Evaluation of Scientific Evidence Based on Likelihood Ratio as a Function of Measurement Uncertainty for Different Continuous Distribution Functions

Abstract

Likelihood Ratio  serves as a substantial basis in evaluating scientific evidence in Bayesian analysis and is a fundamental engine in terms of understanding the forensic decision-making in legal matters. As for continuous probability distribution types, the accuracy of calculations of likelihood ratios is determined not only by the form of unknown probability distribution but also by the maximum measurement error of the measured amount as uncertainty of measurement. In this paper the effect of the value of measurement uncertainty on the likelihood ratio is considered by means of the exact calculation procedure based on probability integral and the probability density function approximation technique as well as the approximation technique in regard to several models of continuous distributions. Normal and Student’s t-distributions are considered to demonstrate the cases of large and limited sample sizes correspondingly. The developed extended derivative of  formulas is applied in a numerical example of a real legal case concerning the glass refractive index measurement and the results are evaluated in regard to the range of uncertainty in the measurement and the degrees of freedom of different samples. The obtained results indicate that the approximation technique tends to converge to the exact solution as the uncertainty in measurement becomes lower. For normally distributed continuous data the methods provide close likelihood ratios when the uncertainty in measurement is equal to approximately 2% or less of the measured value. For the t-distribution with 2 degrees of freedom that is typically used in practice, the convergence occurs at approximately 4.5% or less. Moreover, as the degrees of freedom increase the t-distribution converges to the normal one and provides similar estimates of the likelihood ratio. The suggested approach provides practical guidelines for choosing the appropriate formula of likelihood ratio calculation in regard to the underlying distribution, sample size and the uncertainty of measurement.