Generative Models and Probability Evaluation for Forensic Evidence

Abstract

Generative approaches to pattern recognition and machine learning involve two aparts: first describing the underlying probability distributions and then using such models to compute probabilities or make classificatory decisions. We consider generative models for forensic evidence where the goal is to describe the distributions using graphical models and to use such models to compute probabilistic metrics for measuring the degree of individuality of a forensic modality or of a piece of evidence. The metrics are defined as variations of the probability of random correspondence (PRC) when evidence consists of a set of measurements and correspondence is within a tolerance. Three metrics are defined, the first two of which concern the modality and the third concerns evidence: 1) PRC of two samples, 2) PRC among a random set of n samples (nPRC), and 3) PRC between a specific sample among n others (specific nPRC). Computation of these probabilities are described using graphical models which makes all the variables explicit. The metrics are evaluated for several cases-some of which are illustrative (birthdays and heights) and others concern fingerprints. For birthdays, which are discretevalued and exact, assuming uniformly distributed birthdays, while nPRC rapidly approaches unity with higher n (which is the well-known birthday paradox), specific nPRC grows much less rapidly. For human heights, which are continuous-valued scalars, a quantization is needed and results are genderspecific: assuming Gaussian distributed heights, the PRC for males is higher than for females due to lower variance. Two forms of fingerprint representation are considered: ridge flow and minutiae. Gaussian mixtures are used to model location and orientation of minutiae. With parameters estimated from standard databases (using expectation maximization), fingerprint PRCs are determined for given numbers of available and matching minutiae (which correspond to quantization), for the case of 36 available minutiae where 24 of them match, the PRC is 4.0 × 10 −34 . The methodology put forward should be value to establish the relative value of different types of forensic evidence in courtroom scenarios.

Quantifying the Weight of Evidence from a Forensic Fingerprint Comparison: A New Paradigm

“…Finally, Srihari and Su (2011) recently proposed probability of random correspondence as a way of quantifying fingerprint matching. Their model development is based on capturing the distribution of minutiae locations and their orientations.…”

“…Although the model developed by Srihari and Su (2011) appears to be easier to implement, its performance on large data sets has not yet been put to test. On balance, comparison of such competing models for quantification of the weight of evidence goes a long way in the process of refining the methodology proposed.…”

A Bayesian hierarchical model for evaluating forensic footwear evidence

“…where y ≡ s indicates that shoe s exhibits features consistent with those of the print y, and s ∼ A is shorthand for s being chosen uniformly at random from all shoes in the set A. Further discussion of random match probabilities with examples from forensic science is available in Srihari and Su (2011).…”