Pr(Gj)(Gj) is computed under a standard population genetics model

Pr(Gj)(Gj) is computed under a standard population genetics model [1]. The unknown parameters ϕ can be replaced with estimates, or eliminated by maximisation or integration with respect to a prior distribution. Currently, there are only limited possibilities to check the validity of an algorithm for evaluating an LTDNA

LR (henceforth ltLR). One approach is to evaluate the ltLR when Q is repeatedly replaced by a random profile [3]. In that case H  p is false and we expect the majority of computed ltLRs to be find more small. Here, we propose to investigate a performance indicator for ltLR algorithms when H  p is true. Under H  d, it may occur that GX=GQGX=GQ, where GXGX and GQGQ denote the genotypes of X and Q. This occurs with probability π  Q, the match probability for Q. Since Pr(E|Hd,GX=GQ)=Pr(E|Hp)(E|Hd,GX=GQ)=Pr(E|Hp), it follows that [4] equation(3) ltLR=Pr(E|Hp)Pr(E|Hd,GX=GQ)πQ+Pr(E|Hd,GX≠GQ)(1−πQ)≤1πQ.We will refer to 1/πQ as the inverse match probability (IMP). Consider first that Q is the major contributor to an LTDNA profile. Intuitively, if E   implies that GX=GQGX=GQ then equality should

be achieved in Eq. (3). The key idea of this paper is that if H  p is true then increasing numbers of LTDNA replicates should provide increasing evidence that GX=GQGX=GQ, and so the ltLR should converge to the IMP. Compound C in vivo This holds even for mixtures Myosin if Q is the major contributor, since differential dropout rates should allow the alleles of Q to be identified from multiple replicates. However, any inadequacies in the underlying mathematical model or numerical approximations may become more pronounced with increasing numbers of replicates, preventing the ltLR from approaching the IMP. Therefore we propose to consider convergence of the ltLR towards the IMP as the number

of replicates increases as an indicator of the validity of an algorithm to compute the ltLR when Q is the major contributor. If Q is not the major contributor, even for many replicates there may remain ambiguity about the alleles of Q so that there remains a gap between the ltLR and IMP. However, the bound (3) still holds, and there is a useful guide to the appropriate value of the ltLR provided by the mixture LR for good-quality CSPs computed using only presence/absence of alleles [5]. If under Hp the contributors are Q and U, where U denotes an unknown, unprofiled individual, and Hd corresponds to two unknown contributors X and U, an example of a mixture LR is equation(4) mixLR=Pr(CSP=ABC,GQ=AB|Q,U)Pr(CSP=ABC,GQ=AB|X,U)=Pr(GUisoneofAC,BC,CC)Pr((GX,GU)isoneof(AA,BC),(AC,BB),(AB,CC),(AB,AC),(AB,BC),(AC,BC)),where within-pair ordering is ignored in the denominator.

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