Posterior likelihood ratios for evaluation of forensic trace evidence given a two-level model on the data

Lead isotope ratios for bullets, forensic evaluation in a Bayesian paradigm

“…Later work [12], [13] centered on the development of a likelihood ratio in the more general multivariate case, and where the distribution 3 of between item means may not necessarily be normal in form. Further advances have come from Alberink et al where a Bayesian approach has been taken to model parameter estimation [14]. The dearth of proper statistical treatment, the lack of experimental planning and other factors, were some of the reasons the Federal Bureau of Investigation in the United States of America (FBI) abandoned the chemical analysis of trace elements in bullet lead (CABL) [15].…”

Bayesian Hierarchical Random Effects Models in Forensic Science

“…Later work on evidence evaluation has extended the work done in Lindley ( 1977 ) to cover other data types, allowing for different forms of the within and between source distributions (Aitken and Lucy, 2004 ; Aitken et al, 2006 , 2007a ). In Bozza et al ( 2008 ) and Alberink et al ( 2013 ), extensions are given so that the between-source distribution in Equation (9) becomes a function of both the mean and the variance. This allows for variation in the variance of samples from different sources.…”

Avoiding overstating the strength of forensic evidence: Shrunk likelihood ratios/Bayes factors

“…The first stage of a score-based procedure can be considered a procedure for extracting information about the similarity of the questioned-origin data with respect to the known source and their typicality with respect to the relevant population, 8 and projecting the complex multidimensional feature space down to a univariate score space. The second stage then fits simple models to the univariate scores.…”

“…Such procedures are also potentially problematic because they are not monotonic and do not naturally produce well calibrated results by estimating a small number of parameters using a relatively large amount of data. 8 Scores should be calculated in a manner which captures information about both similarity and typicality, i.e., the procedure for calculating a score should be an attempt to estimate a log likelihood ratio value. Scores based only on similarity lack information about the typicality of the questioned-origin data with respect to the relevant population, and that information cannot be adequately incorporated as part of the score to likelihood ratio conversion stage [26].…”

“…1 If the amount of sample data is large, the coverage interval will be small, hence there will be little adjustment to the reported value, but if the amount of sample data is small, the coverage interval will be large, hence the reported value will be substantially closer to 1. For discussion and examples of approaches of this sort, see [1][2][3][4][5][6][7][8][9]. From a subjectivist Bayesian perspective, there are no true but unknown population distributions, and the value of a Bayes factor (which is the Bayesian counterpart of the frequentist likelihood ratio) is a state of belief, not an estimate of a true but unknown value (in the context of evaluation of forensic evidence, this position is espoused in, for example [10,11]).…”