Transcript Statistical Reasoning in Court

 G | g  ( g )
  ( g | G)
 (G )
Bayes Rule
 g | GP(G)   (G | g ) g    g  G
N total people: This is
the much bigger blue
box including G and g
Everyone with
trait G
Everyone with
trait g
G
g
gG
The basic idea here:
no one cares about “G”
except as it affects
the probability that
the person is “g”
Prior odds
ratio of a random
person being a spy:
from Scotchmer 1998 Statistical Reasoning in Court
Statistical Reasoning in Court: Collins
• LA mugging. Yellow car. Interracial couple, she with a blond ponytail,
he with a moustache.
Court case: The defendants match.
What should we make of this? Convict them?
• N=couples in population; Priors
Prior odds of guilt:
N 1
 (i ) 
N
(g) 
1
N
(g)
1

 (i ) N  1
Posterior odds of guilt? Posterior to what? Answer, the match.
E.g., the pink area.
from Scotchmer 1998 Statistical Reasoning in Court
Statistical Reasoning: More on Collins
Suppose the only evidence is the match to the description.
Suppose there are 2 couples in LA that match: G, the pink circle
 (G)  2/N
Posterior probability of (g)uilt:
 (G | g ) ( g ) 1 ( g ) 1 1
1


   g | G 
 (G )
 (G ) N  (G ) 2
Posterior odds ratio,
(g)uilt to (i)nnocence:
 (g | G)
 (g | G)
1/ 2


1
 (i | G ) 1   ( g | G ) 1 / 2
Preponderance of evidence: odds ratio is larger than one
Beyond a reasonable doubt: odds ratio is very large
from Scotchmer 1998 Statistical Reasoning in Court
Statistical Reasoning
Red buses and blue buses
• City has one bus line but two bus companies.
Pedestrian is run over; no witnesses.
There are 4 times as many blue buses as red buses.
• Prior probabilities of guilt and innocence:
(g) = 4/5 and (i)=1/5
• With no evidence, the posterior probability is the same as the
prior probability. The probability ratio of guilt to innocence is
4/1. By the preponderance-of-evidence standard, trier of fact
should convict.
• Victim’s family sues the blue-bus company on grounds that it
is 4-to-1 likely that the perpetrator was a blue bus.
• Should the court hold the company liable?
from Scotchmer 1998 Statistical Reasoning in Court
Statistical Reasoning
More on red buses and blue buses
Hm…. No one likes this outcome!! In the court case that this
is based on, the court disallowed such a conclusion.
Observations:
(1) The whole point of liability is to deter negligence. But if the blue
company is always convicted, there is no deterrence of negligence by
the red company.
(2) If the blue bus company is always convicted, then the blue drivers will
be more careful, leading to the conclusion that they do not provide four
times as many accidents. Thus the premise is wrong. The statistics
must account for equilibrium behavior.
(3) Is it fair to convict the bus company without identifying the driver?
Homework: What is the odds ratio for a particular blue-bus driver?
from Scotchmer 1998 Statistical Reasoning in Court
Statistical reasoning: Probabilistic Effects
• 1950’s, nuclear testing in Nevada.
• Later, cancer appeared (leukemia).
• Epidemiological data: (these numbers are slightly wrong)
The leukemia rate is 3 cases per 1000 people.
In Nevada the rate is 6 cases per 1000 people.
• For an individual case, is the AEC liable?
There is no specific evidence that the AEC caused the victim’s
leukemia. The prior probability that the AEC is guilty is (g)=1/2.
The posterior is the same, so the probability ratio is (g)/ (i)=1,
which just meets the preponderance of evidence standard.
• Problem: No standard of evidence will assign liability correctly. The
AEC is liable either for all the cancers or for none of them.
from Scotchmer 1998 Statistical Reasoning in Court
Statistical Reasoning: Order Statistics
• Cancer Clusters: Woburn has a leather-tanning plant
and a chemical plant, and it turns out that the town has a
leukemia cluster.
• The town has five times the ordinary cancer rate.
• Should the plants be liable?
• Some town will be the highest order statistic. What do
we make of this?
from Scotchmer 1998 Statistical Reasoning in Court
Apply to evidence techniques
• Fingerprinting?
What does a match mean?
What are the dark blue, light blue and pink areas?
• DNA match
• Repressed memory? Is this in the same
category?
from Scotchmer 1998 Statistical Reasoning in Court