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.

Comment on the Use of "Likelhood Ratios" for Fingerprints

It may be proposed that for an arrangement of ridge formations in a latent or crime scene print that matches a known suspect, fingerprint examiners need to consider two hypotheses, known as the prosecution hypothesis and the defense hypothesis. The prosecution hypothesis is that the fingerprint evidence came from the suspect, who left it at the crime scene; while the defense hypothesis is that the fingerprint didn't come from the suspect, but from a different person in the population. The likelihood ratio is a statistical measure that it seems may consider the relative probabilities of these two hypotheses being true, and in doing so provide an indication of the value of the fingerprint evidence.

However, the numerator takes into account discrepancies which speaks to the strength of the non-corresponding ridge features found in two impressions.  For example, on rare occasion, both corresponding and non-corresponding ridge features (absent clear distortion markers) are observed in fingerprints (see Non-Corresponding Ridge Events).  Dissimilarities observed in two impressions which also bear similar features should not be "explained away" without ability to demonstrate how the dissimilarity occurred, e.g., reproduce it to the satisfaction of the trier of fact and in court, if necessary.

Due to difficulties associated with defining a quantitative-qualitative value of the numerator for the likelihood ratio, e.g., in particular qualitative values for reduced levels of clarity, reliability and quality of agreement, as well as values for various levels of non-corresponding ridge features found in two impressions, especially in a corresponding ridge feature framework, the use of likelihood ratios to express the significance of a match is not applied to the T-Model. 

The T-Model expresses the significance of a match and the total quantitative-qualitative value of an arrangement of fingerprint ridge features present in two impressions as a "T-Value".  

"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 probabilistic inference for identification to a single source is justified." 

Henry Templeman

 

 

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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]