Inverse Inference Practise Question: was Hinckley crazy?

This is a sample exam question from a few years back on inverse inference. It begins with a short description from the press…then moves on to the question.

In 1982 Robert Hinckley went on trial for attempted murder of President Ronald
Reagan. As the press reported:

On March 30, 1981, in broad daylight, among a crowd of supporters and onlookers,
Hinckley fired six bullets at Reagan in the space of three seconds, hitting Reagan,
a police officer and a Secret Service agent, and seriously wounding Press Secretary
James Brady. [Reagan was wounded but not killed. ] ….Hinckley’s trial in 1982
ended in a not-guilty verdict, by reason of insanity. The assassination attempt
won him notoriety and media attention, and also led to legislation [during the
next decade] limiting the use of the insanity plea in several states

During Hinckley’s trial the defense argued that Hinckley had a mental illness,
schizophrenia. The prosecution argued that schizophrenia was rare, with only
around 1 in 100 of the adult population suffering from schizohprenia. The defense
lawyer didn’t dispute this claim, but wanted to introduce as evidence a brain
scan of Hinckley’s that showed substantial brain atrophy (“atrophy” is
a decrease in size or wasting away of a body part or tissue). The defense also
presented evidence in the form of expert testimony that when people diagnosed
as schizophrenics have brain scans, about 30% show signs of substantial brain
atrophy, whereas when normal, non-schizophrenic people have the scan only about
2% show signs of substantial brain atrophy. The defense then argued that on
the evidence it was 15 times more likely, that Hinckley suffered from schizophrenia
compared to a normal person.

 

C1 What would you, as an advisor to the jury, say about the
defense lawyer’s argument, quantitatively and qualitatively? In your
answer use the Gigerenzer natural frequency method (table or graphical) to
calculate and briefly explain how much more or less likely it is that Hinkley
has schizophrenia after seeing all this evidence from the defense than before
seeing it.
Hint:Let
S be the proposition that a person has schizophrenia and BA the proposition
that a person has brain atrophy. Since either proposition can be true (1) or
false (0) there are 4 logical possibities which you can represent in a truth
table. Suppose we assess a prior probability that P(S=1)=0.01  meaning
that without any other information our  probability  for
some adult in the US having scizophrenia is the same as the proportion 1 in
100 of the adult population that suffers from schizohprenia. Suppose we accept
expert testomony on conditionals, that P(BA=1|S=1)=0.3  [when
people diagnosed as schizophrenics have brain scans, about 30% show signs of
substantial brain atrophy
], whereas P(BA=1|S=0)=0.02 [when normal, non-schizophrenic
people have the scan only about 2% show signs of substantial brain atrophy
].

C2 2 The defense lawyer’s evidence is couched
in “abouts”, implying that the experts are far from certain about
the proportions cited. As a prosecutor your researchers have found out
that Hinckley had a history of drinking alcohol to excess, bordering on
alcoholism. They also discovered that brain atrophy is common amongst alcoholics.
Amongst “normal” (non schizophrenic) alcoholics, the chances
of brain atrophy are estimated to be between 25% and as as high as 40%
in some males but the combination of alcoholism and schizophrenia doesn’t
change the 30% figure for brain atrophy that the defense lawyer used. No
one really knows the percentage of alcoholics who are schizophrenic. How
do these changed bits of evidence alter the defense lawyers case?
Explain your reasoning.

Suggested answer and explanation: DO TRY THE QUESTION BEFORE CHECKING OUT THE SUGGESTED ANSWER VIDEO!!

 

the original file for this article is at pbs.org

Postscript:

On March 30, 1981, John Hinckley Jr. set out to win actress Jodie Foster’s

heart. As “the greatest love offering in the history of the world,” the 25-year-old
attempted to assassinate Ronald Reagan outside the Washington Hilton hotel.
Though flanked by administration members, police officers, and Secret Service
agents, Reagan was shot under the left arm. The bullet malfunctioned and failed
to explode on impact, seriously wounding but not killing Reagan.

The youngest of three children, Hinckley was born in Ardmore, Oklahoma, on
May 29, 1955. The family moved several times, first to Texas, then to Colorado.
Like Reagan’s mother, Hinckley’s mother also belonged to the Disciples of Christ;
his father became a born-again Christian in 1977. A well-adjusted, privileged
child, as a teenager Hinckley became withdrawn and obsessed with public figures,
including John Lennon. In 1976 Hinckley left home for Hollywood, hoping to
become a famous songwriter.

In Hollywood, Hinckley saw Martin Scorsese’s film “Taxi Driver” at least 15
times. A confirmed loner, he apparently identified strongly with the Robert
DeNiro character Travis Bickle. In the film, Travis is infatuated with Cybill
Shepherd’s Betsy, a political campaign worker who rejects him after he takes
her to see a porn film. To regain Betsy’s attention, Bickle plans, but fails,
to assassinate the candidate she works for. Bickle then shifts gears, obsessively
devoting himself to protecting 12-year-old prostitute Iris, played by Foster.
He decides to shoot Iris’ pimp, thereby ensuring his status as a hero to Iris
and the media.

Screenwriter Paul Schrader based his portrayal of Travis on Arthur Bremer,
the would-be assassin of George Wallace whose diaries equate political assassination
with celebrity and revenge against impotence and invisibility. Consciously
or not, Hinckley began to emulate Bickle, accumulating an arsenal of weapons
and fixating on Jodie Foster. Foster would not be his target, but his inspiration:
To “rescue” her, he began to stalk Jimmy Carter during the 1979 presidential
campaign.

Following
his arrest for possession of firearms in the Nashville airport, where
he had followed Carter to a campaign stop, Hinckley’s parents sent their
youngest son to psychiatrist John Hopper. Hinckley was already taking
prescribed antidepressants, and Hopper didn’t detect mental illness;
instead, he attributed Hinckley’s problems to “emotional immaturity,” recommending
the Hinckleys cut off their son financially, which they did.

In May 1980, after reading that Foster was enrolled at Yale University, Hinckley
began to criss-cross the country regularly to be near her. Establishing contact
with her twice, he believed the relationship would go nowhere unless he could
catch Foster’s attention with a grand gesture.

So, on March 30, 1981, in broad daylight, among a crowd of supporters and onlookers,
Hinckley fired six bullets at Reagan in the space of three seconds, hitting
Reagan, a police officer and a Secret Service agent, and seriously wounding
Press Secretary James Brady. Upon his arrest, Hinckley asked the arresting
officers if news of the shooting would preempt that night’s Academy Awards
broadcast. (It did; the ceremony aired the next night, the Academy paying its
respects to one of its own.)

Hinckley’s trial in 1982 ended in a not-guilty verdict, by reason of insanity.
The assassination attempt won him notoriety and media attention, and also led
to legislation limiting the use of the insanity plea in several states. Twelve
years and two administrations later, President Clinton signed the Brady Bill,
which requires a waiting period and background check on all handguns purchased
through licensed dealers. The bill has come under fire both from supporters,
who believe its requirements are too lenient, and opponents, who say it infringes
on the constitutional right to bear arms.

Confined to St. Elizabeth’s Hospital in Washington, DC, since his trial, Hinckley’s
obsession with Foster continued. In 1999, however, after significant progress
in his psychiatric treatment, Hinckley was allowed to leave the grounds for
supervised visits.

In April 2000 he won the right to unsupervised furloughs. The following month
these rights were revoked when guards found in his room a smuggled book about
Jodie Foster. (He is banned from having any material about the star.)

He has always seemed aware of his motivations, even immediately after the shooting.
In 1981 he told “Newsweek”: “The line dividing life and art can be invisible.
After seeing enough hypnotizing movies and reading enough magical books, a
fantasy life develops which can either be harmless or quite dangerous.”

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