Importance sampling allows H_d true tests of highly discriminating DNA profiles

H_d true testing is a way of assessing the performance of a model, or DNA profile interpretation system. These tests involve simulating DNA profiles of non-donors to a DNA mixture and calculating a likelihood ratio (LR) with one proposition postulating their contribution and the alternative postulating their non-contribution. Following Turing it is possible to predict that "The average LR for the H_d true tests should be one"[1]. This suggests a way of validating softwares. During discussions on the ISFG software validation guidelines [2] it was argued by some that this prediction had not been sufficiently examined experimentally to serve as a criterion for validation. More recently a high profile report [3] has emphasised large scale empirical examination. A limitation with H_d true tests, when non-donor profiles are generated at random (or in accordance with expectation from allele frequencies), is that the number of tests required depends on the discrimination power of the evidence profile. If the H_d true tests are to fully explore the genotype space that yields non-zero LRs then the number of simulations required could be in the 10s of orders of magnitude (well outside practical computing limits). We describe here the use of importance sampling, which allows the simulation of rare events to occur more commonly than they would at random, and then adjusting for this bias at the end of the simulation in order to recover all diagnostic values of interest. Importance sampling, whilst having been employed by others for H_d true tests, is largely unknown in forensic genetics. We take time in this paper to explain how importance sampling works, the advantages of using it and its application to H_d true tests. We conclude by showing that employing an importance sampling scheme brings H_d true testing ability to all profiles, regardless of discrimination power.