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) Tuesday, 7 January 14
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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)
Tuesday, 7 January 14
Standard ABC
ABC approximation to likelihood
is exact if 1) summary statistics are sufficient 2) upper and lower tolerances coincide
summary stat
tolerance
(Beaumont 2002)
Tuesday, 7 January 14
Standard ABC
ABC approximation to likelihood
is exact if 1) summary statistics are sufficient 2) upper and lower tolerances coincide
summary stat
tolerance
(Beaumont 2002)
in practice not feasible, ‘asymptotic’ argument
Tuesday, 7 January 14
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)
even with sufficient summary statistics (Fernhead & Prangle 2012)
Standard ABC is noisy
Tuesday, 7 January 14
ABC*
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=97
π(σ2|x)
argmaxσ2
π(σ2|x)
Can we construct ABC such that inference is accurate• wrt point estimate, eg MAP• wrt overall similarity in distribution, eg KL
divergence• and maintain computational feasibility
Modeling summary valuesconstructs an auxiliary probability space
Discussion wrt indirect inference (Gouriéroux 1993)• difficulty in indirect inference: which aux space chosen
here constructed empirically from distr of summary values
Tuesday, 7 January 14
ABC* indirect inference
⇡
true posterior
(✓|x) / `(x|✓) ⇡(✓)
/ `(s1:nkk
(x), k = 1, . . . ,K|✓) ⇡(✓)
= `(s1:nkk
(x), k = 1, . . . ,K|⇢) ⇡(⇢) |@L(✓)|
⇡
abc
(✓|x) / P
x
(ABC accept|⇢) ⇡(⇢) |@L(✓)|
using assumptions A1, A2:
Tuesday, 7 January 14
⇡
true posterior
(✓|x) / `(x|✓) ⇡(✓)
/ `(s1:nkk
(x), k = 1, . . . ,K|✓) ⇡(✓)
= `(s1:nkk
(x), k = 1, . . . ,K|⇢) ⇡(⇢) |@L(✓)|
⇡
abc
(✓|x) / P
x
(ABC accept|⇢) ⇡(⇢) |@L(✓)|
using assumptions A1, A2:
ABC* indirect inference
Tuesday, 7 January 14
ABC* approximation on -space is⇢
⇡
true posterior
(✓|x) / `(x|✓) ⇡(✓)
/ `(s1:nkk
(x), k = 1, . . . ,K|✓) ⇡(✓)
= `(s1:nkk
(x), k = 1, . . . ,K|⇢) ⇡(⇢) |@L(✓)|
⇡
abc
(✓|x) / P
x
(ABC accept|⇢) ⇡(⇢) |@L(✓)|
using assumptions A1, A2:
ABC* indirect inference
Tuesday, 7 January 14
ABC* approximation on -space is⇢
⇡
true posterior
(✓|x) / `(x|✓) ⇡(✓)
/ `(s1:nkk
(x), k = 1, . . . ,K|✓) ⇡(✓)
= `(s1:nkk
(x), k = 1, . . . ,K|⇢) ⇡(⇢) |@L(✓)|
⇡
abc
(✓|x) / P
x
(ABC accept|⇢) ⇡(⇢) |@L(✓)|
using assumptions A1, A2:match through calibrationof ABC tolerances and m
ABC* indirect inference
Tuesday, 7 January 14
ABC* approximation on -space is⇢
⇡
true posterior
(✓|x) / `(x|✓) ⇡(✓)
/ `(s1:nkk
(x), k = 1, . . . ,K|✓) ⇡(✓)
= `(s1:nkk
(x), k = 1, . . . ,K|⇢) ⇡(⇢) |@L(✓)|
⇡
abc
(✓|x) / P
x
(ABC accept|⇢) ⇡(⇢) |@L(✓)|
using assumptions A1, A2:match through calibrationof ABC tolerances and m
ABC* indirect inference
Regularity conditions on the link functionA3 the link function is bijective and continuously differentiable
Tuesday, 7 January 14
Example: moving average no sufficient statistics other than data, simple enough so that link function is analytically known
x
t
= u
t
+ au
t�1, u
t
⇠ N (0,�2)
✓ = (a,�2)
⌫1 = (1 + a
2)�2
⌫2 = a/(1 + a
2)
⇢1 = (1 + a
2)�2/⌫
x1
⇢2 = atanh(a/(1 + a
2))� atanh(⌫x2),
Tuesday, 7 January 14
Example: moving average no sufficient statistics other than data, simple enough so that link function is anlytically known
x
t
= u
t
+ au
t�1, u
t
⇠ N (0,�2)
✓ = (a,�2)
⌫1 = (1 + a
2)�2
⌫2 = a/(1 + a
2)
⇢1 = (1 + a
2)�2/⌫
x1
⇢2 = atanh(a/(1 + a
2))� atanh(⌫x2),
Tuesday, 7 January 14
Example: moving average no sufficient statistics other than data, simple enough so that link function is anlytically known
x
t
= u
t
+ au
t�1, u
t
⇠ N (0,�2)
✓ = (a,�2)
⌫1 = (1 + a
2)�2
⌫2 = a/(1 + a
2)
⇢1 = (1 + a
2)�2/⌫
x1
⇢2 = atanh(a/(1 + a
2))� atanh(⌫x2),
Tuesday, 7 January 14
Example: moving average no sufficient statistics other than data, simple enough so that link function is anlytically known
x
t
= u
t
+ au
t�1, u
t
⇠ N (0,�2)
✓ = (a,�2)
⌫1 = (1 + a
2)�2
⌫2 = a/(1 + a
2)
⇢1 = (1 + a
2)�2/⌫
x1
⇢2 = atanh(a/(1 + a
2))� atanh(⌫x2),
−0.4 −0.2 0.0 0.2 0.4
0.6
0.8
1.0
1.2
1.4
aσ
2
1
1
1.5
2
1 3
5
10
Testing only variance:link not bijective
exact posterior
−0.4 −0.2 0.0 0.2 0.4
0.6
0.8
1.0
1.2
1.4
a
σ2
1
3
5
1
3
5
10
Testing variance and autocorrelation with even values:summary values not sufficient
Tuesday, 7 January 14
Example: moving average no sufficient statistics other than data, simple enough so that link function is anlytically known
x
t
= u
t
+ au
t�1, u
t
⇠ N (0,�2)
✓ = (a,�2)
⌫1 = (1 + a
2)�2
⌫2 = a/(1 + a
2)
⇢1 = (1 + a
2)�2/⌫
x1
⇢2 = atanh(a/(1 + a
2))� atanh(⌫x2),
−0.4 −0.2 0.0 0.2 0.4
0.6
0.8
1.0
1.2
1.4
aσ
2
1
1
1.5
2
1 3
5
10
Testing only variance:link not bijective
exact posterior
−0.4 −0.2 0.0 0.2 0.4
0.6
0.8
1.0
1.2
1.4
a
σ2
1
3
5
1
3
5
10
Testing variance and autocorrelation with even values:summary values not sufficient
Tuesday, 7 January 14
Example: moving average no sufficient statistics other than data, simple enough so that link function is anlytically known
x
t
= u
t
+ au
t�1, u
t
⇠ N (0,�2)
✓ = (a,�2)
⌫1 = (1 + a
2)�2
⌫2 = a/(1 + a
2)
⇢1 = (1 + a
2)�2/⌫
x1
⇢2 = atanh(a/(1 + a
2))� atanh(⌫x2),
−0.4 −0.2 0.0 0.2 0.4
0.6
0.8
1.0
1.2
1.4
aσ
2
1
1
1.5
2
1 3
5
10
exact posterior
−0.4 −0.2 0.0 0.2 0.4
0.6
0.8
1.0
1.2
1.4
a
σ2
1
3
5
1
3
5
10
−0.4 −0.2 0.0 0.2 0.4
0.6
0.8
1.0
1.2
1.4
a
σ2
1
3
5
10
1
3
5
10
5 tests: link bijective and summary values sufficient
Tuesday, 7 January 14
Example: flu time series datastochastic transmission model, derived from ODEs
three parameters of interest: reproductive number R0, duration of immunity, reporting rate
6 sets of iid summary values, from 3 time series, subsetting odd and even values
Tuesday, 7 January 14
Example: flu time series datastochastic transmission model, derived from ODEs
three parameters of interest: reproductive number R0, duration of immunity, reporting rate
6 sets of iid summary values, from 3 time series, subsetting odd and even values
Tuesday, 7 January 14
Example: flu time series dataTest if linkbijective from ABC* output
previous standard MCMC ABC
MCMCABC* with calibrated tolerances
Tuesday, 7 January 14
Example: flu time series dataTest if linkbijective from ABC* output
previous standard MCMC ABC
MCMCABC* with calibrated tolerances
Tuesday, 7 January 14
Example: flu time series dataTest if linkbijective from ABC* output
previous standard MCMC ABC
MCMCABC* with calibrated tolerances
Tuesday, 7 January 14
Conclusions
using statistical decision theory in ABC,
• we can entirely avoid previous asymptotic arguments
• and construct accurate ABC algorithms by calibrating the decision tests appropriately
necessary to understand the distribution of the data on a summary levelidentifying replicate structures and modeling them is key in ABC as in any other approaches for which the likelihood is tractable
Tuesday, 7 January 14
Thank you
co-workers on this projectAnton 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)
acknowledgementsIoanna Manolopoulou (University College London)