ABC convergence under well- and mis-specified models

ABC: convergence and misspeciﬁcation
Christian P. Robert
Universit´e Paris-Dauphine PSL, Paris & University of Warwick,
Coventry
Joint work with D. Frazier, G. Martin, and J. Rousseau

Outline
motivatoy example
Approximate Bayesian computation
ABC for model choice
[some] asymptotics of ABC
ABC under misspeciﬁcation

A motivating if pedestrian example
paired and orphan socks
A drawer contains an unknown number of socks, some of which
can be paired and some of which are orphans (single). One takes
at random 11 socks without replacement from this drawer: no pair
can be found among those. What can we infer about the total
number of socks in the drawer?
sounds like an impossible task
one observation x = 11 and two unknowns, nsocks and npairs
writing the likelihood is a challenge [exercise]

Feller’s shoes
A closet contains n pairs of shoes. If 2r shoes are chosen
at random (with 2r < n), what is the probability that
there will be (a) no complete pair, (b) exactly one
complete pair, (c) exactly two complete pairs among
them?
[Feller, 1970, Chapter II, Exercise 26]

Feller’s shoes
them?
Resolution as
pj =
n
j
22r−2j n − j
2r − 2j
2n
2r
being probability of obtaining js pairs among those 2r shoes, or for
an odd number t of shoes
pj = 2t−2j n
j
n − j
t − 2j
2n
t

Feller’s shoes
them?
If one draws 11 socks out of m socks made of f orphans and g
pairs, with f + 2g = m, number k of socks from the orphan group
is hypergeometric H(11, m, f ) and probability to observe 11
orphan socks total is
11
k=0
f
k
2g
11−k
m
11
×
211−k g
11−k
2g
11−k

A prioris on socks
Given parameters nsocks and npairs, set of socks
S = s1, s1, . . . , snpairs , snpairs , snpairs+1, . . . , snsocks
and 11 socks picked at random from S give X unique socks.
Rassmus’ reasoning
If you are a family of 3-4 persons then a guesstimate would be that
you have something like 15 pairs of socks in store. It is also
possible that you have much more than 30 socks. So as a prior for
nsocks I’m going to use a negative binomial with mean 30 and
standard deviation 15.
On npairs/2nsocks I’m going to put a Beta prior distribution that puts
most of the probability over the range 0.75 to 1.0,
[Rassmus B˚a˚ath’s Research Blog, Oct 20th, 2014]

Simulating the experiment
Given a prior distribution on nsocks and npairs,
nsocks ∼ Neg(30, 15) npairs|nsocks ∼ nsocks/2Be(15, 2)
possible to
1. generate new values
of nsocks and npairs,
2. generate a new
observation of X,
number of unique
socks out of 11.
3. accept the pair
(nsocks, npairs) if the
realisation of X is
equal to 11

Meaning
ns
Density
0 10 20 30 40 50 60
0.000.010.020.030.040.050.06
The outcome of this simulation method returns a distribution on
the pair (nsocks, npairs) that is the conditional distribution of the
pair given the observation X = 11
Proof: Generations from π(nsocks, npairs) are accepted with probability
P {X = 11|(nsocks, npairs)}

Meaning
ns
Density
0 10 20 30 40 50 60
0.000.010.020.030.040.050.06
The outcome of this simulation method returns a distribution on
the pair (nsocks, npairs) that is the conditional distribution of the
pair given the observation X = 11
Proof: Hence accepted values distributed from
π(nsocks, npairs) × P {X = 11|(nsocks, npairs)} ∝ π(nsocks, npairs|X = 11)

motivatoy example
ABC basics
Automated summary selection

Untractable likelihoods
Cases when the likelihood function
f (y|θ) is unavailable and when the
completion step
f (y|θ) =
Z
f (y, z|θ) dz
is impossible or too costly because of
the dimension of z
c MCMC cannot be implemented

The ABC method
Bayesian setting: target is π(θ)f (x|θ)
When likelihood f (x|θ) not in closed form, likelihood-free rejection
technique:
ABC algorithm
For an observation y ∼ f (y|θ), under the prior π(θ), keep jointly
simulating
θ ∼ π(θ) , z ∼ f (z|θ ) ,
until the auxiliary variable z is equal to the observed value, z = y.
[Tavar´e et al., 1997]

A as A...pproximative
When y is a continuous random variable, equality z = y is
replaced with a tolerance condition,
ρ(y, z)
where ρ is a distance
Output distributed from
π(θ) Pθ{ρ(y, z) < } ∝ π(θ|ρ(y, z) < )
[Pritchard et al., 1999]

ABC algorithm
Algorithm 1 Likelihood-free rejection sampler 2
for i = 1 to N do
repeat
generate θ from the prior distribution π(·)
generate z from the likelihood f (·|θ )
until ρ{η(z), η(y)}
set θi = θ
end for
where η(y) deﬁnes a (not necessarily suﬃcient) statistic

Output
The likelihood-free algorithm samples from the marginal in z of:
π (θ, z|y) =
π(θ)f (z|θ)IA ,y (z)
A ,y×Θ π(θ)f (z|θ)dzdθ
,
where A ,y = {z ∈ D|ρ(η(z), η(y)) < }.
The idea behind ABC is that the summary statistics coupled with a
small tolerance should provide a good approximation of the
posterior distribution:
π (θ|y) = π (θ, z|y)dz ≈ π(θ|η(y)) .

Dogger Bank re-enactment
Battle of Dogger Bank on Jan 24, 1915, between British and
German ﬂeets : how likely was the British victory?
[MacKay, Price, and Wood, 2016]

Dogger Bank re-enactment
Battle of Dogger Bank on Jan 24, 1915, between British and
German ﬂeets : ABC simulation of posterior distribution
[MacKay, Price, and Wood, 2016]

ABC advances
Simulating from the prior is often poor in eﬃciency
Either modify the proposal distribution on θ to increase the density
of x’s within the vicinity of y...
[Marjoram et al, 2003; Bortot et al., 2007, Beaumont et al., 2009]
...or by viewing the problem as a conditional density estimation
and by developing techniques to allow for larger
[Beaumont et al., 2002; Blum & Fran¸cois, 2009]
.....or even by including in the inferential framework [ABCµ]
[Ratmann et al., 2009]

ABC consistency
Recent studies on large sample properties of ABC posterior
distributions and ABC posterior means
[Li & Fearnhead, 2018a, b; Frazier et al., 2018]
Under regularity conditions on summary statistics,
incl. convergence at speed dT , characterisation of rate of posterior
concentration as a function of tolerance convergence
less stringent condition on tolerance decrease than for
asymptotic normality of posterior;
asymptotic normality of posterior mean does not require
asymptotic normality of posterior itself
Cases for limiting ABC distributions
1. dT T −→ +∞;
2. dT T −→ c;
3. dT T −→ 0
and limiting ABC mean convergent for 2
T = o(1/dT )
[Frazier et al., 2018]

Noisily exact ABC
Idea: Modify the data from the start
˜y = y0 + ζ1
with the same scale as ABC
[ see Fearnhead-Prangle ]
run ABC on ˜y
Then ABC produces an exact simulation from π(θ|˜y) = π(θ|˜y)
[Dean et al., 2011; Fearnhead and Prangle, 2012]

Consistent noisy ABC
Degrading the data improves the estimation performances:
Noisy ABC-MLE is asymptotically (in n) consistent
under further assumptions, the noisy ABC-MLE is
asymptotically normal
increase in variance of order −2
likely degradation in precision or computing time due to the
lack of summary statistic [curse of dimensionality]

Semi-automatic ABC
Fearnhead and Prangle (2012) study ABC and the selection of the
summary statistic
ABC then considered from a purely inferential viewpoint and
calibrated for estimation purposes
Use of a randomised (or ‘noisy’) version of the summary statistics
˜η(y) = η(y) + τ
Derivation of a well-calibrated version of ABC, i.e. an algorithm
that gives proper predictions for the distribution associated with
this randomised summary statistic [calibration constraint: ABC
approximation with same posterior mean as the true randomised
posterior]
Optimality of the posterior expectation E[θ|y] of the parameter of
interest as summary statistics η(y)!

Fully automatic ABC
Implementation of ABC still requires input of collection of
summaries
Towards automation
statistical projection techniques (LDA, PCA, NP-GLS, &tc.)
variable selection
machine learning approaches
bypassing summaries altogether

ABC with Wasserstein distance
Use as distance between simulated and observed samples the
Wasserstein distance:
Wp(y1:n, z1:n)p
= inf
σ∈Sn
1
n
n
i=1
ρ(yi , zσ(i))p
, (1)
covers well- and mis-speciﬁed cases
only depends on data space distance ρ(·, ·)
covers iid and dynamic models (curve matching)
computional feasible (linear in dimension, cubic in sample size)
Hilbert curve approximation
[Bernton et al., 2017]

Consistent inference with Wasserstein distance
As ε → 0 [and n ﬁxed]
If either
1. f
(n)
θ is n-exchangeable and D(y1:n, z1:n) = 0 if and only if
z1:n = yσ(1:n) for some σ ∈ Sn, or
2. D(y1:n, z1:n) = 0 if and only if z1:n = y1:n.
then, at y1:n ﬁxed, ABC posterior converges strongly to posterior
as ε → 0.

As n → ∞ [at ε ﬁxed]
WABC distribution with a ﬁxed ε does not converge in n to a
Dirac mass

As εn → 0 and n → ∞
Under range of assumptions, if fn(εn) → 0, and
P(W(^µn, µ ) εn) → 1, then WABC posterior with threshold
εn + ε satisﬁes
πεn+ε
{θ ∈ H : W(µ , µθ) > ε + 4εn/3 + f −1
n (εL
n/R)} |y1:n
P
δ

A bivariate Gaussian illustration
100 observations from bivariate Normal with variance 1 and
covariance 0.55
Compare WABC with ABC based on raw Euclidean distance and
Euclidean distance between sample means on 106 model
simulations.

motivatoy example

Bayesian model choice
Several models M1, M2, . . . are considered simultaneously for a
dataset y and the model index M is part of the inference.
Use of a prior distribution. π(M = m), plus a prior distribution on
the parameter conditional on the value m of the model index,
πm(θm)
Goal is to derive the posterior distribution of M, challenging
computational target when models are complex.

Generic ABC for model choice
Algorithm 2 Likelihood-free model choice sampler (ABC-MC)
for t = 1 to T do
repeat
Generate m from the prior π(M = m)
Generate θm from the prior πm(θm)
Generate z from the model fm(z|θm)
until ρ{η(z), η(y)} <
Set m(t) = m and θ(t)
= θm
end for
[Cornuet et al., DIYABC, 2009]

ABC estimates
Posterior probability π(M = m|y) approximated by the frequency
of acceptances from model m
1
T
T
t=1
Im(t)=m .

Limiting behaviour of B12 (under sufficiency)
If η(y) sufficient statistic for both models,
fi (y|θi ) = gi (y)f η
i (η(y)|θi )
Thus
B12(y) =
Θ1
π(θ1)g1(y)f η
1 (η(y)|θ1) dθ1
Θ2
π(θ2)g2(y)f η
2 (η(y)|θ2) dθ2
=
g1(y) π1(θ1)f η
1 (η(y)|θ1) dθ1
g2(y) π2(θ2)f η
2 (η(y)|θ2) dθ2
=
g1(y)
g2(y)
Bη
12(y) .
[Didelot, Everitt, Johansen & Lawson, 2011]
c No discrepancy only when cross-model sufficiency
c Inability to evaluate loss brought by summary statistics

A stylised problem
Central question to the validation of ABC for model choice:
When is a Bayes factor based on an insuﬃcient statistic
T(y) consistent?
Note/warnin: c drawn on T(y) through BT
12(y) necessarily diﬀers
from c drawn on y through B12(y)
[Marin, Pillai, X, & Rousseau, JRSS B, 2013]

A benchmark if toy example
Comparison suggested by referee of PNAS paper [thanks!]:
[X, Cornuet, Marin, & Pillai, Aug. 2011]
Model M1: y ∼ N(θ1, 1) opposed
to model M2: y ∼ L(θ2, 1/
√
2), Laplace distribution with mean θ2
and scale parameter 1/
√
2 (variance one).
Four possible statistics
1. sample mean y (suﬃcient for M1 if not M2);
2. sample median med(y) (insuﬃcient);
3. sample variance var(y) (ancillary);
4. median absolute deviation mad(y) = med(|y − med(y)|);

A benchmark if toy example
Comparison suggested by referee of PNAS paper [thanks!]:
[X, Cornuet, Marin, & Pillai, Aug. 2011]
Model M1: y ∼ N(θ1, 1) opposed
to model M2: y ∼ L(θ2, 1/
√
2), Laplace distribution with mean θ2
and scale parameter 1/
√
2 (variance one).
q
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Gauss Laplace
0.00.10.20.30.40.50.60.7
n=100
q
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Gauss Laplace
0.00.20.40.60.81.0
n=100

Framework
Starting from sample
y = (y1, . . . , yn)
the observed sample, not necessarily iid with true distribution
y ∼ Pn
Summary statistics
T(y) = Tn
= (T1(y), T2(y), · · · , Td (y)) ∈ Rd
with true distribution Tn
∼ Gn.

Assumptions
A collection of asymptotic “standard” assumptions:
[A1] is a standard central limit theorem under the true model with
asymptotic mean µ0
[A2] controls the large deviations of the estimator Tn
from the
model mean µ(θ)
[A3] is the standard prior mass condition found in Bayesian
asymptotics (di eﬀective dimension of the parameter)
[A4] restricts the behaviour of the model density against the true
density
[Think CLT!]

Asymptotic marginals
Asymptotically, under [A1]–[A4]
mi (t) =
Θi
gi (t|θi ) πi (θi ) dθi
is such that
(i) if inf{|µi (θi ) − µ0|; θi ∈ Θi } = 0,
Cl vd−di
n mi (Tn
) Cuvd−di
n
and
(ii) if inf{|µi (θi ) − µ0|; θi ∈ Θi } > 0
mi (Tn
) = oPn [vd−τi
n + vd−αi
n ].

Between-model consistency
Consequence of above is that asymptotic behaviour of the Bayes
factor is driven by the asymptotic mean value µ(θ) of Tn
under
both models. And only by this mean value!

under
Indeed, if
inf{|µ0 − µ2(θ2)|; θ2 ∈ Θ2} = inf{|µ0 − µ1(θ1)|; θ1 ∈ Θ1} = 0
then
Cl v
−(d1−d2)
n m1(Tn
) m2(Tn
) Cuv
−(d1−d2)
n ,
where Cl , Cu = OPn (1), irrespective of the true model.
c Only depends on the diﬀerence d1 − d2: no consistency

under
Else, if
inf{|µ0 − µ2(θ2)|; θ2 ∈ Θ2} > inf{|µ0 − µ1(θ1)|; θ1 ∈ Θ1} = 0
then
m1(Tn
)
m2(Tn
)
Cu min v
−(d1−α2)
n , v
−(d1−τ2)
n

Checking for adequate statistics
Run a practical check of the relevance (or non-relevance) of Tn
null hypothesis that both models are compatible with the statistic
Tn
H0 : inf{|µ2(θ2) − µ0|; θ2 ∈ Θ2} = 0
against
H1 : inf{|µ2(θ2) − µ0|; θ2 ∈ Θ2} > 0
testing procedure provides estimates of mean of Tn
under each
model and checks for equality

Checking in practice
Under each model Mi , generate ABC sample θi,l , l = 1, · · · , L
For each θi,l , generate yi,l ∼ Fi,n(·|ψi,l ), derive Tn
(yi,l ) and
compute
^µi =
1
L
L
l=1
Tn
(yi,l ), i = 1, 2 .
Conditionally on Tn
(y),
√
L { ^µi − Eπ
[µi (θi )|Tn
(y)]} N(0, Vi ),
Test for a common mean
H0 : ^µ1 ∼ N(µ0, V1) , ^µ2 ∼ N(µ0, V2)
against the alternative of diﬀerent means
H1 : ^µi ∼ N(µi , Vi ), with µ1 = µ2 .

Toy example: Laplace versus Gauss
qqqqqqqqqqqqqqq
qqqqqqqqqq
q
qq
q
q
Gauss Laplace Gauss Laplace
010203040
Normalised χ2 without and with mad

motivatoy example
asymptotic setup
consistency of ABC posteriors
asymptotic posterior shape
asymptotic behaviour of EABC [θ]

asymptotic setup
asymptotic: y = y(n) ∼ Pn
θ and = n, n → +∞
parametric: θ ∈ Rk, k ﬁxed
concentration of summary statistics η(zn):
∃b : θ → b(θ) η(zn
) − b(θ) = oP
θ
(1), ∀θ
Objects of interest:
posterior concentration and asymptotic shape of π (·|η(y(n)))
(normality?)
convergence of the posterior mean ^θ = EABC[θ|η(y(n))]
asymptotic acceptance rate
[Frazier et al., 2018]

ABC algorithm Bayesian consistent at θ0 if for any δ > 0,
Π ( θ − θ0 > δ| η(y) − η(z) ε) → 0
as n → +∞, ε → 0
Bayesian consistency implies that sets containing θ0 have posterior
probability tending to one as n → +∞, with implication being the
existence of a speciﬁc rate of concentration

ABC algorithm Bayesian consistent at θ0 if for any δ > 0,
Π ( θ − θ0 > δ| η(y) − η(z) ε) → 0
as n → +∞, ε → 0
Concentration around true value and Bayesian consistency
impose less stringent conditions on the convergence speed of
tolerance n to zero, when compared with asymptotic
normality of ABC posterior
asymptotic normality of ABC posterior mean does not require
asymptotic normality of ABC posterior

Concentration of summary η(z): there exists b(θ) such that
η(z) − b(θ) = oP
θ
(1)
Consistency:
Π n ( θ − θ0 δ|η(y)) = 1 + op(1)
Convergence rate: there exists δn = o(1) such that
Π n ( θ − θ0 δn|η(y)) = 1 + op(1)

Rate of convergence
Π (·| η(y) − η(z) ε) concentrates at rate λn → 0 if
lim sup
ε→0
lim sup
n→+∞
Π ( θ − θ0 > λnM| η(y)η(z) ε) → 0
in P0-probability when M goes to inﬁnity.
Posterior rate of concentration related to rate at which information
accumulates about true parameter vector

Related results
existing studies on the large sample properties of ABC, in which
the asymptotic properties of point estimators derived from ABC
have been the primary focus
[Creel et al., 2015; Jasra, 2015; Li & Fearnhead, 2018a]

Related results
assumption of CLT for summary statistic plus regularity
assumptions on convergence of its density of the summary
statistics to normal limit (incl. existence of Edgeworth
expansion with exponential tail control)
convergence rate of posterior mean if εT = o(1/v
3/5
T )
acceptance probability chosen as arbitrary density
η(y) − η(z)

Related results
stronger conditions on η than our own weak convergence of
vT {η(z) − b(θ)}, with non-uniform polynomial deviations
excludes non compact parameter space
when εT v−1
T , only deviation bounds and weak convergence
are required (weaker than convergence of densities)
when εT = o(1/vT ) no requirement on convergence rate

Related results
characterisation of asymptotic behavior of ABC posterior for
all εT = o(1) with posterior concentration
asymptotic normality and unbiasedness of posterior mean
remain achievable even when limT vT εT = ∞, provided
εT = o(1/v
1/2
T )

Related results
Li and Fearnhead (2018a) point out that if kη > kθ 1 and if
εT = o(1/v
3/5
T ), posterior mean asymptotically normal, unbiased,
but not asymptotically eﬃcient
if vT εT → ∞ and vT ε2
T = o(1), for kη > kθ 1,
EΠε {vT (θ−θ0)} = { θb(θ0) θb(θ0)}−1
θb(θ0) vT {η(y)−b(θ0)}+op(1)

Convergence when n σn
Under (main) assumptions
(A1) ∃σn → 0
Pθ σ−1
n η(z) − b(θ) > u c(θ)h(u), lim
u→+∞
h(u) = 0
(A2)
Π( b(θ) − b(θ0) u) uD
, u ≈ 0
posterior consistency
posterior concentration rate λn that depends on the deviation
control of d2{η(z), b(θ)}
posterior concentration rate for b(θ) bounded from below by O( n)

Convergence when n σn
Under (main) assumptions
(A1) ∃σn → 0
Pθ σ−1
n η(z) − b(θ) > u c(θ)h(u), lim
u→+∞
h(u) = 0
(A2)
Π( b(θ) − b(θ0) u) uD
, u ≈ 0
then
Π n b(θ) − b(θ0) n + σnh−1
( D
n )|η(y) = 1 + op0 (1)
If also θ − θ0 L b(θ) − c(θ0) α, locally and θ → b(θ) 1-1
Π n ( θ − θ0
α
n + σα
n (h−1
( D
n ))α
δn
|η(y)) = 1 + op0 (1)

Comments
(A1) : if Pθ σ−1
n η(z) − b(θ) > u c(θ)h(u), two cases
1. Polynomial tail: h(u) u−κ
, then δn = n + σn
−D/κ
n
2. Exponential tail: h(u) e−cu
, then δn = n + σn log(1/ n)
E.g., η(y) = n−1
i g(yi ) with moments on g (case 1) or
Laplace transform (case 2)

Comments
(A2) : Π( b(θ) − b(θ0) u) uD : If Π regular enough then
D = dim(θ)
no need to approximate the density f (η(y)|θ).
Same results holds when n = o(σn) if (A1) replaced with
inf
|x| M
Pθ σ−1
n (η(z) − b(θ)) − x u uD
, u ≈ 0

proof
Simple enough proof: assume σn δ n and
η(y) − b(θ0) σn, η(y) − η(z) n
Hence
b(θ) − b(θ0) > δn ⇒ η(z) − b(θ) > δn − n − σn := tn
Also, if b(θ) − b(θ0) n/3
η(y) − η(z) η(z) − b(θ) + σn
n/3
+ n/3
and
Π n ( b(θ) − b(θ0) > δn|y)
b(θ)−b(θ0) >δn
Pθ ( η(z) − b(θ) > tn) dΠ(θ)
|b(θ)−b(θ0)| n/3
Pθ ( η(z) − b(θ) n/3) dΠ(θ)
−D
n h(tnσ−1
n )
Θ
c(θ)dΠ(θ)

Assumptions
Applicable to broad range of data structures
[A1] ensures that η(z) concentrates on b(θ), unescapable
[A2] controls degree of prior mass in a neighbourhood of θ0,
standard in Bayesian asymptotics
[A2] If Π absolutely continuous with prior density p bounded,
above and below, near θ0, then D = dim(θ) = kθ
[A3] identiﬁcation condition critical for getting posterior
concentration around θ0, b being injective depending on true
structural model and particular choice of η.

Summary statistic and (in)consistency
Consider the moving average MA(2) model
yt = et + θ1et−1 + θ2et−2, et ∼i.i.d. N(0, 1)
and
−2 θ1 2, θ1 + θ2 −1, θ1 − θ2 1.
summary statistics equal to sample autocovariances
ηj (y) = T−1
T
t=1+j
yt yt−j j = 0, 1
with
η0(y)
P
→ E[y2
t ] = 1 + (θ01)2
+ (θ02)2
and η1(y)
P
→ E[yt yt−1] = θ01(1 + θ02)
For ABC target pε (θ|η(y)) to be degenerate at θ0
0 = b(θ0) − b (θ) =
1 + (θ01)2
+ (θ02)2
θ01(1 + θ02)
−
1 + (θ1)2
+ (θ2)2
θ1(1 + θ2)
must have unique solution θ = θ0
Take θ01 = .6, θ02 = .2: equation has two solutions
θ1 = .6, θ2 = .2 and θ1 ≈ .5453, θ2 ≈ .3204

Concentration for the MA(2) model
True value θ0 = (0.6, 0.2)
Summaries ﬁrst three autocorrelations
Tolerance proportional to εT = 1/T0.4
Rejection of normality of these posteriors

Asymptotic shape of posterior distribution
Shape of
Π (·| η(y), η(z) εn)
for several connections between εn and rate at which η(yn) satisfy
CLT
Three diﬀerent regimes:
1. σn = o( n) −→ Uniform limit
2. σn n −→ perturbated Gaussian limit
3. σn n −→ Gaussian limit

scaling matrices
Introduction of sequence of (k, k) p.d. matrices Σn(θ) such that
for all θ near θ0
c1 Dn ∗ Σn(θ) ∗ c2 Dn ∗, Dn = diag(dn(1), · · · , dn(k)),
with 0 < c1, c2 < +∞, dn(j) → +∞ for all j’s
Possibly diﬀerent convergence rates for components of η(z)
Reordering components so that
dn(1) · · · dn(k)
with assumption that
lim inf
n
dn(j)εn = lim sup
n
dn(j)εn

New assumptions
(B1) Concentration of summary η: Σn(θ) ∈ Rk1×k1 is o(1)
Σn(θ)−1
{η(z)−b(θ)} ⇒ Nk1 (0, Id), (Σn(θ)Σn(θ0)−1
)n = Co
(B2) b(θ) is C1 and
θ − θ0 b(θ) − b(θ0)
(B3) Dominated convergence and
lim
n
Pθ(Σn(θ)−1{η(z) − b(θ)} ∈ u + B(0, un))
j un(j)
= ϕ(u)

main result
Set Σn(θ) = σnD(θ) for θ ≈ θ0 and
Zo = Σn(θ0)−1(η(y) − b(θ0)), then under (B1) and (B2)
when nσ−1
n → +∞
Π n [ −1
n (θ−θ0) ∈ A|y] ⇒ UB0 (A), B0 = {x ∈ Rk
; b (θ0)T
x 1

main result
when nσ−1
n → c
Π n [Σn(θ0)−1
(θ − θ0) − Zo
∈ A|y] ⇒ Qc(A), Qc = N

main result
when nσ−1
n → 0 and (B3) holds, set
Vn = [b (θ0)]n
Σn(θ0)b (θ0)
then
Π n [V −1
n (θ − θ0) − ˜Zo
∈ A|y] ⇒ Φ(A),

intuition (?!)
Set x(θ) = σ−1
n (θ − θ0) − Zo (k = 1)
πn := Π n [ −1
n (θ − θ0) ∈ A|y]
=
|θ−θ0| un
Ix(θ)∈A
Pθ ( σ−1
n (η(z) − b(θ)) + x(θ) σ−1
n n)p(θ)dθ
|θ−θ0| un
Pθ ( σ−1
n (η(z) − b(θ)) + x(θ) σ−1
n n)p(θ)dθ
+ op(1)
If n/σn 1 :
Pθ σ−1
n (η(z) − b(θ)) + x(θ) σ−1
n n = 1+o(1), iﬀ x σ−1
n n+o(1)
If n/σn = o(1)
Pθ σ−1
n (η(z) − b(θ)) + x σ−1
n n = φ(x)σn(1 + o(1))

more comments
Surprising : U(− n, n) limit when n σn but not that
surprising since n = o(1) means concentration around θ0
and σn = o( n) implies that b(θ) − b(θ0) ≈ η(z) − η(y)
again, no need to control approximation of f (η(y)|θ) by a
Gaussian density: merely a control of the distribution
generalisation to the case where eigenvalues of Σn are
dn,1 = · · · = dn,k
behaviour of EABC (θ|y) consistent with Li & Fearnhead
(2016)

even more comments
If (also) p(θ) is Hölder β
EABC (θ|y) − θ0 = σn
Zo
b(θ0)
score for f (η(y)|θ)
+
β/2
j=1
2j
n Hj (θ0, p, b)
bias from threshold approx
+o(σn) + O( β+1
n )
with
if 2
n = o(σn) : Efficiency
EABC (θ|y) − θ0 = σn
Zo
b(θ0)
+ o(σn)
the Hj (θ0, p, b)’s are deterministic
c we gain nothing by getting a first crude ^θ(y) = EABC (θ|y)
for some η(y) and then rerun ABC with ^θ(y)

Illustration in the MA(2) setting
Sample sizes of T = 500, 1000
Asymptotic normality rejected for εT = 1/T0.4 and for θ1,
T = 500 and εT = 1/T0.55

asymptotic behaviour of EABC [θ]
When p = dim(η(y)) = d = dim(θ) and n = o(n−3/10)
EABC [dT (θ − θ0)|yo
] ⇒ N(0, ( bo
)T
Σ−1
bo −1
[Li & Fearnhead (2018a)]
In fact, if β+1
n
√
n = o(1), with β H¨older-smoothness of π
EABC [(θ−θ0)|yo
] =
( bo)−1Zo
√
n
+
k
j=1
hj (θ0) 2j
n +op(1), 2k = β
Iterating for ﬁxed p mildly interesting: if
˜η(y) = EABC [θ|yo
]
then
EABC [θ|˜η(y)] = θ0 +
( bo)−1Zo
√
n
+
π (θ0)
π(θ0)
2
n + o()
[Fearnhead & Prangle, 2012]

more asymptotic behaviour of EABC [θ]
Li and Fearnhead (2018a,b) consider that
EABC [dT (θ − θ0)|yo
]
not optimal when p > d
If
√
n 2
n = o(1) and n
√
n = o(1)
√
n[EABC (θ) − θ0] = P bo Zo
+ op(1)
Zo
=
√
n(η(y) − bo
)
P bo Zo
= (( bo
)T
bo
)−1
( bo
)T
Zo
and Vas(P bo Zo
) ( bo
)T
Vas(Zo
)−1
( bo
)
−1
If n
√
n = o(1)
√
n[EABC (θ)−θ0] = ( bo
)T
Σ−1
bo −1
( bo
)T
Σ−1
Zo
+op(1)

impact of the dimension of η
dimension of η(.) does not impact above result, but impacts
acceptance probability
if n = o(σn), k1 = dim(η(y)), k = dim(θ) & k1 k
αn := Pr ( y − z n) k1
n σ−k1+k
n
if n σn
αn := Pr ( y − z n) k
n
If we choose αn
αn = o(σk
n) leads to n = σn(αnσ−k
n )1/k1
= o(σn)
αn σn leads to n α
1/k
n .

Illustration in the MA(2) setting
Sample sizes of T = 500, 1000
Asymptotic normality accepted for all graphs

Practical implications
In practice, tolerance determined by quantile (nearest neighbours):
Select all θi associated with the α = δ/N smallest distances
d2{η(zi ), η(y)} for some δ
Then (i) if εT v−1
T or εT = o(v−1
T ), acceptance rate associated
with the threshold εT is
αT = pr ( η(z) − η(y) εT ) (vT εT )kη
× v−kθ
T v−kθ
T
(ii) if εT v−1
T ,
αT = pr ( η(z) − η(y) εT ) εkθ
T v−kθ
T

Curse of dimensionality
For reasonable statistical behavior, rate of decline of αT the faster
the larger the dimension of θ, kθ, but unaﬀected by dimension of
η, kη
Theoretical justiﬁcation for dimension reduction methods that
process parameter components individually and independently of
other components
[Beaumont & al., 2002; Fearnhead & Prangle, 2012; Martin& al., 2016]
importance sampling approach of Li & Fearnhead (2018a) yields
acceptance rates αT = O(1), when εT = O(1/vT )

Monte Carlo error
Link the choice of εT to Monte Carlo error associated with NT
draws in Algorithm
Conditions (on εT ) under which
^αT = αT {1 + op(1)}
where ^αT = NT
i=1 1l [d{η(y), η(z)} εT ] /NT proportion of
accepted draws from NT simulated draws of θ
Either
(i) εT = o(v−1
T ) and (vT εT )−kη ε−kθ
T MNT
or
(ii) εT v−1
T and ε−kθ
T MNT
for M large enough;

conclusion on ABC consistency
asymptotic description of ABC: diﬀerent regimes depending
on n & σn
no point in choosing n arbitrarily small: just n = o(σn)
no asymptotic gain in iterative ABC
results under weak(er) conditions by not studying g(η(z)|θ)

Mis-ABC-ﬁed
motivatoy example
Misspeciﬁcation
Consequences

Illustration
Assumed data generating process (DGP) is z ∼ N(θ, 1) but actual
DGP is y ∼ N(θ, ˜σ2)
Use of summaries
sample mean η1(y) = 1
n
n
i=1 yi
centered summary η2(y) = 1
n−1
n
i=1(yi − η1(y))2 − 1
Three ABC:
ABC-AR: accept/reject approach with
K (d{η(z), η(y)}) = 1l [d{η(z), η(y)} ] and
d{x, y} = x − y
ABC-K: smooth rejection approach, with K (d{η(z), η(y)})
univariate Gaussian kernel
ABC-Reg: post-processing ABC approach with weighted linear
regression adjustment

Illustration
Posterior means for ABC-AR, ABC-K and ABC-Reg as
misspeciﬁcation σ2 increases (N = 50, 000 simulated data sets)
αn = n−5/9 quantile for ABC-AR
ABC-K and ABC-Reg bandwidth of n−5/9

Framework
Data y with true distribution P0
Model P := {θ ∈ Θ ⊂ Rkθ : Pθ}
Summary statistic η(y) = (η1(y), ..., ηkη (y))
Misspeciﬁcation
inf
θ∈Θ
D(P0||Pθ) = inf
θ∈Θ
− log
dP0(y)
dPθ(y)
dP0(y) > 0,
with
θ∗
= arg inf
θ∈Θ
D(P0||Pθ)
[Muller, 2013]
ABC misspeciﬁcation
for b0 (resp. b(θ)) limit of η(y) (resp. η(z))
inf
θ∈Θ
d{b0, b(θ)} > 0
ABC pseudo-true value
θ∗
= arg inf
θ∈Θ
d{b0, b(θ)}.

Compulsory tolerance
Under standard identiﬁcation conditions on b(·) ∈ Rkη , there exists
∗ such that
∗
= inf
θ∈Θ
d{b0, b(θ)} > 0
c For tolerances n = o(1), once n < ∗ no draw of θ to be
selected and posterior Π [A|η(y)] ill-conditioned
But appropriately chosen tolerance sequence ( n)n allows
ABC-based posterior to concentrate on distance-dependent
pseudo-true value θ∗

ABC Posterior Concentration Under Misspeciﬁcation
Assumptions
[A0] There exist continuous b : Θ → B and decreasing ρn(·) such
that ρn(u) → 0 at inﬁnity and
Pθ [d{η(θ), b(θ)} > u] c(θ)ρn(u),
Θ
c(θ)dΠ(θ) < +∞
with existence of either
(i) Polynomial deviations: positive sequence vn → +∞ and
u0, κ > 0 such that ρn(u) = v−κ
n u−κ
, for u u0.
(ii) Exponential deviations: hθ(·) > 0 such that
Pθ[d{η(z), b(θ)} > u] c(θ)e−hθ(uvn)
and there exist c, C > 0
such that
Θ
c(θ)e−hθ(uvn)
dΠ(θ) Ce−c(uvn)τ

Assumptions
[A1] There exist D > 0 and M0, δ0 > 0 such that, for all
δ0 δ > 0 and M M0, there exists
Sδ ⊂ {θ ∈ Θ : d{b(θ), b0} − ∗ δ} for which
In case (i), D < κ and
Sδ
1 −
c(θ)
M
dΠ(θ) δD
.
In case (ii),
Sδ
1 − c(θ)e−hθ(M)
dΠ(θ) δD
.

Under [A0] and [A1], for n ↓ ∗ with
n
∗
+ Mv−1
n + v−1
0,n
if Mn sequence going to inﬁnity and
δn Mn{( n − ∗) + ρn( n − ∗)}, then
Π [d{b(θ), b0} ∗
+ δn|η(y)] = oP0 (1), (2)
provided
ρn( n − ∗
) ( n − ∗
)−D/κ
in case (i)
ρn( n − ∗
) | log( n − ∗
)|1/τ
in case (ii)
Further
Π [d{θ, θ∗
} > δ|η(y)] = oP0 (1)
[Berton & al., 2017; Frazier & al., 2017]

ABC Asymptotic Posterior
Under further assumptions
existence of positive deﬁnite matrix Σn(θ0) such that for all
θ − θ0 δ and all 0 < u δvn
Pθ [ Σn(θ0){η(z) − b(θ)} > u] c0u−κ
parametric CLT: existence of sequence of positive deﬁnite
matrices Σn(θ) such that for all θ in a neighbourhood of θ0
Σn(θ) ∗ vn, with vn → +∞ and
Σn(θ){η(z) − b(θ)} ⇒ N(0, Ikη ),
classical concentration assumptions
If limn vn( n − ∗) = 0, then for Z0
n = Σn(θ∗){η(y) − b0} and
Φkη (·) standard Normal
lim
n→+∞
Π Σn(θ∗
){b(θ) − b(θ∗
)} − Z0
n ∈ B|η(y) = Φkη (B)

Accept/Reject ABC revisited
Given link between n and αn in Frazier & al. (2018), posterior
concentration achieved through sequence of αn converging to zero
ABC algorithm based on appropriate αn always yields (under
appropriate regularity) approach that concentrates asymptotically,
regardless of misspeciﬁcation.
(i) if ( n − ∗) v−1
n or ( n − ∗) = o(v−1
n ), then
αn = Pr ( η(z) − η(y) n) (vn{ n − ∗
})kη
× v−kθ
n v−kθ
n
(ii) if ( n − ∗) v−1
n , then
αn = Pr ( η(z) − η(y) n) { n − ∗
}kθ
v−kθ
n

Regression adjustment
In case of scalar θ, ABC-Reg ﬁrst runs ABC-AR, with tolerance n,
and obtains selected draws and summaries {θi , η(zi )}
ABC-Reg second uses linear regression to predict accepted values
of θ from η(z) through
θi
= α + β {η(y) − η(zi
)} + νi
ABC-Reg deﬁnes adjusted parameter draw as
˜θi
= θi
+ ^β {η(y) − η(zi
)}
where
^β =
1
N
N
i=1
η(zi
) − ¯η η(zi
) − ¯η
−1
1
N
N
i=1
η(zi
) − ¯η θi
− ¯θ =

ABC-Reg pseudo-consistency
If
n
∗
+ Mv−1
n + v−1
0,n ,
with
(i) ∗ = infθ∈Θ d{b(θ), b0} > 0
(ii) θ∗ = arg infθ∈Θ d{b(θ), b0} exists.
(iii) For some β0 with β0 > 0, ^β − β0 = oPθ
(1).
then
Π [|˜θ − ˜θ∗
| > δ|η(y)] = oP0 (1),
[Frazier & al., 2017]

ABC convergence under well- and mis-specified models

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