accurate abc oliver ratmann
TRANSCRIPT
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Centre for Outbreak Analysis and Modelling
Statistical modeling of summary values leads to accurate Approximate
Bayesian Computations
Oliver Ratmann (Imperial College London, UK)Anton Camacho (London School of Hygiene & Tropical Medicine, UK)
Adam Meijer (National Institute of the Environment & Public Health, NL)Gé Donker (Netherlands Institute for Health Services Research, NL)
Thursday, 30 May 13
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Noisy ABC
σ2
n-A
BC
est
imat
e of
πτ(σ
2 |x)
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0 n=60
naivetolerancesτ-=0.35τ+=1.65
π(σ2|x)
argmaxσ2π(σ2|x)
0.90 0.95 1.00 1.05 1.10
050
010
0015
00
estimated mean of σ2
n−AB
C re
petit
ions
S2(y)−S2(x)
[c−,c+]=[−0.5,0.5]
[c−,c+]=[−0.3,0.3]
[c−,c+]=[−0.1,0.1]
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Accurate ABC
σ2n−
ABC
est
imat
e of
πτ(σ
2 |x)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
n=60
calibratedtolerancesτ−=0.572τ+=1.808m=97
π(σ2|x)
argmaxσ2
π(σ2|x)
Can we construct ABC sth inference is accurate• wrt posterior mean / MAP /
• wrt to posterior variance
If yes, under which conditions?
How general are these?
✓0
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Accurate ABC - overview
1. m sim and n obs data points on summary level “summary values” ➣ can model their distribution, eg
s
1:n(x) ⇠ N (µx
,�
2x
)
Three elements for accurate ABC
3. indirect inference ➣ link auxiliary space back to original space
2. classification on auxiliary space ➣ given , is the underlying small ? s
1:n(x) s1:m(y) ⇢ = µ(✓)� µx
s1:m(y) ⇠ N (µ(✓),�2(✓))
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Summary valuesm sim and n obs data points on summary level➣ can model their distribution
data
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Summary valuesm sim and n obs data points on summary level➣ can model their distribution
data summary values
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Summary valuesm sim and n obs data points on summary level➣ can model their distribution
data summary values modeled distribution
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Summary valuesm sim and n obs data points on summary level➣ can model their distribution
data summary values modeled distribution
eg Normal, Exponential,Gamma, Chi-Square;or data transformation eg Log-Normal
Sufficient statistics available on auxiliary space
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Constructing -space⇢modeling summary values defines an auxiliary probability space
s
1:n(x) ⇠ N (µx
,�
2x
)
s1:n(y) ⇠ N (µ(✓),�2(✓))
⇢ = µ(✓)� µx
obs
simpopulation error
L : ⇥ ⇢ RD ! � ⇢ RK
✓ ! (⇢1, . . . , ⇢K)
⇢k
= �k
(⌫xk
, ⌫k
(✓))
⇢ = (⇢1, . . . , ⇢K)✓ = (✓1, . . . , ✓D)D orig parameters
K error parametersLink function
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Indirect inference on -space⇢transform sufficiency problem into change of variable problem⇡
true posterior
(✓|x) / `(x|⇢)⇡(⇢) |@L(✓)|`(S(x)|⇢)⇡(⇢) |@L(✓)|
⇡
abc
(✓|x) / P
x
(ABC accept|⇢)⇡(⇢) |@L(✓)|
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Indirect inference on -space⇢transform sufficiency problem into change of variable problem
ABC approximation on -space is⇢
⇡
true posterior
(✓|x) / `(x|⇢)⇡(⇢) |@L(✓)|`(S(x)|⇢)⇡(⇢) |@L(✓)|
⇡
abc
(✓|x) / P
x
(ABC accept|⇢)⇡(⇢) |@L(✓)|
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Indirect inference on -space⇢transform sufficiency problem into change of variable problem
ABC approximation on -space is⇢
⇡
true posterior
(✓|x) / `(x|⇢)⇡(⇢) |@L(✓)|`(S(x)|⇢)⇡(⇢) |@L(✓)|
⇡
abc
(✓|x) / P
x
(ABC accept|⇢)⇡(⇢) |@L(✓)|
match through calibrationof ABC tolerances and m
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Indirect inference on -space⇢transform sufficiency problem into change of variable problem
ABC approximation on -space is⇢
⇡
true posterior
(✓|x) / `(x|⇢)⇡(⇢) |@L(✓)|`(S(x)|⇢)⇡(⇢) |@L(✓)|
⇡
abc
(✓|x) / P
x
(ABC accept|⇢)⇡(⇢) |@L(✓)|
0.5 1.0 1.5 2.0 2.5 3.0
01
23
45
σ2
density
match through calibrationof ABC tolerances and m
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Indirect inference on -space⇢transform sufficiency problem into change of variable problem
ABC approximation on -space is⇢
⇡
true posterior
(✓|x) / `(x|⇢)⇡(⇢) |@L(✓)|`(S(x)|⇢)⇡(⇢) |@L(✓)|
⇡
abc
(✓|x) / P
x
(ABC accept|⇢)⇡(⇢) |@L(✓)|
Discussion wrt indirect inference (Gouriéroux 1993)• difficulty in indirect inference: which aux space chosen
here constructed empirically from distr of summary values• MLE invariant under parameter transformation, only need bijective
for posterior distribution, entersL
|@L(✓)|
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Accept/reject on -spaceinterpret accept/reject as hypothesis testing procedureR =
�c
� T
�s
1:n(x), s1:m(y)� c
+
⇢
T-testobjective: declare , unequalH0: , equalH1: , unequalrejection region:
µ(✓) µx
µ(✓) µx
µ(✓) µx
(�1, c�] [ [c+,1)
ABCobjective: declare , equalH0: , unequalH1: , equalrejection region:
µ(✓) µx
µ(✓) µx
µ(✓) µx
[c�, c+]
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Accept/reject on -spaceinterpret accept/reject as hypothesis testing procedureR =
�c
� T
�s
1:n(x), s1:m(y)� c
+
⇢
T-testobjective: declare , unequalH0: , equalH1: , unequalrejection region:
µ(✓) µx
µ(✓) µx
µ(✓) µx
(�1, c�] [ [c+,1)
ABCobjective: declare , equalH0: , unequalH1: , equalrejection region:
set , sth
µ(✓) µx
µ(✓) µx
µ(✓) µx
[c�, c+]
c� c+ P (R |H0 ) ↵
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Accept/reject on -spaceinterpret accept/reject as hypothesis testing procedureR =
�c
� T
�s
1:n(x), s1:m(y)� c
+
⇢
T-testobjective: declare , unequalH0: , equalH1: , unequalrejection region:
µ(✓) µx
µ(✓) µx
µ(✓) µx
(�1, c�] [ [c+,1)
ABCobjective: declare , equalH0: , unequalH1: , equalrejection region:
set , sth
µ(✓) µx
µ(✓) µx
µ(✓) µx
[c�, c+]
c� c+ P (R |H0 ) ↵critical region depends on summary values
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Accept/reject on -spaceinterpret accept/reject as hypothesis testing procedureR =
�c
� T
�s
1:n(x), s1:m(y)� c
+
⇢
T-testobjective: declare , unequalH0: , equalH1: , unequalrejection region:
µ(✓) µx
µ(✓) µx
µ(✓) µx
(�1, c�] [ [c+,1)
ABCobjective: declare , equalH0: , unequalH1: , equalrejection region:
set , sth
is power function
µ(✓) µx
µ(✓) µx
µ(✓) µx
[c�, c+]
c� c+
⇢ ! P (R | ⇢ )
P (R |H0 ) ↵critical region depends on summary values
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Accept/reject on -spaceinterpret accept/reject as hypothesis testing procedureR =
�c
� T
�s
1:n(x), s1:m(y)� c
+
⇢
T-testobjective: declare , unequalH0: , equalH1: , unequalrejection region:
µ(✓) µx
µ(✓) µx
µ(✓) µx
(�1, c�] [ [c+,1)
ABCobjective: declare , equalH0: , unequalH1: , equalrejection region:
set , sth
is power function
µ(✓) µx
µ(✓) µx
µ(✓) µx
[c�, c+]
c� c+
⇢ ! P (R | ⇢ )
P (R |H0 ) ↵critical region depends on summary values
power known, so we know ABC accept probability
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Accept/reject on -spaceinterpret accept/reject as hypothesis testing procedureR =
�c
� T
�s
1:n(x), s1:m(y)� c
+
⇢
T-testobjective: declare , unequalH0: , equalH1: , unequalrejection region:
µ(✓) µx
µ(✓) µx
µ(✓) µx
(�1, c�] [ [c+,1)
ABCobjective: declare , equalH0: , unequalH1: , equalrejection region:
set , sth
is power function
µ(✓) µx
µ(✓) µx
µ(✓) µx
[c�, c+]
c� c+
⇢ ! P (R | ⇢ )
P (R |H0 ) ↵
data fixed, so one-sample two-sided test
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Example: test variance
x
1:n ⇠ N (0,�2x
) y1:m ⇠ N (0,�2)
suppose
then
for simplicity, summary values equal data
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Example: test variance
x
1:n ⇠ N (0,�2x
) y1:m ⇠ N (0,�2)
suppose
⇢ = �2/�2x
⇢? = 1
H0 : ⇢ /2 [⌧�, ⌧+]
H1 : ⇢ 2 [⌧�, ⌧+]
T = S2(y1:m)/S2(x1:n) = ⇢1
n� 1
mX
i=1
(yi
� y)2
�2
⇠ ⇢
n� 1�2m�1
then
for simplicity, summary values equal data
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Example: test variance
x
1:n ⇠ N (0,�2x
) y1:m ⇠ N (0,�2)
suppose
⇢ = �2/�2x
⇢? = 1
H0 : ⇢ /2 [⌧�, ⌧+]
H1 : ⇢ 2 [⌧�, ⌧+]
T = S2(y1:m)/S2(x1:n) = ⇢1
n� 1
mX
i=1
(yi
� y)2
�2
⇠ ⇢
n� 1�2m�1
then
for simplicity, summary values equal data
point of equality
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Example: test variance
x
1:n ⇠ N (0,�2x
) y1:m ⇠ N (0,�2)
suppose
⇢ = �2/�2x
⇢? = 1
H0 : ⇢ /2 [⌧�, ⌧+]
H1 : ⇢ 2 [⌧�, ⌧+]
T = S2(y1:m)/S2(x1:n) = ⇢1
n� 1
mX
i=1
(yi
� y)2
�2
⇠ ⇢
n� 1�2m�1
then
for simplicity, summary values equal data
point of equality
tolerances on population level
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Example: test variance
x
1:n ⇠ N (0,�2x
) y1:m ⇠ N (0,�2)
suppose
⇢ = �2/�2x
⇢? = 1
H0 : ⇢ /2 [⌧�, ⌧+]
H1 : ⇢ 2 [⌧�, ⌧+]
T = S2(y1:m)/S2(x1:n) = ⇢1
n� 1
mX
i=1
(yi
� y)2
�2
⇠ ⇢
n� 1�2m�1
then
for simplicity, summary values equal data
point of equality
tolerances on population level
know distribution of T,can work out , andpower function
c� c+
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Example: test variance
x
1:n ⇠ N (0,�2x
) y1:m ⇠ N (0,�2)
suppose
⇢ = �2/�2x
⇢? = 1
H0 : ⇢ /2 [⌧�, ⌧+]
H1 : ⇢ 2 [⌧�, ⌧+]
T = S2(y1:m)/S2(x1:n) = ⇢1
n� 1
mX
i=1
(yi
� y)2
�2
⇠ ⇢
n� 1�2m�1
then
know distribution of T,can work out , andpower function
c� c+
0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
ρpowe
r
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Example: test variance
x
1:n ⇠ N (0,�2x
) y1:m ⇠ N (0,�2)
suppose
⇢ = �2/�2x
⇢? = 1
H0 : ⇢ /2 [⌧�, ⌧+]
H1 : ⇢ 2 [⌧�, ⌧+]
T = S2(y1:m)/S2(x1:n) = ⇢1
n� 1
mX
i=1
(yi
� y)2
�2
⇠ ⇢
n� 1�2m�1
then
0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
ρpowe
r
increase
increase
tighten
move mode
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Example: test variance
x
1:n ⇠ N (0,�2x
) y1:m ⇠ N (0,�2)
suppose
⇢ = �2/�2x
⇢? = 1
H0 : ⇢ /2 [⌧�, ⌧+]
H1 : ⇢ 2 [⌧�, ⌧+]
T = S2(y1:m)/S2(x1:n) = ⇢1
n� 1
mX
i=1
(yi
� y)2
�2
⇠ ⇢
n� 1�2m�1
then
calibrated tol
σ2
n−AB
C e
stim
ate
of π
τ(σ2 |x
)0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
n=60
calibratedtolerancesτ−=0.477τ+=2.2naivetolerancesτ−=0.35τ+=1.65
π(σ2|x)
argmaxσ2
π(σ2|x)
likelihood on -space⇢
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Example: test variance
x
1:n ⇠ N (0,�2x
) y1:m ⇠ N (0,�2)
suppose
⇢ = �2/�2x
⇢? = 1
H0 : ⇢ /2 [⌧�, ⌧+]
H1 : ⇢ 2 [⌧�, ⌧+]
T = S2(y1:m)/S2(x1:n) = ⇢1
n� 1
mX
i=1
(yi
� y)2
�2
⇠ ⇢
n� 1�2m�1
then
calibrated m=97
σ2
n−AB
C e
stim
ate
of π
τ(σ2 |x
)0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
n=60
calibratedtolerancesτ−=0.572τ+=1.808m=97calibratedtolerancesτ−=0.726τ+=1.392m=300
π(σ2|x)
argmaxσ2
π(σ2|x)
likelihood on -space⇢
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Calibration Lemmaswhen is it possible and easy to calibrate?
depends on distribution family
T = S
2(y1:m)/S2(x1:n) = ⇢
1
n� 1
mX
i=1
(yi � y)2
�
2
⇠ ⇢
n� 1�
2m�1
main condition:
• if family continuous in and strictly totally positive of order 3, then power function is unimodal
⇢
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Calibration Lemmaswhen is it possible and easy to calibrate?
depends on distribution family
T = S
2(y1:m)/S2(x1:n) = ⇢
1
n� 1
mX
i=1
(yi � y)2
�
2
⇠ ⇢
n� 1�
2m�1
main condition:
• if family continuous in and strictly totally positive of order 3, then power function is unimodal
⇢
Discussionmany tests satisfying these criteria available, see
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Combining test statisticsequivalent to combining summary statistics
very briefly:
• Mahalanobis approach possible, corresponds to KT location tests for normal summary values
• Intersection approach possible,
can combine KT tests arbitrarily
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Back to indirect inference
transform sufficiency problem into change of variable problem
ABC approximation on -space is⇢
⇡
true posterior
(✓|x) / `(x|⇢)⇡(⇢) |@L(✓)|`(S(x)|⇢)⇡(⇢) |@L(✓)|
⇡
abc
(✓|x) / P
x
(ABC accept|⇢)⇡(⇢) |@L(✓)|
now calibrated to match closely
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Back to indirect inference
transform sufficiency problem into change of variable problem
ABC approximation on -space is⇢
⇡
true posterior
(✓|x) / `(x|⇢)⇡(⇢) |@L(✓)|`(S(x)|⇢)⇡(⇢) |@L(✓)|
⇡
abc
(✓|x) / P
x
(ABC accept|⇢)⇡(⇢) |@L(✓)|
we are left with the change of variables
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Conditions on link function
if tolerances & m calibrated, main condition isthat must be bijective
can be tested from ABC output
L : ✓ ! (⇢1, . . . , ⇢K)
a
−0.4−0.2
0.00.2
0.4
sigma^2
0.9
1.0
1.1
log(rho[1])
−0.2
0.0
0.2
−0.4 −0.2 0.0 0.2 0.4
0.9
1.0
1.1
1.2
a
σ2
2
2 4
6
8
10
moving average example
two model parameters
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Conditions on link function
if tolerances & m calibrated, main condition isthat must be bijective
can be tested from ABC output
L : ✓ ! (⇢1, . . . , ⇢K)
a
−0.4−0.2
0.00.2
0.4
sigma^2
0.9
1.0
1.1
log(rho[1])
−0.2
0.0
0.2
−0.4 −0.2 0.0 0.2 0.4
0.9
1.0
1.1
1.2
a
σ2
2
2 4
6
8
10
moving average example
only one test
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Conditions on link function
if tolerances & m calibrated, main condition isthat must be bijective
can be tested from ABC output
L : ✓ ! (⇢1, . . . , ⇢K)
a
−0.4−0.2
0.00.2
0.4
sigma^2
0.9
1.0
1.1
rho[2]
−0.4
−0.2
0.0
0.2
a
−0.4−0.2
0.00.2
0.4
sigma^2
0.9
1.0
1.1
log(rho[1])
−0.2
0.0
0.2
−0.4 −0.2 0.0 0.2 0.4
0.9
1.0
1.1
1.2
a
σ2
2
2 4
6
8
10
−0.4 −0.2 0.0 0.2 0.4
0.9
1.0
1.1
1.2
a
σ2
10
20
30
40 ●
moving average example
adding one more test
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Conditions on link function
if tolerances & m calibrated, main condition isthat must be bijective
can be tested from ABC output
L : ✓ ! (⇢1, . . . , ⇢K)
• bijectivity easy to check1. record estimate of , eg 2. reconstruct link function with regression3. link bijective if and only if
is a single point
⇢ s
1:m(y)� s
1:n(x)
Discussion
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Conclusions
possible to set up ABC such that the ABC mean or MAP are exactly those of the true posterior(calibrate , )
possible to set up ABC such thatthe KL divergence of the ABC approximation to the true posterior is very small(calibrate m)
⌧� ⌧+
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Conclusions
possible to set up ABC such that the ABC mean or MAP are exactly those of the true posterior(calibrate , )
possible to set up ABC such thatthe KL divergence of the ABC approximation to the true posterior is very small(calibrate m)
⌧� ⌧+
To achieve this, need to
1. identify summary values2. use a suitable test statistic for calibrations3. calibrate4. test if link function meets conditions
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Resources
code on githubmanuscript on arxiv
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Thank you!
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Time series application
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Time series application
first patch, strong seasonality
second patch, weak seasonality.Re-seeds first patch.
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Time series application
first patch, strong seasonality
second patch, weak seasonality.Re-seeds first patch.
parameters to estimate + reporting rate
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Time series application
3 model parameters3 tests
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Time series application100 replicate runs
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