Likelihood Ratios Given Activity-Level Propositions for DNA Transfer Evidence: Theoretical Foundations of the HaloGen Framework (Part I)

The interpretation of trace DNA evidence at activity level requires explicit modelling of transfer, persistence, and failure to detect a person of interest. We present the theoretical foundations of HaloGen, an open-source hierarchical Bayesian framework for evaluating biological results under competing activity-level propositions, such as direct versus secondary transfer. HaloGen accounts for dropout, multiple contributors, and multiple stains. Evidence is evaluated using an exhaustive-propositions likelihood ratio frame-work that combines information across contributors and stains, while fully accounting for uncertainty in transfer and detection. Observed DNA quantities and non-detects are handled consistently within a single probabilistic model, avoiding reliance on fixed parameter estimates. The framework yields intuitive and robust behaviour: strong support for direct transfer when DNA quantities are informative, and appropriately neutral or defence-leaning likelihood ratios in low-information or non-detect scenarios. An empirically constrained fail-rate parameter prevents spurious inflation of likelihood ratios when offender detection is unlikely, providing stability across laboratories and experimental conditions. This paper establishes the theoretical basis of HaloGen; a companion paper addresses validation and applied casework examples.