## Likelihood Ratios And Bayesian Statistics

A likelihood ratio (LR) involves a comparison of the probabilities of the evidence under two alternative propositions. As will be described later in this chapter, these two propositions are often referred to as the null hypothesis and the alterative hypothesis. In forensic DNA settings, these mutually exclusive hypotheses represent the position of the prosecution - namely that the DNA from the crime scene originated from the suspect - and the position of the defense - that the DNA just happens to coincidently match the defendant and is instead from an unknown person out in the population at large (see Chapter 21). In mathematical terms, the likelihood ratio is written as:

LR = Hp/Hd where Hp represents the hypothesis of the prosecution and Hd represents the hypothesis of the defense. The likelihood ratio is used in Bayes' theorem to relate the probabilities of the propositions after the evidence to the probabilities prior to the evidence (Weir 2003):

Likelihood Ratio which can be stated as the posterior odds on Hp are equal to the LR times the prior odds on Hp.

Prior odds relates to the relative guilt or innocence of the suspect. Thus, in order to perform this calculation, one must make assumptions about the prior odds of guilt or innocent. As you might imagine, this approach has not caught on in the United States where the judicial system tries to maintain 'innocent until proven guilty.' However, there is nothing wrong with using the likelihood ratio by itself and have the judge and jury decide on the prior and post odds of guilt or innocence.

A good DNA typing system should provide large likelihood ratios when the defendant and the perpetrator of a crime is the same person. Likewise, if they are different people, then the likelihood ratio will be less than 1. Relative levels of likelihood ratios will be discussed in more detail in Chapter 21.

### STATISTICS

Statistics is the science of uncertainty and its measurement. It also provides a sense of how reliable a measurement is when the measurement is made multiple times. Statistics involves using samples to make inferences about populations. A population is considered in this context to be a set of objects of interest, which may be infinite or otherwise unmeasurable in their entirety. An observable subset of a population can be referred to as a sample with a statistic being some observable property of the sample. In the context of DNA testing, the 'population' would be the entire group of individuals that could be considered (e.g., billions of people around the world or those living within a particular country or region). The 'sample' would be a set of individuals from the population at large (e.g., 100 males) that were selected at random and tested at particular genetic markers to try and establish a reliable representation of the entire population. The 'statistic' examined might be the observed allele or genotype frequencies for the tested genetic markers.