IOSR Jou nal o Ma hema ics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 14, Issue 6 Ve . I (No - Dec 2018), PP 01-06
www.ios jou nals.o g
DOI: 10.9790/5728-1406010106 www.ios jou nals.o g 1 | Page
Ma ix Fo m o The Bayes Theo em And Diagnos ic Tes s
Ma ía Magdala Pé ez-Nimo1 José A. Camúñez-Ruiz1
1Depa amen o de Economía Aplicada I Uni e sidad de Se illa Facul ad de Ciencias Económicas y
Emp esa iales A da. Ramón y Cajal 1 41008 Se illa Spain
Co esponding Au ho ; Ma ía Magdala Pé ez-Nimo
Abs ac : In o de o sol e ce ain p oblems in calcula ing p obabili ies, such as Ma ko chains o condi ional
speci ica ion o disc e e dis ibu ions, he use o ma ix and ec o ea men o condi ioned p obabili ies and o
ec o s o ma ginal p obabili ies is common. Following hese ideas, he p esen s udy ob ains ma ix o ms o
some elemen a y esul s o p obabili y heo y, such as he o al p obabili y and Bayes heo ems. These esul s
and me hodology a e applied o he ma ix s udy o esul s o diagnos ic es s, allowing an immedia e
gene aliza ion o es s wi h mo e han wo esul s. In addi ion,we p opose sa e y and alidi y measu es o a es
based on ma ix ules, which in some cases a e ela ed o he well-known Youden index.
Key Wo d: Condi ioned P obabili ies, BayesTheo em, Ma ix, Diagnos ic Tes s
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Da e o Submission: 01-11-2018 Da e o accep ance: 15-11-2018
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I. In oduc ion
In he Ma ko chain heo y ([1];[2]) as well as in condi ional speci ica ion p oblems o ini e andom
a iables [3], he ma ix ea men o condi ional p obabili ies is common. This line is ollowed in he p esen
s udy, and he main esul s o condi ional p obabili ies a e ob ained, including a ma ix o m o he Bayes'
heo em. The esul s ob ained, a e applied o he s udy o diagnos ic es .
II. Ma ix App oach
Le us conside wo comple e se s o e en s,
12
, ,..., ,...,
in
A A A A
and
12
, ,..., ,...,
jm
B B B B
, bo h wi h non-
ze o p obabili ies
12
( ), ( ),..., ( ),..., ( )
jm
P B P B P B P B
.
We de ine he
12
( ), ( ),..., ( ),..., ( )
in
P A P A P A P A
ec o s.
1
2
()
()
()
n
PA
PA
PA






α
1
2
()
()
()
m
PB
PB
PB






β
And he ma ices
(( )) ( | ), 1,2,..., ; 1,2,...,
ij ij i j
a a P A B i n j m   A
1 1 1 2 1
2 1 2 2 2
12
( | ) ( | ) ... ( | )
( | ) ( | ) ... ( | )
( | ) ( | ) ... ( | )
m
m
n n n m
P A B P A B P A B
P A B P A B P A B
P A B P A B P A B






A   
(( )) ( | ), 1,2,..., ; 1,2,...,
ij ij j i
b b P B A j m i n   B
Ma ix Fo m O The Bayes Theo em And Diagnos ic Tes s
DOI: 10.9790/5728-1406010106 www.ios jou nals.o g 2 | Page
1 1 2 1 1
1 2 2 2 2
12
( | ) ( | ) ... ( | )
( | ) ( | ) ... ( | )
( | ) ( | ) ... ( | )
m
m
n n m n
P B A P B A P B A
P B A P B A P B A
P B A P B A P B A






B   
Bo h ma ices ha e n×m dimension.
We deno e
n
1
he column ec o o n dimension wi h all i s componen s equal o one, and by
n
0
he column
ec o o n dimension wi h all componen s equal o ze o. On he o he hand, i should be emembe ed ha a
ma ix is said o be s ochas ic by ows (columns) i i is non-nega i e and he sum o he elemen s in each ow
(column) is equal o one.
No e ha

n
m
1 A 1
and

mn
B1 1
, he e o e, A is s ochas ic by columns and B is s ochas ic by ows.
In [3] A nold and P ess ob ain he heo em 1:
Theo em 1. (To al p obabili y heo em, ma ix o m).
Aβα

Bαβ
Demons a ion: we ely on he o al p obabili y heo em in i s usual o m:
1 1 1 2 1 1
2 1 2 2 2 2
12
( | ) ( | ) ... ( | ) ( )
( | ) ( | ) ... ( | ) ( )
( | ) ( | ) ... ( | ) ( )
m
m
n n n m m
P A B P A B P A B P B
P A B P A B P A B P B
P A B P A B P A B P B
  
  
  

  
  
  
Aβ    
1
1
1
22
1
1
( | ) ( )
()
( | ) ( ) ()
()
( | ) ( )
m
jj
j
m
jj
j
n
m
n j j
j
P A B P B
PA
P A B P B PA
PA
P A B P B











  












α


Simila ly, i is shown ha

Bαβ
.
Theo em 2. The ma ices
AB
and
BA
ha e eigen alue one wi h
α
and
β
eigen ec o s associa ed wi h
ha alue and, espec i ely, each ma ix.
Demons a ion: combining he o mulas o he p e ious heo em we ha e:
  
Aβ α AB α α
  
Bα β B Aβ β
As a esul :
( ) de ( ) 0    
n n n
AB α α AB - I α 0 AB - I
( ) de ( ) 0    
m m m
BAβ β B A-I β 0 B A-I
Theo em 3.
2
2

α Aβ α
2
2

β B α β
2
2

α Bβ β
Demons a ion: Jus need o apply Theo em 1.
2
2

α Aβ α α α
Ma ix Fo m O The Bayes Theo em And Diagnos ic Tes s
DOI: 10.9790/5728-1406010106 www.ios jou nals.o g 3 | Page
2
2

β B α β β β
On he o he hand, he las equali y is ob ained by ansposing he second.
Le M be he ma ix o he p obabili ies o he in e sec ions o he wo comple e se s o e en s:
1 1 1 2 1
2 1 2 2 2
12
( ) ( ) ... ( )
( ) ( ) ... ( )
( ) ( ) ... ( )
m
m
n n n m
P AB P AB P AB
P A B P A B P A B
P A B P A B P A B






M   
Theo em 4. (Non-condi ional o al p obabili y heo em).

n
1M β
n
M1 β

m
M1 α
Demons a ion: Simply ope a e on he i s membe o each equa ion,
 
1 1 1 2 1
2 1 2 2 2
12
12
1 1 1
( ) ( ) ... ( )
( ) ( ) ... ( )
1 1 ... 1
( ) ( ) ... ( )
( ) ( ) ... ( )
m
m
n n n m
n n n
i i i m
i i i
P AB P AB P AB
P A B P A B P A B
P A B P A B P A B
P AB P AB P AB
  











  
n
1M
β
   
The second equali y is ob ained by ansposi ion o he i s . The hi d equali y is demons a ed la e is
es ed as he i s one.□ Le
()
nM
be he se o squa e ma ices o n dimension. Le us now de ine he
applica ion
: ( )
n
nn
DM
, which associa es o each ec o
n
 
he squa e ma ix whose diagonal is
and he es o elemen s a e null. I he e is no oom o con usion we will omi he n subsc ip , ha is,
n
DD
. In his way, we ob ain:
1
2
( ) 0 ... 0
0 ( ) ... 0
0 0 ... ( )
n
PA
PA
PA






D(α)    
1
2
( ) 0 ... 0
0 ( ) ... 0
0 0 ... ( )
m
PB
PB
PB






D(β)    
Theo em 5.
AD(β) M D(α)B
Demons a ion:
1 1 1 2 1 1
2 1 2 2 2 2
12
( | ) ( | ) ... ( | ) ( ) 0 ... 0
( | ) ( | ) ... ( | ) 0 ( ) ... 0
( | ) ( | ) ... ( | ) 0 0 ... ( )
m
m
n n n m m
P A B P A B P A B P B
P A B P A B P A B P B
P A B P A B P A B P B
  
  
  

  
  
  
AD(β)        
Ma ix Fo m O The Bayes Theo em And Diagnos ic Tes s
DOI: 10.9790/5728-1406010106 www.ios jou nals.o g 4 | Page
1 1 1 1 2 2 1
2 1 1 2 2 2 2
1 1 2 2
1 1 1 2 1
2 1 2 2 2
( | ) ( ) ( | ) ( ) ... ( | ) ( )
( | ) ( ) ( | ) ( ) ... ( | ) ( )
( | ) ( ) ( | ) ( ) ... ( | ) ( )
( ) ( ) ... ( )
( ) ( ) ... ( )
(
mm
mm
n n n m m
m
m
P A B P B P A B P B P A B P B
P A B P B P A B P B P A B P B
P A B P B P A B P B P A B P B
P AB P AB P AB
P A B P A B P A B
PA








   
   
12
) ( ) ... ( )
n n n m
B P A B P A B







M
The demons a ion o he o he equali y is analogous.
Theo em 6.(Bayes' heo em, ma ix o m).
-1
A D(α)BD(β)
-1
B D(α) AD(β)
I s p oo is an immedia e consequence o Theo em 5.
3.- Diagnos ic es s.
Conside a clinical es o he diagnosis o a ce ain disease (see, [4]). Suppose, as is usual in his con ex , ha
each pa ien can be sick (e en D) o heal hy and ha he es can only p esen posi i e esul s (e en R) i i
de ec s he disease, and nega i e esul s o he wise. De ine he ec o s:
Disease p e alence ec o :
()
()
PD
PD



α
.
I con ains, among o he elemen s, he p e alence o he disease P(D)
Tes esul s ec o :
()
()
PR
PR



β
.
Con aining he p obabili ies o posi i e and nega i e esul s,
()PR
and
()PR
.
We also de ine he ma ices:
Tes secu i y ma ix:
( | ) ( | )
( | ) ( | )
P D R P D R
P D R P D R



A
I con ains, among o he elemen s, he p edic i e alues o he es ,
( | )P D R
(posi i e) and
( | )P D R
(nega i e).
Tes alidi y ma ix:
( | ) ( | )
( | ) ( | )
P R D P R D
P R D P R D



B
.
I con ains, among o he elemen s,
( | )P R D
sensi i i y and
( | )P R D
speci ici y.
Tes Ma ix:
( ) ( )
( ) ( )
P D R P D R
P D R P D R





M
.
Con aining he ue posi i es
()P D R
, ue nega i es
()P D R
, alse posi i es
()P D R
and alse
nega i es
()P D R
.
The ela ion be ween he es ma ix and he p e alence and esul s ec o s is hen ob ained in ma ix o m

2
M1 α
and

2
M1 β
: as well as he ela ion be ween all he elemen s o a diagnos ic es :
AD(β) M D(α)B
.
On he o he hand, applying he o ms o he o al p obabili y heo em and he Bayes' heo em, we ob ain
 -1
A D(B α) BD(α)
. This is he ma ix e sion o well-known o mulas o he ype
Ma ix Fo m O The Bayes Theo em And Diagnos ic Tes s
DOI: 10.9790/5728-1406010106 www.ios jou nals.o g 5 | Page
  
1 1
p e alencexsensibili y
TPV p e alencexsensibili y p e alence speci ici y
  
,
which co ela es p e alence, sensi i i y and speci ici y wi h p edic i e alues.
No e ha he use o ma ices allows an easy gene aliza ion o es s wi h se e al le els o diagnosis o o diseases
wi h di e en ypologies. In addi ion, i allows de ining es secu i y measu es and indexes in e ms o dis ances.
The sa e y o a es usually equi es he s udy o i s p edic i e alues. A es is comple ely sa e i
( | ) ( | ) 1P D R P D R
.Tha is, bo h p edic i e alues a e wo h one. Fo hese es s, he sa e y ma ix A
de ined abo e is he uni . In his case, a sa e y measu e o a es wi h he sa e y ma ix A, can be de ined as a
ce ain dis ance om he ma ix A o he iden i y
( ) ( , ) || ||MS d  A I A I A
, o some ma ix no m
(See, o ins ance, [5]). Fo a comple ely unsa e es hese measu es a e wo h ze o. Fo a comple ely useless
es , ha is,
( | ) ( | ) 0P D R P D R
, he sa e y ma ix is
01
10



J
and MS(J)mus be maximal. I s
alue will depend on he no m used. No malizing and ans e ing he MS measu e,secu i y indexes -SI- can be
ob ained, so ha i he maximum alue o MS is

hen
1MS
SI


, aking alues be ween 0 and 1. The
es is comple ely sa e i i akes he alue 1 and comple ely useless when i akes he alue 0. Le 's look a some
speci ic cases.
i.-Row no m.
 
1 ( | ) ( | )
( ) || ||
( | ) 1 ( | )
|1 ( | )| | ( | )|,| ( | )| |1 ( | )|
P D R P D R
MS P D R P D R
máx P D R P D R P D R P D R







      
A
( ) ( | ) ( | )
0 ( ) 2
MS P D R P D R
MS




A
A
By no malizing, we ob ain he associa ed sa e y index,
( | ) ( | )
( ) 1 2
P D R P D R
SI

A
ii.-Column no m.
 
11
1 ( | ) ( | )
( ) || ||
( | ) 1 ( | )
|1 ( | )| | ( | )|,| ( | )| |1 ( | )|
P D R P D R
MS P D R P D R
máx P D R P D R P D R P D R






      
A
 
1
1
( ) 2 ( | ), ( | )
0 ( ) 2
MS máx P D R P D R
MS


A
A
By no malizing we ob ain he associa ed sa e y index,
 
1( ) 1 max ( | ), ( | )SI P D R P D RA
iii.-Euclidean o spec al no m.
1 ( | ) ( | )
( ) || - || || ||
( | ) 1 ( | )
s s s
P D R P D R
MS P D R P D R






A I A

Ma ix Fo m O The Bayes Theo em And Diagnos ic Tes s
DOI: 10.9790/5728-1406010106 www.ios jou nals.o g 6 | Page
which is he squa e oo o he g ea es singula alue. The singula alues o ha ma ix a e 0 and
 
22
2 ( | ) ( | )P D R P D R
hus:
 
22
( ) || || 2 ( | ) ( | )
0 ( ) 2
ss
s
MS P D R P D R
MS
   

A I A
A
The associa ed sa e y index is
22
( | ) ( | )
( ) 1 2
s
P D R P D R
SI 
A
.
The same easoning may be used wi h he alidi y ma ix Band de ine es alidi y measu es and indexes. I
mus be aken in o accoun ha o sensi i i y and speci ici y equal o one, we ha e ha B = I,
( ) ( , ) || ||MV d  B I B I B
and he maximum alidi y is ound o ze o MV, while he maximum
alidi y is gi en in he case o B = J.
No m
MV
Index
Row
( | ) ( | )P R D P R D
( | ) ( | )
12
P R D P R D

Column
 
2 ( | ), ( | )máx P R D P R D
 
1 ( | ), ( | )máx P R D P R D
Spec al
 
22
2 ( | ) ( | )P R D P R D
22
( | ) ( | )
12
P R D P R D

MV is bounded be ween 0 and 2 in he h ee cases s udied.
No e ha in he case o he ow no m, he alidi y index
1
2
Y
VI


,whe e Y is he well-known Youden
index ([6]).
In addi ion, his way o dealing wi h condi ional p obabili ies can be used in eaching as a nexus be ween
subjec s con aining elemen a y heo y o p obabili y and hose con aining elemen s o linea algeb a, which a e
usually sepa a ed.
Re e ences.
[1]. Be man A. Plemmons R.J. (1994) Nonnega i e Ma ices in he Ma hema ical Sciences. SIAM.
Piladelphia
[2]. Isaacson D.L. y Madsen R.W. (1976) Ma ko Chain: Theo y and aplica ions. John Wiley and Sons. New
Yo k.
[3]. A nold B.C. P ess S. J. (1989) Compa ible condi ional dis ibu ions. Jou nal o he Ame ican S a is ical
Associa ion. 84, 152-156
[4]. ] Go dis, L. (2014) Epidemiology. 5ª edición. Else ie Saunde s Philadelphia
[5]. Scho J.R.(1997) Ma ix Analysis o S a is ics John Wiley and sons New Yo k.
[6]. Youden, W.J. (1950). Index o a ing diagnos ic es s. Cance 3: 32–35.
Ma ía Magdala Pé ez-Nimo. " Ma ix Fo m O The Bayes Theo em And Diagnos ic Tes s."
IOSR Jou nal o Ma hema ics (IOSR-JM) 14.6 (2018): 01-06.