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

Abstract 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 Halo-Gen, 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. Competing Interest Statement The authors have declared no competing interest. Footnotes We now state explicitly that activity-level propositions concern alleged activities or actions; direct and secondary transfer are modelled as transfer pathways implied by, or compatible with, those activities. We replaced potentially confusing use of 'structural dropout' with 'zero transfer' / 'no-transfer event', to avoid confusion with allele dropout at the sub-source level. We clarified that the conditional density for a detected contributor quantity does not imply that the probability of observing DNA under an activity is one. Instead, it separates the magnitude-of-detected-DNA question from the probability that an actor leaves no detectable DNA, which is represented by F0. We clarified the rationale for equal prior probabilities per elemental hypothesis convention in section 2.4. We expanded the limitations section to emphasise that TPPR data must be case-relevant, that experimental designs rarely match case circumstances perfectly, and that the model should not be applied mechanically without proposition- and data-relevance assessment. We removed Section 3.5 on 'interpretable AI' and retained only a modest statement that the model is transparent and auditable as a structured Bayesian model. We added explicit practitioner reporting rules for no-detect-only situations and for cases where contributor-level quantities cannot be reliably assigned across stains.