Response to: DNA identification by pedigree likelihood ratio accommodating population substructure and mutations- authors' reply

In their letter to the editor, Egeland et al. [1] criticize the mutation model used in our paper [2], and propose that our comments about the mutation model used by Dawid et al. [3] are not convincing, because we do not provide any data in support of our assertions. Their criticisms are primarily based on three premises: 1) that our mutation model, presented on page 5 of our paper [2], is mathematically incorrect, because our equation 8 does not define a proper probability distribution (that is, the probabilities do not add to 1); 2) that our mutation model allows for production of alleles of zero or negative repeat sizes, which are not meaningful; and 3) that the model used in the paper by Dawid et al [3] uses the relationship between mutational transition probabilities and allele frequency on the basis that allele frequencies are representative of the stationary distribution of a mutation process, and hence, in the absence of natural selection, is presumably applicable to the sequence tagged repeat (STR) loci used in DNA forensics. Each of these issues needs further discussion, and we thank the authors for giving us an opportunity to explain them further.First, the mutation model, explained by equation 8 of page 5 of our paper [2], clearly states that the geometric distribution for Pr (X = x) applies to 'alleles to change by adding or subtracting an absolute number of x repeat units'. Hence, by definition x > 0, and as noted just after equation 8 'equal probabilities for gaining or losing repeats are assumed', it is incorrect to multiply the geometric terms by a factor of 2, as Egeland et al. have done [1]. Following this logic, our equation 8 mathematically represents a valid probability distribution, because the total probability of mutation (that is, X ≠ 0) becomes μ, by summing the individual terms over all non-zero positive integer values of X. In addition, we are not the first to use such formulations of a mutation model. Estoup et al. [4] used exactly the s