S p e c t r a l f i t t i n g m e t h o d s Wo - - PowerPoint PPT Presentation
S p e c t r a l f i t t i n g m e t h o d s Wo r k s h o p a t t h e Ma x P l a n c k I n s t i t u t e f o r e x t r a t e r r e s t r i a l P h y s i c s 2 4 - 2 5 .
Wo r k s h
a t t h e Ma x P l a n c k I n s t i t u t e f
e x t r a t e r r e s t r i a l P h y s i c s 2 4
5 . 9 . 2 1 9
g a n i s e d b y J
a n n e s B u c h n e r , J Mi c h a e l B u r g e s s , J
r n Wi l ms
S p e c t r a l f i t t i n g m e t h
s
Probabilities of discrete events Probabilities of hypotheses Scientific application
Probabilities of discrete events Probabilities of hypotheses Poisson distribution (by Abraham de Moivre) Bayesian inference (by Pierre-Simon Laplace) Scientific application Horse kicks (by Ladislaus Bortkiewicz) Stigler's law of eponymy (Robert K. Merton)
r a c t i c a l i n f
m a t i
– o
r g a n i s e r s
– w
i f i
– l
a l i n f
m a t i
– r
g h a g e n d a & g
l
– d
i n n e r
i r s t c
t e n t b l
k
n i n g
– M
e a s u r e m e n t p r
e s s & s t a t i s t i c s i n v
v e d
– B
a c k g r
n d t r e a t m e n t
– L
a l b e s t f i t s
u n c h C a n t i n e
f t e r n
– G
l
a l f i t s & p r
a b i l i t y d i s t r i b u t i
s
– M
e l c
p a r i s
– C
f e e 1 5 : 3
– C
b i n i n g i n f
m a t i
– B
e y
d X
a y s p e c t r a
– D
i s c u s s i
& Q u e s t i
s
n d : 1 7 : 3 . D i n n e r 1 9 :
n i n g
– E
x t e n d e d s
r c e s , c a l i b r a t i
– D
i s c u s s i
& Q u e s t i
s
– P
s s
k n
l e d g e & p r a c t i c a l p
n t e r s
G
l : w h a t m e t h
s e x i s t w h a t a r e t h e i r b e n e f i t s & l i m i t a t i
s w h a t t
a y a t t e n t i
t
6 + B u s 5 9
– D
i e t l i n d e n s t r .
– B
u s 5 9 G i e s i n g
– R
i c h a r d
t r a u s s
t r .
6 + U 4
– O
d e
s p l a t z
– U
4 A r a b e l l a p a r k
– R
i c h a r d
t r a u s s
t r .
6 + B u s 5 9
– D
i e t l i n d e n s t r .
– B
u s 5 9 G i e s i n g
– R
i c h a r d
t r a u s s
t r .
6 + U 4
– O
d e
s p l a t z
– U
4 A r a b e l l a p a r k
– R
i c h a r d
t r a u s s
t r .
v e r v i e w & I n t r
u c t i
– M
e a s u r e m e n t p r
e s s
– B
a c k g r
n d & s
r c e r e g i
s
– L
i n e a r a l g e b r a a p p r
i m a t i
– L
i k e l i h
& s t a t i s t i c s
f t e r : b a c k g r
n d
f t e r : f i t t i n g
e r n
l l i c
n f l i p
– k
= ( p )
– k
= 1 ( 1
)
i n
i a l
– n
t r i e s , f i r s t k s u c c e s s f u l
– P
= p
k
( 1
)
( n
)
s s
– n i
n f b u t p n =
r u l e
t h u m b : i f n > 2 a n d p < . 5 n > 1 a n d n p < 1
s s
– k
: i n t e g e r
–
: r e a l ( m e a n & v a r i a n c e ) λ
– A
s y m m e t r i c
– I
n t e g e r
– P
i t i v e
c a l i n g
d d i t i
u b t r a c t i
S a m p l e s E l e c t r
i c s ( s h
n
s e ) P h
c
n t i n g ( P
s s
n
s e )
( S k e l l a m d i s t r i b u t i
) ( P
s s
d i s t r i b u t i
) V a r i a b i l i t y !
λ
s h a p e c h a n g e s
s s
– k
: i n t e g e r
–
: r e a l ( m e a n & v a r i a n c e ) λ
a u s s i a n
– M
e a n ( µ ) & v a r i a n c e (² ) = σ²) =! λ
– M
e a n ( µ ) & v a r i a n c e (² ) = k σ²) =!
– r
e a l , c a n b e n e g a t i v e
λ
K n
n d a t a U n k n
n r a t e L i k e l i h
P r
a b i l i t y ( f r e q u e n c y ) t
r
u c e e x a c t l y t h i s d a t a
K n
n d a t a U n k n
n r a t e L i k e l i h
P r
a b i l i t y ( f r e q u e n c y ) t
r
u c e e x a c t l y t h i s d a t a
K n
n d a t a U n k n
n r a t e L i k e l i h
P r
a b i l i t y ( f r e q u e n c y ) t
r
u c e e x a c t l y t h i s d a t a
K n
n d a t a U n k n
n r a t e L i k e l i h
P r
a b i l i t y ( f r e q u e n c y ) t
r
u c e e x a c t l y t h i s d a t a
K n
n d a t a U n k n
n r a t e L i k e l i h
P r
a b i l i t y ( f r e q u e n c y ) t
r
u c e e x a c t l y t h i s d a t a
K n
n d a t a U n k n
n r a t e L i k e l i h
P r
a b i l i t y ( f r e q u e n c y ) t
r
u c e e x a c t l y t h i s d a t a
a i l s h a v e d i f f e r e n t s l
e s
– G
a u s s h i g h
n d m
e p e r m i s s i v e
– P
s s
l
n d m
e p e r m i s s i v e
i g h t w a y : P
s s
i s t
i c a l l y : G a u s s f a s t e r t
v a l u a t e
s s
– L
i k e l i h
* l
a u s s i a n
– L
i k e l i h
* l
s s
– L
i k e l i h
* l
a u s s i a n
– L
i k e l i h
* l
S t a t , C a s h C h i ²
D
s n
m e a n t h e y f
l
a c h i ² d i s t r i b u t i
!
C a s h ( 1 9 7 9 )
s s
a u s s i a n
k
1, λ1 k
2, λ2
k
1, λ1 k
2, λ2 R e m e m b e r : = n u m b e r / c m ² / s / k e V * d E * d t * d A λ k = n u m b e r F l u x C
n t s
k
1 S, λ1
Sk
1 B, λ1
Bk
s r c, λs
r c, t
s r c, A
s r ck
b k g, λb
k g, t
b k g, A
b k gA s s u m e t i m e , l
a t i
n d e p e n d e n c e
R e m e m b e r : = n u m b e r / c m ² / s / k e V * d E * d t * d A λ + A s s u m p t i
s :
r e a e n e r g y
n d e p e n d e n t
a t e c
s t a n t w i t h a r e a , t i m e , l
a t i
R e m e m b e r : = n u m b e r / c m ² / s / k e V * d E * d t * d A λ A s s u m p t i
s :
r e a e n e r g y
n d e p e n d e n t
a t e c
s t a n t w i t h a r e a , t i m e , l
a t i
r c + b k g G a u s s G a u s s ( s u b t r a c t a b l e , f l a t s / d a r k s )
r c + b k g P
s s
P
s s
i g h c
n t s ( > 1 ) i n e v e r y s i n g l e s r c a n d b k g b i n G a u s s +
u b t r a c t w i t h b k g v a r i a n c e p r
a g a t i
– S
u b t r a c t & m
e l w i t h S k e l l a m d i s t r i b u t i
– D
h e r i g h t t h i n g a n d m
e l b
h a s P
s s
r c + b k g G a u s s G a u s s ( s u b t r a c t a b l e , f l a t s / d a r k s )
r c + b k g P
s s
P
s s
i g h c
n t s ( > 1 ) i n e v e r y s i n g l e s r c a n d b k g b i n G a u s s +
u b t r a c t w i t h b k g v a r i a n c e p r
a g a t i
– S
u b t r a c t & m
e l w i t h S k e l l a m d i s t r i b u t i
– D
h e r i g h t t h i n g a n d m
e l b
h a s P
s s
s s
e s t i m a t e
r a t e i n e a c h b i n , i n d e p e n d e n t l y
u n c t i
a p p r
i m a t i
b a c k g r
n d
– I
n c
n t s ( e m p i r i c a l m
e l )
– P
h y s i c a l b a c k g r
n d f l u x m
e l
– F
i t s i m u l t a n e
s l y w i t h s
r c e
– F
i t b a c k g r
n d m
e l f i r s t , u s e b e s t
i t b a c k g r
n d s h a p e f
s
r c e f i t
R e m e m b e r : = n u m b e r / c m ² / s / k e V * d E * d t * d A λ A s s u m p t i
s :
r e a e n e r g y
n d e p e n d e n t
a t e c
s t a n t w i t h a r e a , t i m e , l
a t i
i f f u s e e m i s s i
– L
a l h
b u b b l e
– G
a l a c t i c d i s k
– G
a l a c t i c h a l
m i c b a c k g r
n d
– U
n r e s
v e d A G N
i g h
n e r g y p a r t i c l e b a c k g r
n d
h t t p s : / / w i k i . m p e . m p g . d e / e R
i t a / S c i e n c e R e l a t e d S t u f f / B a c k g r
n d
S e m i
h y s i c a l b a c k g r
n d m
e l s
M a x i m i z e p
s s
l i k e l i h
a t a l l b i n s s h a p e N u S T A R ( Wi k + 1 4 ) e s p e c i a l l y i m p
t a n t f
e x t e n d e d s
r c e P a r t i c l e b a c k g r
n d C
m i c b a c k g r
n d I n s t r u m e n t a l b a c k g r
n d … L
a t i
& t i m e
e p e n d e n t
E m p i r i c a l b a c k g r
n d m
e l s
M a x i m i z e p
s s
l i k e l i h
a t a l l b i n s s h a p e C h a n d r a ( X M M , C h a n d r a , S w i f t m
e l s i n
s e )
P r
:
a n c
t a i n p h y s i c a l k n
l e d g e & s m
h n e s s
m a l l u n c e r t a i n t i e s
b i n c
n t s
C
s :
e e d t
p e c i f y m
e l
i t c a n b e p
E m p i r i c a l b a c k g r
n d m
e l s
A u t
a t e d s h a p e f i n d i n g S i m m
d s , B u c h n e r e t a l . ( 2 1 7 ) X M M / P N , M O S , C h a n d r a / A C I S , N u S T A R , S u z a k u , R X T E , S w i f t / X R T
B a c k g r
n d : I n d i v i d u a l b i n s
E s t i m a t e m
t l i k e l y b a c k g r
n d r a t e i n e a c h b i n A d d s c a l e d t
r c e r e g i
c
n t s ( w s t a t , X s p e c d e f a u l t i f s e t t
s t a t w i t h n
a c k g r
n d m
e l ) p g s t a t
P r
:
e l s p e c i f i c a t i
n e e d e d C
s :
t i n u i t y
n n e c e s s a r i l y l a r g e u n c e r t a i n t i e s
e e d > c
n t s p e r b i n
H i g h e r L : m
e l u n d e r t h e s e p a r a m e t e r s
t e n m a k e s t h i s d a t a L
e r L : l e s s f r e q u e n t l y F r e q u e n c y
d a t a L i k e l i h
f u n c t i
a t D , a t p a r a m e t e r v a l u e s ( n
a d e n s i t y )
a r a m e t e r r a n g e s a l l
e d
p r
a b l e ( L , T , …, p h y s i c a l p a r a m e t e r s )
I n i n f i n i t e l y s m a l l r e g i
: z e r
r
a b i l i t y
P r
a b i l i t y d e n s i t y V
u m e D e n s i t y P r
a b i l i t y m a s s
F i n d r e g i
s w i t h h i g h p r
a b i l i t y m a s s P a r a m e t e r s p a c e e x p l
a t i
C
d i t i
a l p r
a b i l i t i e s
– B
a y e s t h e
e m
– P
( A | B ) ! = P ( B | A )
– N
m a l i s a t i
– P
a r a m e t e r i n f e r e n c e
– M
e l i n f e r e n c e
– I
n t e r p r e t a t i
B a y e s t h e
e m
P a r a m e t e r s p a c e e x p l
a t i
a l
t i m i z a t i
– L
M , s i m p l e x , … ( m a n y )
– M
t e c a r l
t i m i z a t i
a l s a m p l i n g : M C M C
– T
e m p e r i n g
– L
i m i t a t i
s
l
a l
t i m i z a t i
– G
e n e t i c a l g
i t h m s ( D E )
l
a l s a m p l i n g
– N
e s t e d s a m p l i n g
f a w a y f r
b
n d a r y
f m
e l i s l i n e a r
f n d a t a h i g h
f i s t r u e p a r a m e t e r
h e n
( s y m m e t r i c , s i n g l e g a u s s )
I f m a n y d a t a a r e c r e a t e d u n d e r
L i n t e r v a l
b e l
b e s t f i t C
t a i n s t r u e v a l u e 6 8 %
r e a l i s a t i
s
C
f i d e n c e i n t e r v a l
Wh a t w a s t h e q u e s t i
a g a i n ? A r e c
d i t i
s f u l f i l l e d ? Wh a t d
n e q u a l “ e r r
s ” m e a n ?
f a w a y f r
b
n d a r y
f m
e l i s l i n e a r
f n d a t a h i g h
f i s t r u e p a r a m e t e r
s y m m e t r i c , s i n g l e g a u s s )
C
f i d e n c e i n t e r v a l
I f c
d i t i
s a r e n
m e t
( a l w a y s )
M
t e C a r l
s i m u l a t i
s ( p a r a m e t r i c b
s t r a p )
i
n
C a l i b r a t e a
θ
L
f a w a y f r
b
n d a r y
f m
e l i s l i n e a r
f n d a t a h i g h
f i s t r u e p a r a m e t e r
s y m m e t r i c , s i n g l e g a u s s )
I f c
d i t i
s a r e n
m e t
( a l w a y s )
M
t e C a r l
s i m u l a t i
s ( p a r a m e t r i c b
s t r a p )
i
n
= θ
L L … p
a l u e s
θ
L C
v
u t i
T r u e p a r a m e t e r d i s t r i b u t i
+ M e a s u r e m e n t e r r
& a n a l y s i s m e t h
C
f i d e n c e i n t e r v a l s H i s t
r a m
b e s t f i t s C u m u l a t i v e d i s t r i b u t i
C l e a n s
u t i
: M
e l p
u l a t i
d i s t r i b u t i
( H B M ) M e a n i n g ? U p p e r l i m i t s ? B u c h n e r + 1 7 a
V
u m e D e n s i t y P r
a b i l i t y m a s s
F i n d r e g i
s w i t h h i g h p r
a b i l i t y m a s s
θ
L L
I d e a : S a m p l e p a r a m e t e r s
u t i
s p r
t i
a l l y t
h e i r p r
a b i l i t y F
e x a m p l e w i t h a g r i d
θ
P
p a r a m e t e r s
u t i
s w e i g h t e d b y t h e i r p r
a b i l i t y
C r e d i b l e i n t e r v a l s
D e f i n i t i
s : D e n s i t y c u m u l a t i v e q u a n t i l e s
i g h e s t D e n s i t y I n t e r v a l s B
d e r s ( u p p e r l i m i t s )
d
g r i d i n f e a s i b l e
a m p l e
2 3 …. w
1
w
2w 3
….
e c h n i q u e s :
– I
m p
t a n c e s a m p l i n g
– M
C M C
– N
e s t e d s a m p l i n g
( P
t e r i
c h a i n s )
t e r i
c h a i n
1 2 3 ….
i n d r e g i
s w i t h h i g h p r
p u t e p r
.
r e g i
s
t e r i
p r e d i c t i
s
e r i v e d q u a n t i t i e s
P q10 q50 q90 P P(x>4)= sample fraction F , z L , z
θ
P
D r a w f r
p r
a l d i s t r i b u t i
Q We i g h b y Q ( ) / P ( | D )
e i g h t e d c h a i n
d v a n t a g e s :
f f i c i e n t i n l
a r a l l e l i s a b l e
a n i n t e g r a t e p a r a m e t e r s p a c e D i s a d v a n t a g e s
e e d t
i n d g
p r
a l ( V B )
s c a l i n g t
d
p e r f
m a n c e i f p r
a l i s b a d ( v a r i a n c e i n d i c a t
)
L L
Starting point θ Loop forever:
θ’ = Normal(θ, sigma_p)
if P(θ’|D)/P)D)/P(θ|D)/P)D) > U():
θ = θ’
add θ to chain
x
θ
L L x E m e r g i n g b e h a v i
r :
Starting point θ Loop forever:
θ’ = Normal(θ, sigma_p)
if P(θ’|D)/P)D)/P(θ|D)/P)D) > U():
θ = θ’
add θ to chain
e t r
i s + R a n d
Wa l k
m a n
a r e
( e m c e e )
M C ( H a m i l t
i a n M
t e C a r l
a n i m a t i
h t t p s : / / c h i
e n g . g i t h u b . i
m c m c
e m
a p p . h t m l R a n d
w a l k , H M C
e t r
i s R a n d
Wa l k
– A
d v : s i m p l e
– D
i s a d v : p
m i x i n g
f f i n e
n v a r i a n t e n s e m b l e
– A
d v : a u t
u n i n g f
g a u s s i a n L
– D
i s a d v : p
m i x i n g i n b a n a n a s , c
l a p s e s i n h i g h
( H u i j s e r + 1 5 )
M C ( H a m i l t
i a n M
t e C a r l
– A
d v : t u n e s i t s e l f t
u r f a c e
– D
i s a d v : n e e d g r a d i e n t s
m
e l s
G
m a n & We a r e ( 2 1 ) e m c e e
C M C t h e
y : n i n f
r a c e p l
s
u t
r e l a t i
l e n g t h
v e r g e n c e t e s t s
– D
e t e c t i f u n r e l i a b l e
– G
e l m a n
u b i n d i a g n
t i c
– (
m a n y m
e )
( b y E r i c F
d )
P h a s e s : I d e n t i f i c a t i
M i x i n g ( b u r n
n )
L
E s c a p i n g l
a l m a x i m a : s t r a t e g i e s
u l t i p l e r a n d
s t a r t p
i t i
s
– A
u g m e n t l
a l t e c h n i q u e s
a k e s u r f a c e e a s i e r
– T
e m p e r i n g / A n n e a l i n g
l k e r p
u l a t i
– G
W
– G
e n e t i c a l g
i t h m s ( D E )
L
C r
s
e r I n i t i a l p
u l a t i
M u t a t i
S e l e c t i
O f f s p r i n g v s N e w g e n e r a t i
Wi t h f i t n e s s f u n c t i
( h e r e : L )
i f f e r e n t i a l e v
u t i
s i n t
i g h e s t r e g i
s
A d v a n t a g e s :
l
a l
u s t t
e g e n e r a c i e s , a u t
u n i n g p r
a l
r k s i n h i g h
& n
t i n u
s p a r a m e t e r s D i s a d v a n t a g e s :
e t u n i n g p a r a m e t e r s
t
p i n g c r i t e r i
m a y n
b e m e a n i n g f u l
s n
s a m p l e (
l y b e s t
i t )
moncar
m p i r i c a l m
e l s
– I
n f
m a t i
c
t e n t
– P
r e d i c t i
q u a l i t y
p
e n t p r e s e n c e
– R
e g i
s
p r a c t i c a l e q u i v a l e n c e
h y s i c a l e f f e c t s
– B
a y e s i a n m
e l c
p a r i s
– P
r i
s
t e n w e l l
u s t i f i e d
h t t p s : / / a r x i v .
g / a b s / 1 5 6 . 2 2 7 3 B e t a n c
r t ( 2 1 5 ) B u c h n e r + 1 4
k a i k e i n f
m a t i
c r i t e r i
s m
e c
p l e x w
t h s t
i n g ?
AIC = 2 * d – 2 * Lmax AIC = 2 * d + CStat
A k a i k e ( 1 9 7 3 )
A d v a n t a g e s :
e d i n i n f
m a t i
t h e
y
n d e p e n d e n t
p r i
D i s a d v a n t a g e s :
n c e r t a i n t i e s , t h r e s h
d s u n c l e a r
F l e x i b l e m
e l I n f l e x i b l e m
e l
D a t a L h i g h , V t i n y L m e d i u m , V m e d i u m ( n
n u m b e r
p a r a m e t e r s )
P
t e r i
d s r a t i
r i
d s r a t i
a y e s f a c t
B u c h n e r + 1 4
Missing ingredients
Galilean, RadFriends, PolyChord
A n i m a t i
: h t t p s : / / j
a n n e s b u c h n e r . g i t h u b . i
m c m c
e m
a p p . h t m l # R a d F r i e n d s
S , s t a n d a r d ( v i a c h i
e n g . g i t h u b . i
P
t e r i
d s r a t i
r i
d s r a t i
a y e s f a c t
B u c h n e r + 1 4
e l p r
a b i l i t i e s d e c i s i
s
a l s e d e c i s i
r a t e
( f a l s e p
i t i v e s / n e g a t i v e s )
– M
t e C a r l
i m u l a t i
s ( p a r a m e t r i c b
s t r a p )
B u c h n e r + 1 4
B u c h n e r + 1 4 False negatives Non-decisions w a b s i n p u t p
e r l a w i n p u t w a b s i n p u t p
e r l a w i n p u t
A d v a n t a g e s :
e t r i d
p a r a m e t e r p r i
d e p e n d e n c e s
a v e f r e q u e n t i s t p r
e r t i e s
B a y e s i a n m e t h
p l e t e l y B a y e s i a n t r e a t m e n t + d e c i s i
s D i s a d v a n t a g e s :
a n b e c
p u t a t i
a l l y e x p e n s i v e
a k e d e c i s i
s
– I
s p a r a m e t e r g r e a t e r t h a n C ?
– I
s t h i s m
e l “ b e t t e r ” t h a n t h e
h e r ?
a r a m e t r i c b
s t r a p
– M
t e C a r l
i m u l a t i
a l l
a r b i t r a r y c
p l e x i t y
T e s t mo d e l i n i s
a t i
? P P C P a r a me t r i c b
s t r a p C
a r e p h y s i c a l mo d e l s
e mp i r i c a l d e s c r i p t i
s ? y e s n
r e l a t i v e I n f
ma t i
c
t e n t ( A I C ) P r e d i c t i
q u a l i t y ( C r
s v a l i d a t i
) e m p i r i c a l p h y s i c a l e f f e c t s A d d i t i v e c
e n t P a r a me t e r e s t i ma t i
R e g i
e q u i v a l e n c e B a y e s i a n mo d e l c
a r i s
y e s n
a y e s i a n mo d e l c
a r i s
e s t e r d a y :
– B
a s i c s t a t i s t i c s , p r
l e m s e t u p
– P
a r a m e t e r e s t i m a t i
m e t h
s , c r e d i b l e & c
f i d e n c e i n t e r v a l s
– M
e l c
p a r i s
m e t h
s
– V
i s u a l i s a t i
s
a y :
– E
x t e n d e d s
r c e s , c a l i b r a t i
– S
t a c k i n g i n f
m a t i
– D
i s c u s s i
& Q u e s t i
s
– P
r a c t i c a l p
n t e r s & Wr a p
p
c
e r e d :
– T
s
– P
i l e
p & v a r i a b i l i t y
S p e c t r
c
y + l
e r E + h i g h e r E + i m a g i n g + t i m e O u t s i d e s p e c t r
c
y
r e n
h i n g s p e c i a l
s s
l i k e l i h
+ g
b a c k g r
n d h a n d l i n g
c
n t s
h i n k i n t e r m s
a l l
e d r e g i
s
N T
L, NH from X-ray spectrum
Scattered Powerlaw component CTK flat spectrum + FeK line
L, NH from X-ray spectrum
Probability cloud
e a s u r e m e n t g a v e x 1 = 4 +
x 2 = 5 +
1
e n e r a t e 1 s a m p l e s f
e a c h
100xN matrix of x
v a l u a t e m
e l F ( x )
100xN matrix of F 100xN matrix of x
u m p r
a b i l i t i e s
u l t i p l y p r
a b i l i t i e s
h e n t r y
t
h e r m
e l p a r a m e t e r s
100xN matrix of F N vector L
P(x) x Measurement error F(x) Population distribution
P(x) x Measurement error F(x) Population distribution
P(x) x Measurement error F(x) Population distribution
P(x) x Measurement error F(x) Population distribution
e n e r a t e f r
6 +
w i t h m e a s u r e m e n t e r r
s
c a n d
h i s i n a n y p a c k a g e !
t a t e w h a t y
a r e d
n g
S t a t ( P
s s
)
a c k g r
n d w i t h f u n c t i
s ( c h e c k f i t )
i s u a l i s e , v i s u a l i s e , v i s u a l i s e
h
p
t e r i
d i s t r i b u t i
s & f i t s i n d a t a s p a c e
a r y p r i
s & a s s u m p t i
s
s e n e s t e d s a m p l i n g , M C M C w i t h c a r e
a k e s i m u l a t i
s
s k f
h e l p
i s i s , s h e r p a , s p e x , x s p e c , 3 m l , . . .
C
t a c t p
n t s f
q u e s t i
s
s k a c
l e a g u e
s t r
t a t i s t i c s F a c e b
g r
p
S P E C F a c e b
g r
p
M P E
– J
M i c h a e l B u r g e s s
– J
a n n e s B u c h n e r