ProbLog4Fairness: Modeling Bias in AI

• The framework defines algorithmic bias using explicit probabilistic logic programs that model label, measurement, and historical bias.
• It integrates symbolic reasoning with neural learning via DeepProbLog to predict latent unbiased outcomes from only biased observations.
• Empirical evaluations show that modeling bias explicitly can achieve accuracy close to ideal baselines while preserving non-problematic correlations.

ProbLog4Fairness is a neurosymbolic fairness framework that models algorithmic bias through explicit probabilistic logic programs rather than by imposing a fixed fairness constraint such as statistical parity or equalized odds. It formalizes task-specific bias assumptions in ProbLog, a probabilistic logic programming language that allows the description of probabilistic causal relationships through logic, and integrates those assumptions into neural-network training through DeepProbLog. The framework is designed for settings in which only biased observations are available during training, while the learning objective is to recover predictions with respect to latent unbiased labels or latent unbiased features (Adriaensen et al., 12 Nov 2025).

1. Conceptual orientation

ProbLog4Fairness is motivated by the observation that operationalizing fairness definitions is difficult because multiple definitions can be incompatible while each is arguably desirable. The paper therefore shifts the focus from directly optimizing a fixed fairness notion to explicitly modeling the mechanism by which unfairness enters the data. A canonical fairness formula discussed in the paper is statistical parity,

$$P(\hat Y = 1 \mid A = 1) = P(\hat Y = 1 \mid A = 0),$$

where $A$ is the sensitive attribute and $\hat Y$ is the predicted label. The paper’s position is not that such metrics are useless, but that optimizing them directly can be too coarse for many applications (Adriaensen et al., 12 Nov 2025).

The proposed alternative is to specify bias mechanisms such as label bias, measurement bias, and historical bias. In this formulation, observed variables are biased versions of latent fair variables. The learner is then trained to predict as if it had access to the unbiased world. This makes bias mitigation a model-based correction problem rather than a direct metric-enforcement problem.

The framework is explicitly neurosymbolic. It combines symbolic structure in the form of logic rules, probabilistic semantics in the form of ProbLog programs over possible worlds, and neural learning via DeepProbLog. This yields a system in which fairness-relevant assumptions are inspectable at the level of rules and probabilities, while the predictive component remains a trainable neural classifier.

2. ProbLog representation of bias assumptions

The central formal objects in ProbLog4Fairness are the sensitive variable $A$, unbiased features $\mathbf{X}$, biased observed features $\tilde{\mathbf{X}}$, unbiased label $Y$, biased observed label $\tilde Y$, and prediction $\hat Y$. The goal is not to minimize error on the biased distribution, but to learn a classifier that minimizes error relative to the unbiased labels $Y$, even when only biased observations $A$0 are available during training (Adriaensen et al., 12 Nov 2025).

Bias assumptions are encoded in ProbLog by probabilistic facts and probabilistic rules. For binary variables, the paper states that a bias mechanism is fully specified by four probabilities that distinguish negative distortion versus positive distortion and sensitive-group membership versus non-sensitive-group membership. The generic form is

$A$1

These parameters can be supplied from domain knowledge or estimated from a small subset of data for which both biased and unbiased variables are observed. The paper explicitly avoids jointly learning these parameters with the classifier because that creates identifiability problems: multiple classifier-bias decompositions may fit the observed data equally well, including biased ones.

A notable modeling choice is that ProbLog is treated as a declarative alternative to manually handling Bayesian networks. The paper states that the encoded mechanism is equivalent in spirit to a BN, but uses ProbLog because practitioners may find standard causal or Bayesian models harder to interpret and harder to integrate into data pipelines.

3. Neurosymbolic learning with DeepProbLog

The neural component of ProbLog4Fairness predicts the probability of a latent unbiased atom and inserts that probability into the logic program as a neural probabilistic fact: $A$2 Here the network does not directly predict the observed biased label. Instead, it predicts the latent unbiased label, and the symbolic bias model explains how that latent quantity gives rise to the observed biased outcome (Adriaensen et al., 12 Nov 2025).

Training proceeds through distant supervision. The observed training signal is attached to biased labels or biased features, while DeepProbLog computes the probability of those observations under all latent unbiased interpretations consistent with the program. The paper describes the overall computational graph as

$A$3

Because the query probability is differentiable with respect to the neural probabilities, gradients can be propagated back through the compiled probabilistic circuit.

The paper does not introduce a new bespoke closed-form fairness objective. Instead, it relies on standard DeepProbLog training with standard neural losses. It mentions binary cross-entropy for most experiments and focal loss for the Student experiment due to class imbalance. The structural point is that the program marginalizes over latent unbiased variables. The details describe training as effectively guided by

$A$4

for the relevant bias mechanism. This suggests that the framework functions as a hidden-variable model whose latent structure is explicitly encoded in probabilistic logic.

4. Bias templates

The paper proposes three main templates: label bias, measurement bias, and historical bias. They share the same high-level structure but differ in which variables are treated as latent and which distortion process is modeled.

Label bias

Label bias assumes that unbiased features $A$5 are available, but the observed label $A$6 is a biased proxy for the latent fair label $A$7. The governing conditional distribution is

$A$8

The ProbLog template is

$A$9

This means that the observed biased label is positive either because the latent fair label is negative but gets positively biased, or because the latent fair label is positive and is not negatively biased (Adriaensen et al., 12 Nov 2025).

The paper’s loan example makes the mechanism explicit: $\hat Y$0 This separates discriminatory rejection from random annotation noise.

Measurement bias

Measurement bias assumes that observed features $\hat Y$1 are distorted proxies of fair features $\hat Y$2, while the label depends on the fair features. For a feature $\hat Y$3, the relevant conditional table is

$\hat Y$4

The paper gives the single-feature template

$\hat Y$5

The appendix expands this to multiple features through recursive debiasing. The resulting prediction is a weighted combination over possible debiased feature values. Mitigation thus occurs by probabilistically reconstructing plausible unbiased features and then classifying with respect to them.

Historical bias

Historical bias is modeled as a situation in which the features themselves are historically unfair and the labels are generated from those historically distorted features. The paper describes this as a composition of a measurement-bias-style transformation from historically biased to unbiased features and, if needed, a label-bias-style transformation from unbiased to biased labels (Adriaensen et al., 12 Nov 2025).

A special simplifying assumption plays a central role: if the function from biased features to biased labels is the same as the function from unbiased features to unbiased labels, then a classifier trained on biased features and labels can still make fair predictions when given unbiased features at test time. Under that assumption, mitigation is achieved by training on biased data and applying the measurement-bias debiasing mechanism at test time. This gives historical bias a counterfactual flavor: the predictor is asked to operate on a feature distribution corresponding to a normatively preferable world.

5. Empirical evaluation

The empirical study spans synthetic tabular data, a real-world student dataset, and the CELEB-A image dataset. The baselines are Lower, Upper, Unawareness, Massaging, and Error parity. Evaluation uses accuracy for balanced cases, F1 score when labels are imbalanced, and fairness metrics centered on statistical disparity, with equalized odds additionally reported in appendix analyses (Adriaensen et al., 12 Nov 2025).

For the synthetic tabular data, the generator is adapted from Bias on Demand and uses variables $\hat Y$6, $\hat Y$7, $\hat Y$8, $\hat Y$9, $A$0, $A$1, $A$2, and $A$3. The paper gives the generation equations

$A$4

When the encoded bias type and parameters match the true data-generation mechanism, ProbLog4Fairness achieves accuracy and statistical disparity close to the Upper baseline while being trained only on biased data. An especially important reported result is the case $A$5: ProbLog4Fairness matches the Upper baseline statistical disparity, which is not zero, whereas other fairness baselines incorrectly force disparity toward zero. The paper also reports that sensitivity to moderate misspecification of the assumed bias probability is small near the optimum.

The Student Alcohol Consumption dataset contains 856 students with tabular features such as alcohol consumption, gender, and study time, and the task is to predict whether the student passed the exam. The original pass/fail labels are treated as unbiased labels, while a second set of labels is supplied by human annotators primed to produce bias against male students. The modeled mechanism is label bias under the simplifying assumption

$A$6

The reported outcome is that ProbLog4Fairness approaches the expected statistical disparity from the unbiased labels and outperforms all mitigating baselines in F1 score on unbiased labels. Simpler variants of the program, including “no positive bias” and “no bias on $A$7,” perform even better in some cases.

In CELEB-A, the task is to predict the attribute Mouth Slightly Open using a cleaned version of that attribute from prior work. The correlated attributes Smiling, Blurry, and High Cheekbones are treated as sensitive variables for label bias modeling. The model uses a ResNet-50 pretrained on ImageNet, fine-tunes the full network with learning rate $A$8, batch size 16, and BCE loss, and applies a sequential label-bias program over the three attributes. The Upper baseline improves F1 by 10.18% over training on original labels, and ProbLog4Fairness recovers part of this gain. A particularly notable finding is that jointly modeling multiple sensitive or bias attributes works substantially better than correcting for only one.

6. Interpretation, relation to adjacent frameworks, and limits

ProbLog4Fairness is best understood as a framework for model-based bias mitigation rather than direct fairness verification or direct fairness-constrained empirical risk minimization. Earlier work such as "Fairness as a Program Property" (Albarghouthi et al., 2016) and "Quantifying Program Bias" (Albarghouthi et al., 2017) treats fairness as a probabilistic property of a decision-making program relative to a population model, while "Probabilistic Verification of Fairness Properties via Concentration" (Bastani et al., 2018) provides high-confidence sampling-based certification for black-box predictors. By contrast, ProbLog4Fairness uses explicit bias assumptions to train a predictor of latent unbiased variables. A plausible implication is that it occupies a different methodological niche: it does not primarily prove that a fixed model satisfies a fairness predicate, but instead modifies learning so that the predictor better reflects an assumed unbiased world.

This differs as well from direct fairness-aware probabilistic classification under log loss. "Fairness for Robust Log Loss Classification" (Rezaei et al., 2019) derives a fairness-aware probabilistic classifier from a minimax robust log-loss formulation, whereas ProbLog4Fairness does not derive a new closed-form classifier. Its novelty lies instead in the fairness-oriented modeling layer implemented in ProbLog and DeepProbLog.

Several caveats are explicit. The method is only as good as the encoded bias model. Bias probabilities must be specified externally, either from domain knowledge or from a small unbiased calibration subset, because jointly learning them with the classifier leads to identifiability problems. Real-world performance need not approach the Upper baseline when important biases remain unmodeled. The appendix reports substantial runtimes, including synthetic sweeps up to approximately 72 hours and CELEB-A sequential reproduction time around 100 hours.

The paper also provides a Hoeffding-based sample complexity bound for estimating bias probabilities: $A$9 where $\mathbf{X}$0 is the desired estimation error and $\mathbf{X}$1 the confidence level. The paper gives the example that 10% error and 95% confidence require 184 datapoints in the sensitive group with unbiased positive labels. This operational detail underscores the framework’s practical dependence on reliable calibration data.

At a conceptual level, ProbLog4Fairness does not equate fairness with forcing all group disparities to zero. Its most distinctive empirical and methodological feature is the ability to preserve non-problematic correlations when the assumed unbiased world still contains them. That is why the framework is presented not as a universal fairness criterion, but as a flexible, interpretable, and probabilistically grounded language for stating what kind of bias is believed to be present and for training under that assumption (Adriaensen et al., 12 Nov 2025).