Likelihood Ratios Given Activity-Level Propositions for DNA Transfer Evidence: Theoretical Foundations of the HaloGen...

The interpretation of trace DNA evidence at activity level requires explicit modelling of transfer, persistence, and the possibility that a relevant actor leaves no detectable DNA. We present the theoretical foundations of HaloGen, an open-source hierarchical Bayesian framework for evaluating DNA quantities under competing activity-level propositions. The propositions are framed in terms of alleged activities or actions; direct and secondary transfer are treated as transfer mechanisms through which those activities may give rise to the observed DNA. HaloGen accounts for zero transfer, multiple contributors, specified unknown contributors, unobserved actors, and multiple stains. Evidence is evaluated using an exhaustive-propositions likelihood-ratio framework that combines information across contributors and stains while propagating uncertainty in transfer and detection. Observed DNA quantities and non-detects are handled within a single probabilistic framework: detected quantities are evaluated using conditional-on-detection densities, whereas the probability that a relevant actor leaves no detectable DNA is represented by an empirically constrained fail-rate parameter. The framework yields transparent and stable behaviour: informative DNA quantities can support propositions involving direct transfer, while low-information or no-detect situations are neutral or defence-conservative. The empirical-clamped fail-rate policy prevents spurious inflation of likelihood ratios when non-detection of a relevant actor is plausible. This paper establishes the theoretical basis of HaloGen; a companion paper addresses validation and applied casework examples.