## "Converting" a likelihood ratio to a probability

The result of evaluating DNA evidence – be it a simple stain a mixture, or a kinship situation – is a likelihood ratio. Rightly or wrongly many analysts are reluctant to present a likelihood to the court and sometimes I have been asked how to convert the likelihood ratio to a probability.

At the word "convert" I bristle. As you can infer form my use of quotation marks I am a bit worried about the use of the word "convert" because of the implicit assumption that likelihood ratio and probability are simply two different ways of expressing the same thing, just as 0.3 and 30% are, or as 1 meter and 100 cm are. But it's not like that at all.

Probability (call it W) – meaning posterior probability – depends on likelihood ratio (L) and in addition on prior probability (p). The formula is

W = pL / (pL + 1-p).

So for a given L, you can end up with any W whatever, depending on what p is. And there is no possible "preferred" value of p, so it is impossible for the laboratory to report a W value to the court. One possible approach, though, to avoid talking about likelihood ratios in court, is to consider a range of possible values for p and give a chart showing the various resultant W's.
 If L=10,000,000 for example, if p>1/1,000,000 then w>91% if p=1/1000 then W=99.99%, if p=1/100 then W=99.999%, if p=1/10 then W=99.9999%, if p=1/2 then W=99.99998%.
Deciding which line of the chart is the relevant one is a decision that is made by the court based on its evaluation of the non-DNA part of the evidence. That is true not only in principle, but also and in the practice of every criminal jurisdiction I have ever heard of.

Admittedly, for family court – for routine civil paternity decisions – the custom is somewhat different. The principle is the same but the practice is that the court is used to receiving paternity reports based on the assumption that p=1/2. It does not do much harm, since only money and emotions are at stake and in civil matters there is no principle comparable to "better 10 guilty go free than one innocent be convicted" because in civil matters a wrong decision in either direction is unjust to one of the parties.

If I've overreacted to a word, sorry for the lecture!