bayesian theorem

7/23/2019 Bayesian Theorem

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CONDITIONAL BAYESIAN

PROBABILITY – ANAPPLICATIONDr. Laldinliana

Department of CommerceMioram !ni"er#it$



%&o i# Ba$e#'

Thomas Bayes ()*+) – * April )*-) /a# an

En0li#& #tati#tician1 p&ilo#op&er and Pre#2$terianmini#ter1 3no/n for &a"in0 form4lated a #peci5cca#e of t&e t&eorem t&at 2ear# &i# name.

The probability of any event is the ratio between

the value at which an expectation depending onthe happening of the event ought to be computed,and the value of the thing expected upon itshappening



%&$ Ba$e#ian'

• Bayes’ theorem fnds the actualprobability o an event rom the resultso your tests. 6or e7ample1 $o4 can8 –

Correct or measurement errors. If $o4 3no/t&e real pro2a2ilitie# and t&e c&ance of a fal#epo#iti"e and fal#e ne0ati"e1 $o4 can correct formea#4rement error#.

– Relate the actual probability to the measured

test probability. Ba$e#9 t&eorem let# $o4 relatePr(A:;1 t&e c&ance t&at an e"ent A &appened0i"en t&e indicator ;1 and Pr(;:A1 t&e c&ance t&eindicator ; &appened 0i"en t&at e"ent A occ4rred.



E7ample

• <i"en mammo0ram te#t re#4lt# and 3no/nerror rate#1 $o4 can predict t&e act4al c&anceof &a"in0 cancer.

•

An article de#cri2e# a cancer te#tin0 #cenario8 – )= of /omen &a"e 2rea#t cancer (and t&erefore

>>= do not.

– ?+= of mammo0ram# detect 2rea#t cancer /&en

it i# t&ere (and t&erefore @+= mi## it. – >.-= of mammo0ram# detect 2rea#t cancer /&en

it9# not t&ere (and t&erefore >+.= correctl$ret4rn a ne0ati"e re#4lt.



• P4t in a ta2le1 t&e pro2a2ilitie# loo3li3e t&i#8



o/ do /e read it'

• )= of people &a"e cancer

• If $o4 already have cancer1 $o4 are in

t&e 5r#t col4mn. T&ere9# an ?+= c&ance$o4 /ill te#t po#iti"e. T&ere9# a @+= c&ance$o4 /ill te#t ne0ati"e.

• If $o4 don’t have cancer1 $o4 are in t&e#econd col4mn. T&ere9# a >.-= c&ance $o4/ill te#t po#iti"e1 and a >+.= c&ance $o4/ill te#t ne0ati"e.



How Accurate Is The Test

No/ #4ppo#e $o4 0et a po#iti"e te#t re#4lt. %&atare t&e c&ance# $o4 &a"e cancer' ?+=' >>=' )='

•

O31 /e 0ot a po#iti"e re#4lt. It mean# /e9re#ome/&ere in t&e top ro/ of o4r ta2le it co4ld2e a tr4e po#iti"e or a fal#e po#iti"e.

• T&e c&ance# of a true positive c&ance $o4 &a"ecancer c&ance te#t ca40&t it )= ?+= .++?

• T&e c&ance# of a false positive c&ance $o4don9t &a"e cancer c&ance te#t ca40&t it an$/a$ >>= >.-= +.+>F+



• !robability " desired event # all possibilities

• T&e c&ance of 0ettin0 a real1 po#iti"e re#4lt i# .++?. T&ec&ance of 0ettin0 an$ t$pe of po#iti"e re#4lt i# t&ec&ance of a tr4e po#iti"e pl4# t&e c&ance of a fal#e

po#iti"e (.++? G +.+>F+ .)+H+.• So1 o4r c&ance of cancer i# .++?.)+H+ +.+**-1 or

a2o4t *.?=.

• Intere#tin0 a po#iti"e mammo0ram onl$ mean# $o4&a"e a *.?= c&ance of cancer1 rat&er t&an ?+= (t&e#4ppo#ed acc4rac$ of t&e te#t. It mi0&t #eem #tran0eat 5r#t 24t it ma3e# #en#e8 t&e te#t 0i"e# a fal#e po#iti"e)+= of t&e time1 #o t&ere /ill 2e a ton of fal#e po#iti"e#in an$ 0i"en pop4lation.



A0ain1 let9# te#t o4r int4ition 2$ dra/in0 a concl4#ionfrom #impl$ e$e2allin0 t&e ta2le.

• If $o4 ta3e )++ people1 onl$ ) per#on /ill &a"ecancer ()=1 and t&e$9re nearl$ 04aranteed to te#t

po#iti"e (?+= c&ance. Of t&e >> remainin0 people1a2o4t )+= /ill te#t po#iti"e1 #o /e9ll 0et ro40&l$ )+fal#e po#iti"e#.

• Con#iderin0 all t&e po#iti"e te#t#1 J4#t ) in )) i#

correct1 #o t&ere9# a ))) c&ance of &a"in0 cancer0i"en a po#iti"e te#t. T&e real n4m2er i# *.?=(clo#er to ))H1 comp4ted a2o"e1 24t /e fo4nd area#ona2le e#timate /it&o4t a calc4lator.



• %e can t4rn t&e proce## a2o"e intoan eK4ation1 /&ic& i# Ba$e#9

T&eorem. It let# $o4 ta3e t&e te#t

re#4lt# and correct for t&e #3e/introd4ced 2$ fal#e po#iti"e#. Yo4 0ett&e real c&ance of &a"in0 t&e e"ent.

ere9# t&e eK4ation8



• Pr(A:; C&ance of &a"in0 cancer (A 0i"en apo#iti"e te#t (;. T&i# i# /&at /e /ant to 3no/8 o/li3el$ i# it to &a"e cancer /it& a po#iti"e re#4lt' In o4rca#e it /a# *.?=.

• Pr(;:A C&ance of a po#iti"e te#t (; 0i"en t&at $o4&ad cancer (A. T&i# i# t&e c&ance of a tr4e po#iti"e1?+= in o4r ca#e.

• Pr(A C&ance of &a"in0 cancer ()=.

•

Pr(A C&ance of not &a"in0 cancer (>>=.• Pr(;:A C&ance of a po#iti"e te#t (; 0i"en t&at

$o4 didn9t &a"e cancer (A. T&i# i# a fal#e po#iti"e1>.-= in o4r ca#e.



• It all come# do/n to t&e c&ance of a true positiveresult di"ided 2$ t&e chance o any positive result.%e can #implif$ t&e eK4ation to8

• Pr(; i# a normaliin0 con#tant and &elp# #cale o4reK4ation. %it&o4t it1 /e mi0&t t&in3 t&at a po#iti"e te#tre#4lt 0i"e# 4# an ?+= c&ance of &a"in0 cancer.

• Pr(; tell# 4# t&e c&ance of 0ettin0 any po#iti"e re#4lt1/&et&er it9# a real po#iti"e in t&e cancer pop4lation ()=or a fal#e po#iti"e in t&e noncancer pop4lation (>>=.It9# a 2it li3e a /ei0&ted a"era0e1 and &elp# 4# comparea0ain#t t&e o"erall c&ance of a po#iti"e re#4lt.



Appl$in0 t&e t&eorem in te#tin0 a&$pot&e#i#

$pot&e#i#8

Re#4rrection of a man named e#4# i#&i#toric

Alternate $pot&e#i#8

Re#4rrection of a man named e#4# i#not &i#toric



i#torical criteria Q)M4ltiple1 Independent So4rce#

Criterion• @ a4t&or# of antiK4it$ /it&in )F+$ear# mentioned a2o4t e#4#1 /&ile

• Ti2eri4# Cae#ara1 /a# mentioned 2$)+ a4t&or# /it&in )F+ $ear# after &i#deat&

• 4li4# Cae#ar /&o #po3e t&e famo4#

/ord# I came1 I #a/1 I conK4er9 /a#mentioned in onl$ F /or3# of antiK4it$



i#torical criteria Q@Enem$ Atte#tation Criterion

• Rome ocial#1 S$rian p&ilo#op&er1traditional Je/i#& te7t#

• Te#timon$ of Sa4l of Tar#4#



i#torical criteria QHPrinciple of Em2arra##ment Criterion

• %omen atte#tation

• Di#ciple# &a"e no cate0or$ of4nder#tandin0 a2o4t t&e comin0deat& of e#4#



i#torical criteria QE$e/itne## Te#timon$ Criterion

• Matt&e/1 o&n1 Pa4l1 Peter

• Epi#tle# of 4de and ame#



i#torical criteria QFEarl$ So4rce Criterion

• T&e <o#pel acco4nt# /ere /ritten /it&inHF-F $ear# after t&e e"ent

While,

• Rome9# Cae#ar A404#t4# /ritten in -/or3#1 e7cept &i# f4neral note1 t&e earlie#t/or3 /a# Pl4tarc&9#1 /&ic& /a# at lea#t >+$ear# after e"ent. S4etoni4# and Tacit4#after )++ $ear#1 Appian after )++)F+$ear#1 Dio Ca##i4# after )*F@++ $ear#



Comin0 to t&e G) Minimal6act#

• e#4# died 2$ cr4ci57ion

• i# di#ciple# believed t&e$9"e #eent&e re#4rrected e#4#

• Per#ec4tor of t&e follo/er# of e#4#1Pa4l believed &e enco4ntered t&ere#4rrected e#4#

• S3eptic ame# /a# con"erted

• T&e empt$ tom2



Minimal fact# Q) e#4# died 2$ cr4ci57ion

• 4 Gospels

• Suetonius

• Tacitus

• Pliny The Younger

• Thallus

• Lucian of Samosata

• ara !ar Saparion• "osephus

• Talmud



Minimal fact# Q@i# di#ciple# believed t&e$9"e #een t&e

re#4rrected e#4#

A. T&e$ proclaimed it

• Pa4l9# epi#tle#

• Oral tradition

• <o#pel# and /or3# of earl$ c&4rc& fat&er# – #lement of $ome

– Polycarp

–

Letter of !arnabas



B. T&e$ #tron0l$ 2elie"ed it

• Co/ard# t4rned mart$r# for t&eir 2elief#

• Martardom /ritten in L43e9#1 Clement ofRome1 I0nati4#1 Pol$carp1 Dion$#i4# ofCorint&1 Tert4llian and Ori0en

• )) di#ciple#1 Pa4l and ame#

#4eredmart$red• C&ance of mart$rdom &ad &e not

re#4rrected )8)+H>



Minimal fact# QHPa4l believed &e enco4ntered t&e re#4rrected e#4#

• Sa4l9# con"er#ion acco4nt in &i# o/n/or3# and L43e9# Act#

• Mart$rdom recorded in Clement ofRome () Clem.F8@*1 Pol$carp (P&il.>8@1 Tert4llian (Scorpiace )F1Dion$#i4# of Corint& (Eccle#ia#tical

i#tor$ @8@F8? and Ori0en(Commentar$ on <ene#i# in E H8)



Minimal fact# QS3eptic ame# /a# con"erted

• Brot&er# of e#4# did not 2elie"e e#4#

• T&e$ 2ecame follo/er# after t&ere#4rrection onl$

• ame#1 2rot&er of e#4# 2ecame t&eBi#&op of er4#alem1 ordained 2$

Peter1 2ecame a mart$r



Minimal fact# QFEmpt$ tom2

• er4#alem factor

• Enem$ atte#tation#

•

Te#timonie# of /omen



Pro2a2ilit$ of e#4#9 re#4rrection

• e#4# died 2$ cr4ci57ion M Ual4e )

• Di#ciple# 2elie"ed t&e$ enco4ntered M Ual4e H

• Per#ec4tor Pa4l con"ert# M Ual4e H• S3eptic ame# con"ert# M Ual4e H

• Empt$ tom2 M Ual4e @

$vidence showsresurrection did nothappen %&cale '(')*

$vidence showsresurrection did happen%&cale '(')*

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ORDER O6 MA<NIT!DE (M



PA6TER PBE6ORE ; M PBE6ORE ; M G )++= PBE6ORE

%ERE1 M P(E:RP(E:R

>-=

TH$R$+,R$- TH$ ,CCRR$/C$ ,+ 0$&&’R$&RR$CTI,/ I& 1,R$ !R,BAB2$ B3 456%THAT I&

756(8)6* THA/ /,T

bayesian theorem

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