"Statistics are the only tools by which an opening may be cut through the formidable thicket of difficulties that bars the path of those who pursue the Science of Man."

Francis Galton

Fingerprint match probability (FMP) is the probability that an arrangement of fingerprint ridge features matches to a particular source.  The T-Model infers fingerprint identification when FMP is less then the reciprocal of the relevant population (RP) for the case at hand (see below). 

Qualitative metrics are used to define reduced levels of ridge feature clarity, reliability and quality of agreement.  As a result, the term "T-Value" is used to represent the reciprocal of a quantitative-qualitative FMP. 

The T-Model expresses FMP as 1 divided by the total quantitative-qualitative value, or 1/T-Value, for an arrangement of ridge features present in a single impression or found in agreement between two impressions. 

The T-Model defines Relevant Population (RP) as the plausible number of people who could have left the latent fingerprint mark multiplied by 10 (number of fingers per person) multiplied by the estimated number of parts (P) per finger defined by T^P=10^120 (see The Formulae).

The formula used to establish inference for fingerprint identification may be expressed mathematically.  For example, in terms of FMP, it may be expressed as follows:

If FMP < 1/RP, then ID

where,

FMP = Fingerprint Match Probability (e.g., reciprocal of T-Value (T) or 1/T)

RP = Relevant Population for case at hand

ID = Inference for Identification

Therefore, inference for fingerprint identification is established when the final FMP is less then the reciprocal of the total population defined by the case at hand.   

In terms of a quantitative-qualitative positive T-Value, inference for identification may be expressed as follows:

If T/RP > 1, then ID

where,

RP = Relevant Population for the case at hand

T = T-Value

ID = Inference for Identification

Therefore, inference for fingerprint identification is established when the T-Value exceeds the Relevant Population for the case at hand.

Example #1

Let the T-Value for an arrangement of matching ridge features be 10^11, e.g., 11 clear, reliable ending ridges in a funnel, each with 0-1 intervening ridges to the nearest neighbor, and all are in excellent agreement.

Let the relevant human population for the case at hand be the default 300 million people (roughly the total United States human population -- similar to the default human population group used by the FBI for DNA analysis).

Based on a T-Value of 10^11, the number of fingerprint “parts” is defined by T^P=10^240 (see The Formulae), where T = T-Value and P = Number of fingerprint  “parts” per fingerprint per number of fingers in the relevant population for the case at hand. 

If T = 10^11, then P = 21.8.

As a result the Relevant Population (RP) can be calculated as follows: 

21.8 parts x 10 fingers x 300 million people = 65.4 billion  

Based on the above formula, e.g., "If T/RP > 1, then ID", the value 10^11/ 64.5 billion = 100 billion/65.4 billion, which is greater than 1.  As a result there is valid basis to establish inference for positive identification.

Example #2

Let the T-Value for an arrangement of matching ridge features be 1 less ending ridge feature then in the above example, e.g., 10^10, which represents the aggregate weight for 10 clear, reliable ending ridges in a funnel, each with 0-1 intervening ridges to the nearest neighbor, and all are in excellent agreement.

Based on a T-Value of 10^10, the number of fingerprint “parts” is defined by T^P=10^240 (see The Formulae), where T = T-Value and P = Number of fingerprint  “parts” per fingerprint per number of fingers in the relevant population for the case at hand. 

If T = 10^10, then P = 24.

As a result the Relevant Population (RP) can be calculated as follows: 

24 parts x 10 fingers x 300 million people = 72 billion  

Based on the above formula, e.g., "If T/RP > 1, then ID", the value 10^10/ 64.5 billion = 10 billion/65.4 billion, which is less than 1.  As a result there is not valid basis to establish inference for positive identification.

"Nature permits us to calculate only probabilities." 

Richard P. Feynman

Likelihood Ratio to Express Significance of a Match

The following was written by Christophe Champod on Sept. 28, 2009 for the Fingerprint Inquiry investigating the matter of Shirley McKie [104]:

“The most recent published studies on fingerprint selectivity are all anchored in a likelihood ratio based framework. In these studies, the forensic findings are the outcome of a metric describing the distance between the features under examination (between the mark and the print).”, and “… the aim of all these models is to obtain an assessment of the numerator and the denominator of the likelihood ratio.”

It is significant to note that the T-Model is not exclusively anchored in a likelihood ratio framework, and it does not aim to express the significance of a match based solely on an assessment of the numerical quantitative-qualitative value for arrangements of ridge features.  Instead, the primary aim of the T-Model is make the best, most accurate estimate for the number of look-alikes present in the relevant population for the case at hand, and when the estimate for the number of look-alikes for an arrangement of matching ridge features present in two impressions is “less than 1”, then it establishes inference for identification.  It is not the examiner who defines the ”discrimination value” for individual ridge features and clusters of ridge features, and then decides whether or not the value is high enough to infer identification.  It is the T-Model that makes this estimate and infers a “match”, not the examiner. 

The T-Model establishes and incorporates quantitative-qualitative "T-Values” for arrangements of ridge features, but those values are meaningless without an estimate for the plausible number of people who could have left the latent print at the scene of the crime and what is the correlation between these two variables.  As the plausible number of people who could have left a latent print at a crime scene increases so does the number of look-alikes present in the relevant population for that case.  Therefore, the variable “relevant population” must be taken under consideration when assessing the value, e.g., T-Value or Likelihood Ratio, for arrangements of ridge features and whether or not they are “sufficient” to infer identification.  The T-Model incorporates relevant population into the formulae to establish inference for identification; to the author’s knowledge, only the Likelihood Ratio model developed by Cedric Neumann [96] does the same.

"So far, the T-Model is the only statistical probability model with defined demarcation thresholds to infer identification and the only model to successfully identify the largest and best fingerprint "look-alikes " as insufficient to infer identification."

                                                                                Henry Templeman

Statistical Probability Models and AFIS

The job of AFIS is to find the best “look-alikes” present in its database.  The best look-alikes and most notable erroneous fingerprint identification ever made were the result of AFIS searches, e.g., The Clark non-match [link], the Chesapeake non-match [link], and the Brandon Mayfield non-match [link]).  Since the number of look-alikes for an arrangement of ridge features present in a population correlates with population size (e.g., as population size increases so do the number of look-alikes) the relevant population variable must be included in any fingerprint model.

The following example of likelihood ratio value for an amount of matching ridge features is presented by Christophe Champod (see below image) [104]:  The likelihood ratio for the below arrangement of ridge features is estimated to be 295843, or roughly 300 thousands.  As a result “the forensic findings annotated here [between the mark and the print] are 300 thousand times more likely to be observed if the mark originates from the same source as the print than from a different source”.  The below mark and prints happen to be from the same source, however, if the exemplar is the result of, for example, an FBI IAFIS search bearing a fingerprint population of over half billion individual fingerprint records, then the best look-alikes found by IAFIS for this partial, fragmented latent print mark may very well have significantly high likelihood ratios and thereby provide false or unjustified support to infer identity of source.

Any probabilistic model should clearly define demarcation thresholds to infer identification and be able to accurately ferret out the largest and best look-alikes any AFIS can find as insufficient to infer identification and thereby not provide strong support for the view that the two impressions came from the same source.  The T-Model has shown to be able to ferret out the largest and best look-alikes AFIS can find as insufficient to infer identification [link].  To the author's knowledge, no probabilistic model using a likelihood ratio format can make the same claim.

"The T-Value represents a theory of correspondence between quantitative number and qualitative description.  The product of quantitative numerics and qualitative metrics is the T-Value for the arrangement of ridge features present in two impressions.  When the T-Value exceeds the relevant population for the case at hand, then less than 1 look-alike is estimated, and subsequently probabilistic inference for identification to a single source is justified." 

Henry Templeman

NEXT PAGE >>>

The T-Model establishes inference for fingerprint identification when the fingerprint match probability (FMP) is less then the reciprocal of the relevant population (RP) for the case at hand. 

"The numerator of the likelihood ratio asks for the probability of the evidence if the suspect has left the recovered evidence.  This probability is not systematically equal to one and must be assessed in each case taking into account the intra-variability of the whole process that generates the mark." [34]