ProbLog4Fairness: A Neurosymbolic Approach to Modeling and Mitigating Bias

AI summary

Overview

• Research area: Algorithmic fairness, neurosymbolic AI, and probabilistic logic programming (ProbLog / DeepProbLog).
• Technical level: Intermediate. The paper assumes familiarity with binary classification, fairness metrics such as statistical parity, and the basics of logic programming, but it introduces the probabilistic logic machinery it relies on.
• Scope: The paper proposes ProbLog4Fairness, a method that expresses ad-hoc bias assumptions as ProbLog programs and integrates them into neural network training via DeepProbLog, then evaluates it on synthetic tabular data, a real-world tabular dataset, and image data.

What This Paper Is About

Fairness definitions in machine learning are numerous, often mutually incompatible, and normatively contentious, which makes them hard to operationalize for a specific real-world task. The authors argue it can be easier to describe the concrete mechanisms that make data biased—label bias, measurement bias, and historical bias—and to correct for those mechanisms during training instead of optimizing a fixed fairness constraint. The goal is a framework that is simultaneously principled, flexible, and interpretable for modeling and mitigating such bias.

Key Contributions

1. ProbLog templates for bias mechanisms. The authors define a set of ProbLog templates describing label bias, measurement bias, and historical bias, each defined through four probabilistic facts that specify the conditional probability table of a biased binary variable given the sensitive variable. The templates can be simplified (for example, removing positive bias rules) or extended to multiclass labels and categorical sensitive variables.

2. A neurosymbolic training procedure. Using DeepProbLog, the bias program is integrated into the training of a neural classifier through distant supervision: the program is supervised only through observed biased features and labels, and gradients are backpropagated through the logic so the network is updated according to all unbiased interpretations consistent with the biased data.

3. Two test-time regimes. The method explicitly distinguishes predicting on unbiased features at test time (where the logic can be dropped and the classifier used directly) from predicting on biased features at test time (where the debiasing mechanism is retained).

4. Empirical validation across three data settings. Experiments on synthetic tabular data with known bias, the real-world Student Alcohol Consumption dataset, and CELEB-A image data show improved fairness and predictive performance relative to baselines that uphold a fixed bias type or notion of fairness.

Main Findings

• Synthetic data, correct parameters: When the relevant bias probability is known and the program parameters are set accordingly, ProbLog4Fairness reaches accuracy and statistical disparity comparable to the upper baseline (trained on unbiased features and labels), despite training only on biased labels. The mitigating baselines (Massaging, Error parity, Unawareness) generally achieve fairness but with limited predictive performance on the unbiased data.

• Handling correlation between the sensitive variable and the label: When A is not independent of Y, ProbLog4Fairness removes only the problematic bias, reaching the statistical disparity of the upper baseline—which is not expected to be close to zero in this setting. Other baselines cannot distinguish problematic from non-problematic bias and incorrectly impose a statistical disparity of zero.

• Sensitivity to parameter misspecification: With a fixed label bias probability of 0.3 and A not independent of Y, accuracy is highest and statistical disparity closest to the upper baseline when the program parameters match the actual bias probability. Around that optimum the sensitivity to parameter estimation error is small, which supports estimating parameters from a limited subset of data where unbiased features or labels are also available.

• Real-world tabular results: On the Student Alcohol Consumption data, the method approaches the statistical disparity present in the unbiased data and obtains a higher F1 score on the unbiased labels than the mitigating baselines. Results are similar when equalized odds is used instead of statistical parity. The paper notes that evaluating on biased data makes the baselines incorrectly appear to perform better.

• Simpler programs can help: Two simplified programs ("no positive bias" and "no bias on A=0") achieve a better F1 score, and "no positive bias" achieves a better statistical disparity. One hypothesis is a large estimation error on positive bias parameters, since only 67 girls and 82 boys did not pass and thus have an unbiased negative label.

• Image data results: On CELEB-A, fine-tuning a ResNet-50 pretrained on ImageNet on the cleaned Mouth Slightly Open labels yields a 10.18% improvement in F1 score over fine-tuning on the original labels. ProbLog4Fairness achieves part of that increase under a label bias assumption, and correcting for all three correlated sensitive attributes (Smiling, Blurry, High Cheekbones) at once performs much better in both F1 score and statistical disparity than correcting for a single attribute.

• Real-world gap: Performance on real-world data comes less close to the upper baseline than on synthetic data, which the authors attribute to other biases present in the data that the program does not account for.

Methodology in Plain English

The authors start from the observation that models trained on biased data inherit that bias, and that the path from unbiased variables to observed data can be described probabilistically. Rather than picking a fairness constraint, they describe the data-generating process in ProbLog, a probabilistic logic programming language in which probabilistic facts and rules define a probability distribution over possible worlds and a query's success probability is computed over those worlds.

A classifier is expressed as a probabilistic fact whose probability is predicted by a neural network from the features. On top of that, the authors write logic rules that transform between biased and unbiased variables using four parameters per binary variable—separately for negative and positive bias and for the sensitive and non-sensitive group. For label bias, the observed label is a biased, noisy proxy of the unobserved fair label while the features are unbiased. For measurement bias, observed features are biased proxies of unbiased features, and the label depends on the unbiased features; the classifier's prediction is then a weighted combination over possible unbiased feature vectors, weighted by how probable each is given the observed biased vector. Historical bias combines a biased feature distribution with labels based on those biased features; the paper notes that when the mapping from biased features to biased labels equals the mapping from unbiased features to unbiased labels, simply training on biased data and applying the measurement debiasing mechanism at test time can be enough.

Training uses DeepProbLog's distant supervision. The program is compiled into a circuit that computes the probability of queried facts given the network's predictions, and gradients from the observed biased labels flow back through the circuit to the network. The bias parameters are set from domain knowledge or estimated from a small subset of data with both biased and unbiased labels; the authors explicitly leave joint learning of parameters and classifier out of scope because the optimal classifier then becomes unidentifiable, a known problem in the positive-unlabeled learning literature.

Experiments compare against a lower baseline trained on observed features and labels, an upper baseline trained on unbiased features and labels, an unawareness baseline that omits the sensitive variable, Massaging as a preprocessing approach, and an Error parity postprocessing technique that uses statistical parity as its constraint. Predictive performance is reported as accuracy, or F1 when labels are imbalanced, and fairness is measured with statistical disparity—the difference between mean positive probability for the sensitive and non-sensitive groups. The authors use 5-fold cross-validation and repeat each experiment on at least five different seeds, reporting the mean.

Why This Matters

• Research impact: The paper reframes fairness mitigation from choosing among incompatible fairness definitions to modeling the specific biasing mechanisms in a task. It connects neurosymbolic reasoning (DeepProbLog) with causal fairness work, and the authors claim it is more interpretable than causal fairness methods because assumptions can be added or revised flexibly rather than requiring a fully specified graph. It also generalizes earlier ProbLog template work on labeling mechanisms from positive-unlabeled learning.

• Real-world applications:

  • Credit scoring, where residence, income, or employment history correlate with ethnicity or gender and where annotation itself may be biased.
  • Hiring and job stability proxies, such as "days worked in the past three years," which can be negatively biased against women who took maternity leave.
  • Education and grading, as in the Student Alcohol Consumption setting where human annotators produced biased pass/fail labels against male students.
  • Facial and image attribute detection, where subjective or inconsistent labels (as seen for CELEB-A attributes) affect label quality differently across groups.

• Industry relevance: Practitioners deploying classifiers in regulated or high-stakes domains often have to justify decisions, and the ability to write down explicit, auditable assumptions about how bias entered the data is more actionable than stating a fairness metric. The paper's templates give a concrete, reusable structure, and the DeepProbLog pipeline requires no change of model architecture—only a logic program on top of existing neural components. The finding that correcting one sensitive attribute does not imply fairness for others reinforces that real deployments need to account for several correlated biases at once.

Future Directions

• Estimating bias parameters without ground truth. The method needs parameters for the biasing mechanism, and the authors show accuracy is highest when these match the actual bias probability. Learning them jointly with the classifier is explicitly out of scope because the optimal classifier becomes unidentifiable; resolving this identifiability problem is an open direction.

• Extending beyond binary variables. The paper sketches how to model a multiclass label or a categorical sensitive variable (a probabilistic fact for every transformation and a rule for every value), but does not evaluate these settings experimentally.

• Richer real-world bias programs. The gap between synthetic and real-world performance is attributed to additional unmodeled biases, so more elaborate ProbLog programs that account for multiple interacting mechanisms are a natural next step.

• Choosing the right level of simplification. Simplified programs outperformed the more expressive ones on the Student dataset, and the authors offer competing explanations (parameter estimation error on scarce positive-bias data versus difficulty of learning under an expressive prior). This tension between expressiveness and learnability is left unresolved.

Target Audience

Researchers and practitioners working on algorithmic fairness who are comfortable with probabilistic models and want a toolbox for encoding task-specific bias assumptions rather than adopting a single fairness constraint. It also suits neurosymbolic AI researchers interested in applied DeepProbLog, data scientists working with biased tabular or image datasets, and fairness auditors who value explicit, inspectable models of how bias arises over post-hoc metric optimization. Readers without background in logic programming or probabilistic graphical models will need additional background, since the templates and their semantics are central to the method.

Authors’ abstract

Operationalizing definitions of fairness is difficult in practice, as multiple definitions can be incompatible while each being arguably desirable. Instead, it may be easier to directly describe algorithmic bias through ad-hoc assumptions specific to a particular real-world task, e.g., based on background information on systemic biases in its context. Such assumptions can, in turn, be used to mitigate this bias during training. Yet, a framework for incorporating such assumptions that is simultaneously principled, flexible, and interpretable is currently lacking. Our approach is to formalize bias assumptions as programs in ProbLog, a probabilistic logic programming language that allows for the description of probabilistic causal relationships through logic. Neurosymbolic extensions of ProbLog then allow for easy integration of these assumptions in a neural network's training process. We propose a set of templates to express different types of bias and show the versatility of our approach on synthetic tabular datasets with known biases. Using estimates of the bias distortions present, we also succeed in mitigating algorithmic bias in real-world tabular and image data. We conclude that ProbLog4Fairness outperforms baselines due to its ability to flexibly model the relevant bias assumptions, where other methods typically uphold a fixed bias type or notion of fairness.