Machine learning models are known to memorize samples from their training data, raising concerns around privacy and generalization. Counterfactual self-influence is a popular metric to study memorization, quantifying how the model's prediction for a sample changes depending on the sample's inclusion in the training dataset. However, recent work has shown memorization to be affected by factors beyond self-influence, with other training samples, in particular (near-)duplicates, having a large impact. We here study memorization treating counterfactual influence as a distributional quantity, taking into account how all training samples influence how a sample is memorized. For a small language model, we compute the full influence distribution of training samples on each other and analyze its properties. We find that solely looking at self-influence can severely underestimate tangible risks associated with memorization: the presence of (near-)duplicates seriously reduces self-influence, while we find these samples to be (near-)extractable. We observe similar patterns for image classification, where simply looking at the influence distributions reveals the presence of near-duplicates in CIFAR-10. Our findings highlight that memorization stems from complex interactions across training data and is better captured by the full influence distribution than by self-influence alone.

Our work addresses a fundamental problem in the context of counterfactual inference for Markov Decision Processes (MDPs). Given an MDP path $\tau$, this kind of inference allows us to derive counterfactual paths $\tau'$ describing what-if versions of $\tau$ obtained under different action sequences than those observed in $\tau$. However, as the counterfactual states and actions deviate from the observed ones over time, the observation $\tau$ may no longer influence the counterfactual world, meaning that the analysis is no longer tailored to the individual observation, resulting in interventional outcomes rather than counterfactual ones. Even though this issue specifically affects the popular Gumbel-max structural causal model used for MDP counterfactuals, it has remained overlooked until now. In this work, we introduce a formal characterisation of influence based on comparing counterfactual and interventional distributions. We devise an algorithm to construct counterfactual models that automatically satisfy influence constraints. Leveraging such models, we derive counterfactual policies that are not just optimal for a given reward structure but also remain tailored to the observed path. Even though there is an unavoidable trade-off between policy optimality and strength of influence constraints, our experiments demonstrate that it is possible to derive (near-)optimal policies while remaining under the influence of the observation.

Probabilities of Causation play a fundamental role in decision making in law, health care and public policy. Nevertheless, their point identification is challenging, requiring strong assumptions such as monotonicity. In the absence of such assumptions, existing work requires multiple observations of datasets that contain the same treatment and outcome variables, in order to establish bounds on these probabilities. However, in many clinical trials and public policy evaluation cases, there exist independent datasets that examine the effect of a different treatment each on the same outcome variable. Here, we outline how to significantly tighten existing bounds on the probabilities of causation, by imposing counterfactual consistency between SCMs constructed from such independent datasets ('causal marginal problem'). Next, we describe a new information theoretic approach on falsification of counterfactual probabilities, using conditional mutual information to quantify counterfactual influence. The latter generalises to arbitrary discrete variables and number of treatments, and renders the causal marginal problem more interpretable. Since the question of 'tight enough' is left to the user, we provide an additional method of inference when the bounds are unsatisfactory: A maximum entropy based method that defines a metric for the space of plausible SCMs and proposes the entropy maximising SCM for inferring counterfactuals in the absence of more information.