provide another option.The paper notes that such bounds are generally an open problem, though bounds exist for the NDE with a discrete mediator.
Model fidelity: The authors reject modifying the causal model to force identification because inferences from a model changed away from the truth may no longer track reality.Their alternatives remain within the original model and seek information through bounds or broader pathway effects.

Experiments

Experiments on COMPAS and Adult illustrate constrained fair inference by limiting path-specific effects while retaining sensitive features. The paper also shows that fairness depends on choosing estimators robust to outcome-model misspecification.

The COMPAS Dataset: On COMPAS, the authors defined discrimination as the direct effect of race on recidivism while treating prior convictions as a fair mediator.The model used race as A, prior convictions as M, demographic information as C, and recidivism as Y.
The COMPAS Dataset: 1.3 was the unconstrained COMPAS NDE odds ratio, while constrained BART restricted the NDE to 0.95–1.05 and achieved 66.4% test accuracy.Unconstrained BART achieved 67.8% accuracy, whereas removing treatment from the outcome model reduced accuracy to 64.0%.
The COMPAS Dataset: The COMPAS analysis approximated Northpointe’s proprietary classifier because its exact model and complete input features were unavailable.The authors predicted a binarized COMPAS score and dropped race because Northpointe reportedly did not use it.
The Adult Dataset: For Adult, the disallowed paths included the direct effect of gender on income and the path through marital status, education, occupation, hours, and work class.The path-specific effect was computed using a counterfactual contrast and restricted to 0.95–1.05.
The Adult Dataset: 3.16 was the unconstrained Adult PSE, while constraining it to 0.95–1.05 reduced accuracy from 82% to 72% while assuring fairness.Dropping the sensitive feature produced the same accuracy as the unconstrained model, but the authors argue that this does not remove discrimination under their definition.
Selecting The Outcome Model To Maximize Out Of Sample Predictive Performance: Fairness-preserving outcome-model selection requires PSE estimators that do not depend on the selected outcome model, or correctly specified models on which they depend.In simulation, triply robust and IPW estimators preserved discrimination removal when the A and M models were correctly specified, with an NDE estimate of 3.01.

Discussion And Conclusions

The paper formalizes discrimination as a constrained path-specific effect and learns outcome models by maximizing likelihood subject to limits on that effect. The framework also extends to affirmative action and wage-gap settings, but requires computationally challenging optimization and an unresolved reparameterization alternative.

Discrimination is formalized as a path-specific effect (PSE), and fair inference maximizes likelihood subject to constraints limiting its magnitude.
The framework can model affirmative action by increasing the PSE along domain-defined valid paths from a sensitive feature to an outcome.Examples of valid paths include academic merit, extracurriculars, or the direct path.
Increasing the PSE along prescribed paths may reduce predictive performance.
The approach requires solving a computationally challenging constrained optimization problem.
Reparameterizing the likelihood to include the discrimination PSE could turn minimization into box constraints, but this reparameterization remains an open problem.