1 Finite Population Inference for Latent Values Measured with Error that Partially Account for Identifable Subjects from a Bayesian Perspective Edward

1

Finite Population Inference for Latent Values Measured with Error that

Partially Account for Identifable Subjects from a Bayesian Perspective

Edward J. Stanek IIIDepartment of Public HealthUniversity of Massachusetts

Amherst, MA

2

Collaborators

Parimal Mukhopadhyay, Indian Statistics Institute, Kolkata, IndiaViviana Lencina, Facultad de Ciencias Economicas, Universidad Nacional de Tucumán, CONICET, ArgentinaLuz Mery Gonzalez, Departamentao de Estadística, Universidad Nacional

de Colombia, Bogotá, ColombiaJulio Singer, Departamento de Estatística, Universidade de São Paulo, BrazilWenjun Li, Department of Behavioral Medicine, UMASS Medical School,

Worcester, MARongheng Li, Shuli Yu, Guoshu Yuan, Ruitao Zhang, Faculty and Students

in the Biostatistics Program, UMASS, Amherst

3

Outline

• Review of Finite Population Bayesian Models1. Populations, Prior, and Posterior2. Notation 3. Exchangeable distribution4. Sample Space and Data5. Posterior Distribution, given data

4

Bayesian Model

General Idea

Populations Populations

# Posterior Populations: H

DataPrior Posterior

# Prior Populations: H

Prior

Probabilities

Posterior

Probabilities

L

x

1 2

1 2

1 2

1 2

H

H

H

H

L L L

p p p

1 2

1 2

1 2* * *1 2

H

H

H

H

L L L

p p p

Review

5

Bayesian Model

Population and Data Notation

Populations

2 ; 1,...,h e jj N

; 1,...,h jL j N

1

h

hLN

2e

Label Latent Value

Labels

Parameter

0hλVector

0hμ

Data

2 ; 1,...,t t e sx s n

,It Ix X

Vector

0Iλ ; 1,...,s

L s n

1

1 n

x ssn

Review

Measurement Variance

20ehσ

Measurement Error Variance

* 2 ; 1,...,e sX s n

Expected response

R s sE X

Response

Iμ

Measurement Error Variance

2varR s esX 2eIσ

* 2, , ; 1,..., !Ik Ik ek k n λ X σ2 2ek k eIσ v σ

0Ik k Iλ v λ

Ik k IX v X

6

Exchangeable Prior Populations

General Idea When N=3

11

1 1 12

13

Y

Y Y

Y

Y

p pYY

1p

Each Permutation p of subjects in L(i.e. each different listing)

1p Y

Joint Probability Density

2 11 1

2 2 2 12 3

2 13 2

Y Y

Y Y Y

Y Y

Y 2p

2p Y

6 11 3

6 6 6 12 2

6 13 1

Y Y

Y Y Y

Y Y

Y

6

!

p

N

6p Y

Must beidentical

1

2

3

i

Y

Y Y

Y

YExchangeableRandomVariables

p YThe commondistribution

GeneralNotation

Assigns (usually) equal probability to eachpermutation of subjectsin the population.

Review

7

Exchangeable Prior Populations N=3

Potential Response for Each Listing of subjects

1p

2p

Listings

,

11 0 ,

,

h Rose

h h h Lily

h Daisy

μ u μ

22 0h h

Rose

Daisy

Lily

λ u λ

11 0h h

Rose

Lily

Daisy

λ u λ

Latent Values for Listing

Latent Values for permutations of listing

Review

,

22 0 ,

,

h Rose

h h h Daisy

h Lily

μ u μ

,h Rose

,h Lily

,h Daisy

11 hu μ 1

2 hu μ 15 hu μ 1

3 hu μ 14 hu μ 1

6 hu μ

,h Rose

,h Rose,h Rose

,h Rose ,h Rose,h Lily

,h Lily ,h Lily

,h Lily

,h Lily

,h Daisy

,h Daisy

,h Daisy

,h Daisy,h Daisy

8

Exchangeable Prior Population

Permutations

13hY

Rose

Daisy

Lily

Listing p=1

11hY

12hY

11 hu μ

12 hu μ

11

1 12

13

h

h h

h

Y

Y

Y

Y

Review

,h Rose

,h Lily

,h Daisy

11 hu μ 1

2 hu μ

,h Rose

,h Lily

,h Daisy ,

11 0 ,

,

h Rose

h h h Lily

h Daisy

μ u μ

9


Permutations

Rose

Daisy

Lily

,

11 0 ,

,

10

5

2

h Rose

h h h Lily

h Daisy

μ u μ

Listing p=1

11hY

12hY

13hY

1u

2u

3u

4u

5u

6u

11

1 12

13

h

h h

h

Y

Y

Y

Y

Review

,h Rose

,h Lily

,h Daisy

11 hu μ 1

2 hu μ 15 hu μ 1

3 hu μ 14 hu μ 1

6 hu μ

,h Rose

,h Rose,h Rose


,h Lily ,h Lily

,h Lily

,h Lily

,h Daisy

,h Daisy

,h Daisy

,h Daisy,h Daisy

10


Permutations of Listings

11Y

12Y

13Y

1u

2u

3u

4u

6u

5u

Listing p=1

Same Point in Listing

21Y

22Y

23Y

1u

2u

3u

4u

5u

6u

Listing p=2

31Y

32Y

33Y

1u

2u

3u

4u5u

6u

Listing p=3

41Y

42Y

43Y

1u

2u

3u

4u

5u

6u

Listing p=4

51Y

52Y

53Y

1u

2u3u

4u

5u

6u

Listing p=5

61Y

62Y

63Y

1u

2u

3u

4u

5u

6u

Listing p=6

Review

11

Measurement Error Model

Prior Random Variables

Population h, Prior


; 1,...,h jj N

; 1,...,h jL j N

*

* 0

pp

h p hpλ u u λ

Vectors

Assume Random Variables representinga population are exchangeable

When p=1,define

Sets Prior

11 0h hu λ λ1 Nu I

hp * * * 2

*

, , , ;

, 1,..., !

pp pp pp

h h h ehh

p

p p N

λ μ σ1,...,h H

*

* 0

pp

h p hpμ u u μ

11 0h hu μ μ

*

*

2 20

pp

eh p ehpσ u u σ

1 221 0eh ehu σ σ

12

Prior Random Variables and Datawith Measurement Error

1

2

1

2

n

n

n

N

Y

Y

Y

Y

Y

Y

Y

* *

*

!1

1 1

H Np

h p hp ph p

I I

Y u u μ


Prior Random Variablesthat will correspond to Latent values for subjectsIn the data

Remaining Prior Random Variables

Prior

Data * 2, , ; 1,..., !Ik Ik ek k n λ X σ

* * * 2

*

, , , ;

, 1,..., !

pp pp pp

h h h ehh

p

p p N

λ μ σ1,...,h H

* *

*

!1

1 1

H Np

h p hp ph p

I I

L u u λ

* *

*

!1 22

1 1

H Np

e h p ehp ph p

I I

σ u u σ

13

Bayesian Model Exchangeable Prior Populations N=3

for h when

11hY

12hY

1u

2u

3u

4u

5u

6u

10

10

5

5

2

2

Listing p=1Sample Space n=2

Prior

Review

h

11hY

12hY

13hY

1u

2u

3u

4u

5u

6u

11

1 12

13

h

h h

h

Y

Y

Y

Y

14

Bayesian Model Exchangeable Prior Populations N=3: Sample Point n=2

31Y

32Y

1u

2u

3u

4u

5u

6u

10

10

5

5

2

2

Listing p=3

41Y

42Y

1u

2u

3u

4u

5u

6u

10

10

5

5

2

2

Listing p=4

11Y

12Y

1u

2u

3u

4u

5u

6u

10

10

5

5

2

2

Listing p=1

Listing p=2

21Y

22Y

1u

2u

3u

4u

5u

6u

10

10

5

5

2

2

51Y

52Y

1u

2u

3u

4u

5u

6u

10

10

5

5

2

2

Listing p=5

61Y

62Y

1u

2u

3u

4u

5u

6u

10

10

5

5

2

2

Listing p=6

Review

15


Permutations

Rose

Daisy

Lily

,

11 0 ,

,

10

5

2

h Rose

h h h Lily

h Daisy

μ u μ

Listing p=1

11hY

12hY

13hY

1u

2u

3u

4u

5u

6u

11

1 12

13

h

h h

h

Y

Y

Y

Y

Review

,h Rose

,h Lily

,h Daisy

11 hu μ 1

2 hu μ 15 hu μ 1

3 hu μ 14 hu μ 1

6 hu μ

,h Rose

,h Rose,h Rose


,h Lily ,h Lily

,h Lily

,h Lily

,h Daisy

,h Daisy

,h Daisy

,h Daisy,h Daisy

,L Rose Daisy

16

Bayesian Model Exchangeable Prior Populations N=3

for h when

11hY

12hY

1u

2u

3u

4u

5u

6u

10

10

5

5

2

2

Listing p=1

Prior

Review

h

11hY

12hY

13hY

1u

2u

3u

4u

5u

6u

11

1 12

13

h

h h

h

Y

Y

Y

Y

,L Rose Daisy

Sample Space n=2 when

Listing p=1

,L Rose Daisy

17

Exchangeable Prior Populations N=3: Sample Points n=2

31Y

32Y

1u

2u

3u

4u

5u

6u

10

10

5

5

2

2

Listing p=3

41Y

42Y

1u

2u

3u

4u

5u

6u

10

10

5

5

2

2

Listing p=4

11Y

12Y

1u

2u

3u

4u

5u

6u

10

10

5

5

2

2

Listing p=1

Listing p=2

21Y

22Y

1u

2u

3u

4u

5u

6u

10

10

5

5

2

2

51Y

52Y

1u

2u

3u

4u

5u

6u

10

10

5

5

2

2

Listing p=5

61Y

62Y

1u

2u

3u

4u

5u

6u

10

10

5

5

2

2

Listing p=6

Positive Prob.Review

18

Data n=2

1I

2I

1v

2v

10

10

5

5

2

2

,L Rose Daisy

Review

1

2

IkIk k I

Ik

μ v μ

Axis 1I

Axis 2I

10

2Daisy

Rose

I

μ

19

Data n=2

1I

2I

1

1 0

0 1

v

2v

10

10

5

5

2

2

,L Rose Daisy

Review

1 1

Rose

DaiI

syI

μ v μ

Axis 1I

Axis 2I

10

2Daisy

Rose

I

μ

Rose

Daisy

20

Data n=2

1I

2I

1v

2

0 1

1 0

v

10

10

5

5

2

2

,L Rose Daisy

Review

Axis 1I

Axis 2I

Rose

Daisy

10

2Daisy

Rose

I

μ

2 2Dais

RI I

o

y

se

μ v μ

21

Data n=2Adding Measurement Error

to Rose

1I

2I

10

10

5

5

2

2

,L Rose Daisy

1 1

Rose

DaiI

syI

μ v μ

Axis 1I

Axis 2I

10

2Daisy

Rose

I

μ

Rose

Daisy

2, 8.5

5, 2, 1, 0.5,0.5,1,2,5

e Rose

Rosee

22

Exchangeable Prior Populations N=3 Sample Points with Positive Probability n=2

31Y

32Y

4u

6u

10

10

5

5

2

2

Listing p=3

41Y

42Y

4u

6u

10

10

5

5

2

2

Listing p=4

11Y

12Y

2u

5u

10

10

5

5

2

2

Listing p=1

Listing p=2

21Y

22Y

1u

3u

10

10

5

5

2

2

51Y

52Y

1u

3u

10

10

5

5

2

2

Listing p=5

61Y

62Y

2u

5u

10

10

5

5

2

2

Listing p=6

,L Rose Daisy

Review

23

Exchangeable Prior Populations N=3 Posterior Random Variables

1 2 H

* * * * * *

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

41 42 43 44 45 46

51 52 53 54 55 56

61 62 63 64 65 66

1 1 2 3 4 5 6

1

2

3

4

5

6

h p p p p p p

p

p

p

p

p

p

y y y y y y

y y y y y y

y y y y y y

y y y y y y

y y y y y y

y y y y y y

Prior

Data

Rosey

Daisyy Rosey

Daisyy

,

? , ?

I

ph

hII

Y

Y

Y

Rosey

Daisyy Rosey

Daisyy

If permutations of subjects in listing p are equally likely: * *kkk k

p p p

0

1

II H

h hIIIIh

I

VxY

YY

Random variables representing the data are independent of the remaining random variables.

The Expected Value of random variables for the data is the mean for the data.

Review

*

x nI

h hII N nIIh H

Ep

1Y

1Y

*

2

2varx n n N n

I

N n n h hII N nIIh H

p

P 0 0Y

0 0 PY

when h

24

Posterior Random Variables no Measurement Error

If permutations of subjectsin listing p are equally likely:

0

1

II H

h hIIIIh

I

VxY

YY

*

x nI

h hII N nIIh H

Ep

1Y

1Y

*

2

2varx n n N n

I


p

P 0 0Y

0 0 PY

Review

Data

, ; 1,..., !Ik Ik k n λ x

Populations

Prior


1 2

1 2

1 2

1 2

H

H

H

H

L L L

p p p

25



0

1

II H

h hIIIIh

I

VxY

YY

*

x nI

h hII N nIIh H

Ep

1Y

1Y

*

2

2varx n n N n

I


p

P 0 0Y

0 0 PY

Review

Data

, ; 1,..., !Ik Ik k n λ x

Populations

Prior


1 2

1 2

1 2

1 2

H

H

H

H

L L L

p p p

nxn random permutation matrix

1

20I s

n

x

xx

x

x

Data

26



0

1

II H

h hIIIIh

I

VxY

YY

*

x nI

h hII N nIIh H

Ep

1Y

1Y

*

2

2varx n n N n

I


p

P 0 0Y

0 0 PY

Review

Data

, ; 1,..., !Ik Ik k n λ x

Populations

Prior


1 2

1 2

1 2

1 2

H

H

H

H

L L L

p p p

hII hIIY Wy

27



0

1

II H

h hIIIIh

I

VxY

YY

*

x nI

h hII N nIIh H

Ep

1Y

1Y

*

2

2varx n n N n

I


p

P 0 0Y

0 0 PY

Review

Data

, ; 1,..., !Ik Ik k n λ x

Populations

Prior


1 2

1 2

1 2

1 2

H

H

H

H

L L L

p p p

28

Data without Measurement Error

Data (set)

Vectors

1

2I s

n

x

xx

x

x

1

20I s

n

λ

0Ik k Iλ v λ Ik k Ix v x

; 1,...,s

x s n

; 1,...,s

L s n

permutation matrix, k=1,…,n!

and 1 nv I

kv n nto be anLet

, ; 1,..., !Ik Ik k n λ x

Data (set of vectors)

sx

Latent Value

29

Data with and without Measurement Error

No Measurement Error

1

2I

n

x

x

x

x

1

20I

n

λ

0Ik k Iλ v λ Ik k Ix v x

; 1,...,s

x s n

; 1,...,s

L s n

, ; 1,..., !Ik Ik k n λ x

Latent Value

Data

Data

With Measurement Error

1

2

t

tIt st

nt

x

xx

x

x

st s stx x e

Ik k IX v X

; 1,...,t t sx s n

; 1,...,s

L s n

, ; 1,..., !Ik Ik k n λ xVectors

Sets

Data

Data

* ; 1,...,s

X s n

1

2s I

n

X

XX

X

X

s s sX x E

Response at t Potential Response

* , ; 1,..., !Ik Ik k n λ X

Potential Response

30

Data with Measurement Error

ste the realization of sE

0R sE E

2varR s seE

*

*

0

R s s

s s

E E E

E E

st s stx x e on occasion tThe realization of sX

; 1,...,t t sx s n

; 1,...,s

L s n

Sets Data

* ; 1,...,s

X s n

s s sX x E

the latent valuesxAssume:

Measurement errors are independentwhen repeatedly measured on a subject

31


The Data

0Ik k Iλ v λ Ik k IX v X

1

20I s

n

λ

1

2I s

n

X

XX

X

X

1

22

1

1

1

1

n

x ss

n

x s xs

xn

xn

Vectors

1

2I

n

x

x

x

x

Ik k Ix v x

Define

Latent Values Potential response with error

Data

, ; 1,..., !Ik Ik k n λ x

* , ; 1,..., !Ik Ik k n λ X ; 1,..., !IkL k n λ

32



Populations

Prior


1 2

1 2

1 2

1 2

H

H

H

H

L L L

p p p

Population h

* ; 1,...,h jY j N

; 1,...,h jL j N

1

h

hL

yN

Labels

Parameter

*

* 0

pp

h p hpλ u u λ

Vector*0hY

* *hj hj hjY y E

* 0R hjE E * 2varR hj hjeE

* ; 1,...,h jy E j N


*

*

* *0

pp

h p hpY u u Y

Defines the axes for a cloud of points in the prior

0hy

When p=1, define

* 1*0

10

h h

h h

Y Y

y y

33

Exchangeable Prior Populations N=3 No Measurement Error

Rose

Daisy

Lily

21Y

22Y

23Y

*

* 0

pp

h p hpλ u u λ

0p hu λ

SinglePoint

*

* 0

pp

h p hpy u u y

34

Exchangeable Prior Populations N=3 with Measurement Error

Rose

Daisy

Lily

21Y

22Y

23Y

*

* 0

pp

h p hpλ u u λ

0p hu λ

Cloud of Points

*

*

* *0

pp

h p hpY u u Y

35



Population h,

Prior


; 1,...,h jy j N

; 1,...,h jL j N

*

* 0

pp

h p hpλ u u λ

Vectors

* * * ; 1,...,h jY y E j N


*

*

* *0

pp

h p hpY u u Y

Defines the axes for a cloud of points in the prior

When p=1, define

SetsPrior

11 0h hu λ λ and * 1*

1 0h hu Y Y1 Nu I ,

Labels

Latent Values

Potential Response

Vectors of *pp

hλ

** pp

hY

** pp

hy

hp Population

**; 1,..., !, 1,..., !

pp

h hL p N p N λ

* **, ; 1,..., !, 1,..., !

pp pp

h h h p N p N λ y

* *** *, ; 1,..., !, 1,..., !pp pp

h h h p N p N λ Y

1,..., ; hh H p

36

Prior Random Variables and Datawith Measurement Error


Assume Random Variables representinga population are exchangeablein each population

*

*

*

!**

1

Nppp

h hpp

I

Y Y

*

*

* *0

pp

h p hpY u u YSince

* *

1

H

h hh

I

Y Y

* *hj hj hjY y E

*1

** 1* 2

0

*

h

hh h

hN

Y

Y

Y

Y Y

Response forsubject j hL

or

* *

*

!* 1*

1 1

H Np

h p hp ph p

I I

Y u u Y

Population

**; 1,..., !, 1,..., !

pp

h hL p N p N λ

* **, ; 1,..., !, 1,..., !

pp pp

h h h p N p N λ y

* *** *, ; 1,..., !, 1,..., !pp pp

h h h p N p N λ Y

1,..., ; hh H p

Prior

Data

, ; 1,..., !Ik Ik k n λ x

* , ; 1,..., !Ik Ik k n λ X ; 1,..., !IkL k n λ

37

Posterior Random Variables with Measurement Error


*

1

II H

h hIIIIh

I

VXY

YY

*x nI

RII N nII

E

1Y

1Y

*

2 2

*

2varx n e n n N n

IR


p

P I 0 0Y

0 0 PY

, ; 1,..., !Ik Ik k n λ x

* , ; 1,..., !Ik Ik k n λ X

; 1,..., !IkL k n λ

Data | ;

h

h

h

p

* , ,h h hL

| ;

h

h

h

p

*, ,h h hL

Prior

*

1 *

1,..., !,;

1,..., !

Ik

hhIIk

k nL

k n

λ

w λ

* *

*1 * 1 *

1,..., !,, ;

1,..., !

Ik Ik

hhII hIIk k

k n

k n

λ X

w λ w Y

* *

1 1 *

1,..., !,, ;

1,..., !

Ik Ik

hhII hIIk k

k n

k n

λ x

w λ w y

Points in the Prior that match the data

| hh

2 21e e

Ln

where

1

1 n

x ss

xn

22

1

1

1

n

x s xs

xn

1

h

hIILL

xN n

22 1

1h

hII hIILL

xN n

* | 1hH h d 1 if

0 otherwiseh

hd

*

2 2II h hII

h H

p

*II h hII

h H

p

38

Posterior Random Variables with Measurement Error

*

1

II H

h hIIIIh

I

VXY

YY *

x nIR

II N nII

E

1Y

1Y

*

2 2

*

2varx n e n n N n

IR


p

P I 0 0Y

0 0 PY

, ; 1,..., !Ik Ik k n λ x

* , ; 1,..., !Ik Ik k n λ X

; 1,..., !IkL k n λ

Data

| ;

h

h

h

p

* , ,h h hL

| ;

h

h

h

p

*, ,h h hL

Prior

*

1 *

1,..., !,;

1,..., !

Ik

hhIIk

k nL

k n

λ

w λ

* *

*1 * 1 *

1,..., !,, ;

1,..., !

Ik Ik

hhII hIIk k

k n

k n

λ X

w λ w Y

* *

1 1 *

1,..., !,, ;

1,..., !

Ik Ik

hhII hIIk k

k n

k n

λ x

w λ w y

Points in the Prior that match the data

| hh

Finite Population Mixed Modelfor the subjects in the Data:

where

* *I x n I Y 1 a E

i I x na a V x 1

random subject effect

*R I x nE Y 1

* 2 2var R I x n e n Y P I

Use this model to obtainthe best linear unbiased predictor of the latent valuefor a subject in the data (which we call the BLUP for a realized subject)

39

Capturing Partial Label Information in the Posterior Distribution

*

1

1

II H

h hIIIIh

I

VXY

WyY

*

* x nI

Rh hII N nII

h H

Ep

1Y

1Y

*

2 2

*

2varx n e n n N n

IR


p

P I 0 0Y

0 0 PY

Usual Posterior

* *

*

* * * *

* *

!* 1*

1 1

! !1 * 1

1 11 1

H Np

h p hp ph p

H N H Np p

h p h h p hp p p ph hp p

I I

I I I I

Y u u Y

u u y u u E

Prior

Data

* 1 2

1var

n

R hI hjej

E

* 1 *0h hY Y for listing p=1

* 1

* 1

* 1

hI

h

hII

EE

E

for points where *

*

k

ppk

v 0u u

0 w

where 2 2

1

1 n

e sesn

2

1var

n

R I sej

E

We want to use the label information for the response error, but not use it for the mean.In the posterior, we want to replace

2e n I by 2

1

n

sej

Ik k I k I X v x v E 0Iλ

40


*

1

1

II H

h hIIIIh

I

VXY

WyY

*

* x nI

Rh hII N nII

h H

Ep

1Y

1Y

*

2 2

*

2varx n e n n N n

IR


p

P I 0 0Y

0 0 PY

Usual Posterior

* * * *

* *

! !1 * 1*

1 11 1

H N H Np p


I I I I

Y u u y u u E Prior

Data

We want to define

Ik k I k I X v x v E 0IλIn the posterior, we can list the subjectsthat are in the data in an order, say

0 0Iv λ =realized order for posterior

*

! !

, ,1 1

I I I

n n

I k k I I k k Ik k

I I

Y Vx VE

v x v E

Now

! !*

, , 01 1

n n

I I k k I I k Ik k

I I

Y v x v E

for all k=1,…,n! such that

* 2 20 0

1var

n

R I x n sej

Y P v v

Measurement Error variance willbe block diagonal, in the realizedorder for the posterior.

41

Partially Labeled Posterior Distribution

! !

, , 0*1 1

1

1

n n

I k k I I k Ik kI

HII

h hIIh

I I

I

v x v EY

YWy

*

* x nI

Rh hII N nII

h H

Ep

1Y

1Y

*

2 20 0* 1

2

var

n

x n se n N nj

IR

IIN n n h hII N n

h H

p

P v v 0 0Y

Y 0 0 P

Partially Labeled Posterior

Prior

Data Ik k I k I X v x v E 0Iλ

Latent values for response in the posterior are in random order

0 0Iv λ =realized order for posterior for realized response

???

Measurement error variance is heterogenousand matches the order of the subjects in theposterior.

42

Partially Labeled Posterior Distribution

! !

, , 0*1 1

1

1

n n

I k k I I k Ik kI

HII

h hIIh

I I

I

v x v EY

YWy

*

* x nI

Rh hII N nII

h H

Ep

1Y

1Y

*

2 20 0* 1

2

var

n

x n se n N nj

IR

IIN n n h hII N n

h H

p

P v v 0 0Y

Y 0 0 P

Prior

Data Ik k I k I X v x v E 0Iλ

Latent values for response in the posterior are in random order


???


How do we define a prior distribution that willresult in such a posterior distribution?


43


Usual Posterior

Ik k I k I X v x v E

* *

* * *

* * * *

* 1! !

1*

* 11 11 1

H N H N Ip IIpIp Ip hIp ph p h hp p p

Ip IIph hp p IIp IIp hII

I I I I

u u u u EY u u y

u u u u E

Prior

Data

for points where *

*

k

ppk

v 0u u

0 w

* *

* * *

Ip IIpIp Ip k

Ip IIpIIp IIp k

u u u u v 0

u u u u 0 w

* *

* *

* ** *

* *

* ** *

1 * 1* ! !

1 * 11 11 1

!

1 * 11 1

H N H NIp hI Ip hIIp Ipp pIh hp p

h hp pII IIp hII IIp hIIIIp IIp

H N k I k Ip ph hp p

hII hIIh p pk k

I I I I

I I I I

u u y u u EY

Y u u y u u E

v x v E

v y v E

!

1 1

H N

h

To form a partially labeledposterior, we need

* 1k hIv E to be equal to 0 Iv E

for all k=1,…,n!

44




Prior

Data

for points where *

*

k

ppk

v 0u u

0 w

* *

* * * *

00 0

k Ip IIpIp Ip k k

Ip IIpIIp IIp k k

v v u u u u v v v 0 v 0

u u u u 0 w 0 w

* *

* *

* ** *

* *

* * *

1 * 1* ! ! 0

1 * 11 11 1

! 0

1 * 11 1

H N H NIp hI k Ip hIIp Ipp pIh hp p


H N k I Ip ph hp p

hII hIIh p k k

I I I I

I I I I

u u y v v u u EY

Y u u y u u E

v x v E

v y v E

*

!

1 1

H N

h p

* *

* * *

Ip IIpIp Ip k

Ip IIpIIp IIp k

u u u u v 0

u u u u 0 w

since k k n v v I

??? How do we define a prior distribution that willresult in such a posterior distribution?

45




* *

* * *

* * * *

* * 1! !

1*

* 11 11 1

H N H N k Ip IIpIp Ip hIp ph p h hp p p

h h Ip IIpp p IIp IIp hII

I I I I

v u u u u EY u u y

u u u u E

Partially Labeled Prior

Data

for points where *

*

k

ppk

v 0u u

0 w

* *

** *

*

0k Ip IIpIp Ip

Ip IIp kIIp IIp

v u u u u v 0

0 wu u u u

* *

* *

* ** *

* *

* * *

1 * 1** ! !

1 * 11 11 1

! 0

1 * 11 1

H N H NIp hI k Ip hIIp Ipp pIh hp p


H N k I Ip ph hp p

hII hIIh p k k

I I I I

I I I I

u u y v u u EY

Y u u y u u E

v x v E

v y v E

*

!

1 1

H N

h p

Define *0k kv v v


46




* *

* * *

* * * *

* * 1! !

1*

* 11 11 1

H N H N k Ip IIpIp Ip hIp ph p h hp p p

h h Ip IIpp p IIp IIp hII

I I I I

v u u u u EY u u y

u u u u E

Partially Labeled Prior

Data

* *

* ** *

* ! ! 0

1 * 11 11 1

H N H Nk I Ip pIh hp p

hII hIIh hp pk kII

I I I I

v x v EYv y v EY

*0k kv v v


*

* x nI

Rh hII N nII

h H

Ep

1Y

1Y

*

2 20 0* 1

2

var

n

x n se n N nj

IR

IIN n n h hII N n

h H

p

P v v 0 0Y

Y 0 0 P


47

An ExamplePosterior Random Variables with Measurement Error

where

* *I x n I Y 1 a E

i I x na a V x 1

random subject effect

*R I x nE Y 1

* 2 2var R I x n e n Y P I

Use this model to obtainthe best linear unbiased predictor of the latent valuefor a subject in the data (which we call the BLUP for a realized subject)

PriorPopulation

**; 1,..., !, 1,..., !

pp

h hL p N p N λ

* **, ; 1,..., !, 1,..., !

pp pp

h h h p N p N λ y

* *** *, ; 1,..., !, 1,..., !pp pp

h h h p N p N λ Y

1,..., ; hh H p

Data

, ; 1,..., !Ik Ik k n λ x

* , ; 1,..., !Ik Ik k n λ X ; 1,..., !IkL k n λ

48

1

1

1

1

H

H

H

H

L L

p p

Prior

An Exchangeable Prior N=3No Measurement Error

R

L

D

R

D

L

L

R

D

L

D

R

D

R

L

D

L

R

Rose

Lily

Daisy

Permutation

p*=1 p*=2 p*=3 p*=4 p*=5 p*=6

L

R

D

L

D

R

R

L

D

R

D

L

D

L

R

D

R

L

Lily

Rose

Daisy

p=1

p=2

p=6=N!D

L

R

D

R

L

L

D

R

L

R

D

R

D

L

R

L

D

Daisy

Lily

Rose

Listings

For Population h

* * * *

* *

! !1 * 1*

1 11 1

H N H Np p


I I I I

Y u u y u u E

Documents

1 Finite Population Inference for Latent Values Measured with Error that Partially Account for Identifable Subjects from a Bayesian Perspective Edward