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Dust Temperature and Spectral Index Correlation?
Xianchao Xie
Department of Statistics, Harvard University

Joint work with Brandon Kelly

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1/20

Dust Emission

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2/20

Spectral Energy Distribution Fitting

Modified Blackbody Assumption
S C 0
+3

exp

h T

-1

-1

Parameter in the model: ( , T ). Observations: S1 , · · · , SJ .

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3/20

The Empirical Inverse Correlation

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4/20

A Physical Law or A Statistical One?

A Scientific Discovery
· Similar patterns on various galactic sources. · Confirmed in different exp eriments by indep endent groups.

A Statistical Fallacy
· There are noises in the measurements. · Estimates of parameters with noisy data are usually correlated. · Simulation study has suggested that this might b e the reason.
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Two Types of Correlation
The Thought Process
^ ^ ( , T ) Clean "Data" Dirty Data ( (data), T (data))

The Statistical Correlation
^^ Corr ( , T )

The Scientific Correlation
Corr ( , T )
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Testing the Scientific Hypothesis

A Bayesian Model
Level 1 : p (Data| , T ) Level 2 : p ( , T )

Scientific Hypothesis
T under p ( , T )

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The Statistical Model I
Likelihood
Sij = j Ci
i . i .d

j 0

i +3

exp
i .i .d

hj Ti

-1

-1
ind

e

ij

2 2 j N (0, ), Ci N (µc , c ), eij N (0, i2 ). j

Prior
i |Ti Ti
i .i .d

2 N (ATiB , )

i .i .d

1[2

,150]

(Ti )dTi
2
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Hyper Prior
2 2 2 2 (µc , c , , ) d µc d ln c d ln b eta2 d ln

A dA B1
[-2,2]

(B )dB

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The Graphical Structure of the Model
(A,B) T
1

T

i

j f f

1

i

11

1

f

i1

1j

C

1

c

C

i

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The Plain-Vanilla Gibbs Sampler

Gibbs Components
Step I : (i , Ci )|(Si 1 , 1 ), · · · , (SiJ , J ), Ti , µc , A, B Step II : Ti |(Si 1 , 1 ), · · · , (SiJ , J ), i , Ci , A, B Step III : j |(S1j , T1 , 1 , C1 ), · · · , (Snj , Tn , n , Cn )
2 Step IV : µc , c |C1 , · · · , Cn 2 Step V : |1 , · · · , J n n

Step VI : A|B , T1 , · · · , Tn , 1 , · · · , Step VII : B |A, T1 , · · · , Tn , 1 , · · · ,

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Graphical Illustration of Step I
(A,B) T
1

T

i

j f f

1

i

11

1

f

i1

1j

C

1

c

C

i

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Graphical Illustration of Step II
(A,B) T
1

T

i

j f f

1

i

11

1

f

i1

1j

C

1

c

C

i

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Trace plot for 1
1.65

1.6

1.55

1.5

1.45

1.4

0

200

400

600

800

1000

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Trace plot for T1
13 12.5 12 11.5 11 10.5 10

0

200

400

600

800

1000

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A Better Gibbs Sampler

Gibbs Components
2 2 Step I : (i , Ci )|(Si 1 , 1 ), · · · , (SiJ , J ), Ti , µc , c , , A, B

Step II : Ti |(Si 1 , 1 ), · · · , (SiJ , J ), i , µc , A, B Step III : j |(S1j , T1 , 1 , C1 ), · · · , (Snj , Tn , n , Cn )
2 Step IV : µc , c |C1 , · · · , Cn 2 Step V : |1 , · · · , J n n

Step VI : A|B , T1 , · · · , Tn , 1 , · · · , Step VII : B |A, T1 , · · · , Tn , 1 , · · · ,

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Graphical Illustration of Step II
(A,B) T
1

T

i

j f f

1

i

11

1

f

i1

1j

c

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Trace plot for T1
14

13.5

13

12.5

12

11.5

11
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10.5 0

200

400

600

800

1000
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Application to Real Datasets
4.5

4

3.5

3

2.5

2

1.5

0

200

400

600

800

1000 logo

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2 What's Wrong: Trace Plot for c
30

25

20

15

10

5

0

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0

200

400

600

800

1000

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How to incorporate the prior?

The Form of the Prior
(T , A, B ) (T , A, B )
2 |T , A, B N (AT B , )

The Prior Knowledge
N (2.0, 0.22 ) .
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