The second of two sceencast lectures on how intelligent players reason about states and signals in a game ; the first is here

This lecture uses the two practise problems from class (one on breast cancer screening, the other on witness reports in a courtroom case) to develop a “language” of probability in a way that is (1) easily understandable for ANY type or level of student – whether trained in statistics or not and (2) useful for students of game theory . Simple numerical examples using Gigerenzer style natural frequency/count reasoning are developed to explain a wide range of concepts connecting uncertainties about “states” and “signals” sensitivity, specificity, conditional probabilities, predictive probabilities, inverse probabilities, etc.

## The basics of inverse inference Part One: the game tree, truth tables, states and signals

This is the first part of of two lectures on Bayesian inference, inverse probability inference…or…coherent rational inference, the way any intelligent rational player should think about TWO uncertainties ( about some underlying state and about some diagnostic signal possibly related to or informative about that state) . This first lecture introduces the basic problem of inference via a game tree and introduces the logicians “truth table” to help organise thinking about two interrelated unknowns – states and signals. The second lecture , here , takes these ideas and uses them to develop the language of probability for inference problems – in games against people or in games against nature , easily understandable for any type and level of student. You may also find it helpful to look at the My Inference Tool post to help cement in your understanding of inverse inference ideas.

## 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

**that P(S=1)=0.01 meaning**

*prior probability*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 [

people diagnosed as schizophrenics have brain scans, about 30% show signs of

substantial brain atrophy

*when normal, non-schizophrenic*

people have the scan only about 2% show signs of substantial brain atrophy].

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.”

## Size, Power and discrimination ability of statistical tests

A simple geometric tool for exploring how well hypothesis tests can discriminate between two competing hypotheses, H and not H, based on information or results of a diagnostic test T . The diagnostic (potentially informative) signal T is binary valued, either positive for (favourable to) or negative for (unfavourable to) the truth of H.

**Note: **To enable and use the dynamic features of the figure in your browser you will need to download and install Wolfram’s free computable document format ** CDF** player – it is an app for running dynamic Mathematica programs called CDFs , and this WordPress page you are on enables CDFs to be run from within your browser: click here obtain it from wolfram’s site Installation is straightforward ..but you will have to provide a small amount of information to Wolfram – effectively “registering” that you have downloaded their CDF player software. Wolfram, the makers of Mathematica, have been around as long as Apple and Adobe , much longer than Google, and…are as trustworthy; moreover, once you have downloaded the CDF player there are tens of thousands of fascinating and academically useful user-created Wolfram Demonstration Project programs available to run on it. …. so don’t be put off by the short registration requirement.

[WolframCDF source=”https://strategicecon.com/wp-content/uploads/2013/04/vod-hu-v3.cdf” CDFwidth=”655″ CDFheight=”505″ altimage=”https://strategicecon.com/wp-content/uploads/2013/04/vod-hu-v3.png”]

H is an indicator variable equal to 1 if H is true and equal to 0 if H is false, ie not H is true. T is an indicator variable , taking the value 1 when there is evidence favorable to the truth of H , and the value 0 for evidence unfavourable to the truth of H. Think of H as a “state” variable with possible values 0 or 1, and T as a “signal” for the state variable, a diagnostic test.

The **level of the vertical height** of each curve indicates, for each given prior probability on hypothesis H being true, a posterior probability P(H is true|T=1) -in red – and P(H is true|T=0) – in blue – for each of the two possible values of the test result, T=1 or T=0. Posterior probabilities , the chances of a hypothesis being true AFTER observing a test result T=1 or after observing T=0, will, if the test is in anyway informative, change beliefs, either increasing or decreasing the chances of the hypothesis being true compared to the situation where the testing isn’t done , P(H). The vertical difference between the coloured curves , P(H is true|T=1) – P(H is true|T=0) , is a measure of the amount of information, if any, in the tests

The calculated posterior probabilities are based on three other pieces of information that are manipulable parameters:

**the “size” of the test**, the false negative error probability P(T=0| H is true) conditional on H being true**the “power” of the test**, P(T=0| H is false), 1 minus the false positive error probability P(T=1| H is false) conditional on H being false**the unconditional or prior probability**P(H) of the hypothesis being true

. The size and the power of the test can be independently set using the sliders on the left – or click the small + at the end of the slider to enter a numerical value. The prior probability on hypothesis H also has a slider setting. For comparison purposes a benchmark (BM) setting for these variables is also possible, in the lower left hand panel. The two different test results the ability of test results to discriminate between H and not H

Gigerenzer style natural frequencies are calculated in the accompanying truth table – the simple integer arithmetic of “counts” out of stylized sample sizes n=100,1000, 10000 is helpful in “framing” the inverse inference task by making it easy to identify type 1 errors (false negative rates conditional on H=1) and type 2 errors (false positive rates conditional on H=0).

## Expressing and exploring ambiguous inductive inferences

A simple geometric tool for exploring implications of ambiguity about the basic belief-components of inverse probability reasoning: base rates of states (priors), and sensitivity and specificity of diagnostic tests (likelihoods).

[WolframCDF source=”https://strategicecon.com/wp-content/uploads/2013/01/ambiguityrighthandwp.cdf” CDFwidth=”689″ CDFheight=”524″ altimage=”https://strategicecon.com/wp-content/uploads/2013/01/imageforAmbiguousbeliefs.png”]

(to download a larger version with control bars on the left side click here [wpdm_file id=1] – * you will need Wolfram’s free computable document format cdf player to view and interact with this file. (just like you need a plug in to read pdfs in your browser) *– click here obtain it from wolfram’s site – installation is straightforward and seamless ..but you will have to provide a small amount of “registration” type information to Wolfram – they have been around as long as Apple and Adobe , much longer than Google, and…are as trustworthy …. so don’t be put off by this. Having the cdf player on your local machine is well worthwhile)