1. Thinking About DNA Database Searches William C. Thompson Dept. of Criminology, Law & Society University of California, Irvine

2. Value of DNA Match for Proving Identity Prior Odds x Likelihood Ratio = Posterior Odds May be very low 1 x ------------------ RMP + FPP* *Actually RMP + [FPP x (1-RMP)], see Thompson, Taroni & Aitken, 2003 1:1 million x 1 billion:1 = 1000:1 1:1 million x 1 million:1 = 1:1 1:1000 x 10,000:1 = 10:1

3. Mysterious Clusters and the Law of Truly Large Numbers In a truly large sample space, seemingly unusual events are bound to occur –E.g., double lottery winners; cancer clusters –See, Diaconis & Mosteller (1989). Methods for studying coincidences, JASA, 84 853-861.

4. Taking Account of Coincidence When Searching Truly Large DNA Databases Should the frequency of the matching profile be presented to the jury? Standard answers: –No NRC I – test additional loci; report only freq. of those NRC II—multiply freq. by N (for database) –Yes Friedman/Donnelley—present LR but keep in mind prior odds may be very low Prosecutors Everywhere—jury should hear most impressive number possible “because it’s relevant”

5. My Solution: Present Profile Frequency Only When It Equals the RMP* Multiple Tests of Different Hypotheses –Search unsolved crime evidence against offender database –For each offender, p(match|not source) = frequency Multiple Tests of Same Hypothesis –Search suspect against unsolved crime database to see if he matches any unsolved crime –For this suspect, p(match|not source) = Freq. x N *RMP = p(match|suspect not the source)

6. My Solution: Present Profile Frequency Only When It Equals the RMP* Testing relatives of people who almost match –For most suspects, p(match|not source) = frequency of matching profile –For relatives of people who almost match, p(match|not source) >>>> frequency –Therefore it is misleading to present the frequency of the matching profile in cases where the suspect is selected because a relative almost matches

7. Database Searches and the Birthday Problem The probability that a randomly chosen person will have my birthday is 1 in 365 The probability that any two people in a room share a birthday can be far higher –With 23 people in a room, the likelihood that two will share a birthday exceeds 1 in 2 –With 60 people in the room, the probability is nearly 1 in 1

8. Database Searches and the Birthday Problem Suppose the probability of a random match between any two DNA profiles is between 1 in 10 billion and 1 in 1 trillion What is the probability of finding a match between two such profiles in a database of: –1,000 –100,000 –1,000,000

9. Approximate likelihood that two profiles in a DNA database will match Database Size1 in 10 billion 1 in 100 billion 1 in 1 trillion 10001 in 20,0001 in 200,0001 in 2 million 10,0001 in 2001 in 20001 in 20,000 100,0001 in 2.51 in 201 in 200 1,000,0001 in 1 1 in 2.5 Profile Frequency

10. Why present a birthday statistic in database cases? Because it is relevant…