Stratified Monte Carlo and bootstrapping for approximate Bayesian computation

Stratiﬁed Monte Carlo and bootstrapping for
approximate Bayesian computation
Umberto Picchini
@uPicchini
Chalmers University of Technology
and University of Gothenburg
Sweden
ABC World Seminar, 7 May
2020

Joint work with Richard Everitt:
P and Everitt (2019) Stratiﬁed sampling and bootstrapping for
approximate Bayesian computation, arXiv:1905.07976.
This is ongoing work. Feedback is very welcome.
Umberto Picchini, @uPicchini 2/47

I am going to talk of essentially three main points:
• how to use bootstrap samples to reduce the computational
effort when approximating the ABC likelihood;
• how to reduce the bootstrap-induced bias of the approximated
ABC likelihood using stratified Monte Carlo.
• how to obtain reliable inference with a larger than usual ABC
threshold, via stratified MC.
The above is illustrated using a pseudomarginal ABC-MCMC
approach.
However other samplers could be used.

Notation
• We are interested in Bayesian inference for parameters θ;
• the likelihood p(xobs|θ) for data xobs is “unavailable”;
• we assume the ability to simulate pseudo-data x∗ from a
simulator x∗ ∼ p(x|θ);
• we use an ABC approach (approximate Bayesian computation);
• we provide summary statistics sobs = S(xobs) to data and
s∗ = S(x∗) to simulated data;
• the key idea in ABC is to give higher weights to summaries s∗
close to sobs and with higher probability accept parameters θ∗
generating s∗ ≈ sobs.

A possibility to weight the proximity of s∗ to sobs is to use a kernel
funct Kδ(s∗, sobs) and write
πδ(θ, s∗
|sobs)
augmented posterior
∝ π(θ) × Kδ(s∗
, sobs) × p(s∗
|θ)
for example we may use a Gaussian kernel
Kδ(s∗
, sobs) ∝ exp(−
1
2δ2
(s∗
− sobs) (s∗
− sobs)).
Other kernels are possible, e.g. Epanechnikov’s kernel.
We can then write the (marginal) ABC posterior as
πδ(θ|sobs) ∝ π(θ) Kδ(s∗
, sobs)p(s∗
|θ)ds∗
ABC likelihood

The ABC likelihood: Kδ(s∗, sobs)p(s∗|θ)ds∗.
This can trivially be approximated unbiasedly via Monte Carlo as1
Kδ(s∗
, sobs)p(s∗
|θ)ds∗
≈
1
M
M
r=1
Kδ(s∗r
, sobs), s∗r
∼
iid
p(s∗
|θ).
and plugged in a Metropolis-Hastings ABC-MCMC algorithm,
proposing a move θ∗ ∼ q(θ|θ#), and accepting with probability
min 1,
1
M
M
r=1 Kδ(s∗r, s)
1
M
M
r=1 Kδ(s#r, s)
·
π(θ∗)
π(θ#)
·
q(θ#|θ∗)
q(θ∗|θ#)
Problem: if the model simulator is computationally expensive,
having M large is unfeasible. Often M = 1 is used.
1
Lee, Andrieu, Doucet (2012): Discussion of Fearnhead-Prangle 2012
JRSS-B.

So we have the approximated ABC posterior (up to a constant c)
πδ(θ|sobs) ≈ c · π(θ) ·
1
M
M
r=1
Kδ(s∗r
, sobs)
non-negative unbiased
, s∗r
∼
iid M times
p(s∗
|θ).
• this makes the algorithm an instance of pseudomarginal MCMC
[Andrieu, Roberts 2009]2
• No matter the value of M, the ABC-MCMC will sample exactly
from πδ(θ|sobs)
• Typically ABC-MCMC is computationally intensive and a small M
is chosen, say M = 1;
• the lower the M the higher the variance of the estimate of the ABC
likelihood Kδ(s∗
, s)p(s∗
|θ)ds∗
• the larger the variance the worse the mixing of the chain (due to
occasional overestimation of the likelihood).
2
Andrieu, C., and Roberts, G. O. (2009). The pseudo-marginal approach for
eﬃcient Monte Carlo computations. The Annals of Statistics, 37(2), 697-725.

Dilemma:
a small M will decrease the runtime considerably, however it will
increase the chance to overestimate the likelihood → possibly
high-rejection rates.
Question: is it worth to have M > 1 to reduce the variance of the
ABC likelihood given the higher computational cost?
Bornn et al 20173 found that no, it is not worth and M = 1 is
just ﬁne (when using a uniform kernel).
Basically using M = 1 is so much faster to run that the decreased
variance obtained with M > 1 is not worth given the higher
computational cost.
3
L. Bornn, N. S. Pillai, A. Smith, and D. Woodard. The use of a single
pseudo-sample in approximate Bayesian computation. Statistics and
Computing, 27(3):583590, 2017.

Can we devise a strategy to cheaply generate many artiﬁcial
datasets?

Data resampling (nonparametric bootstrap)
In a similar context (based on synthetic likelihood approaches) Everitt
20174
used the following approach:
At any proposed θ
• simulate say M = 1 datasets x∗
∼ p(x|θ);
• sample with replacement from elements in x∗
to obtain a
bootstrapped dataset (with dimension dim(xobs));
• repeat the resampling R times to obtain x∗1
, ..., x∗R
bootstrap
datasets from x∗
;
• compute the summaries s∗1
, ..., s∗R
for each resampled dataset;
• compute ABC likelihood approximation
1
R
R
r=1
Kδ(s∗r
, sobs)
• plug this into ABC-MCMC
4
Everitt (2017). Bootstrapped synthetic likelihood. arXiv:1711.05825.

This is cheap compared to producing M independent summaries
from the model, when the simulator is computationally intensive.
It reduces the variance of the ABC likelihood compared to using
M = 1 without resampling.
However it is worse than using several M 1 independent samples.

Example: data is 1000 iid observations from N(θ = 0, 1).
Set Gaussian prior on θ → known analytic posterior. Use suﬃcient
stats S(xobs) = ¯xobs throughout.
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
0
2
4
6
8
10
12
14
pseudomarginal ABC
Exact Bayes
(a) pseudomarginal ABC,
M = 500
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
0
2
4
6
8
10
12
14
ABC with resampling
Exact Bayes
(b) M = 1 and R = 500
resamples
• Left: pseudo-marginal ABC with M = 500 independent
datasets;
• Right: M = 1 and R = 500 resampled datasets

What we saw is hardly surprising:
Of course resampling introduces additional variability in the
estimand.
Example: true loglikelihood of the summary statistic ¯x
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
-3
-2
-1
0
1
2
3

We estimate the ABC loglikelihood at several θ-values in [−0.1, 0.1] with
a very small threshold, δ = 3 × 10−5
:
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
105
Figure: Solid lines are mean values over 500 repetitions. Dashed lines are
2.5 and 97.5 percentiles.
In black: analytic loglikelihood (undistinguishable).
In red: ABC loglikelihood approximated via M=500 independent datasets.

Let’s zoom in very central values...
-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01
-50
-40
-30
-20
-10
0
10
Figure: Solid lines are mean values over 500 repetitions. Dashed lines are
2.5 and 97.5 percentiles.
We notice some bias: this can be reduced by increasing model
simulations M to values 500

We now use bootstrap M = 1 simulated datasets R = 500 times:
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
106
1
R
R
r=1
Kδ(s∗r
, sobs)
In blue: ABC loglikelihood approximated via M=1 and R=500
bootstrapped datasets; large bias+overly variable likelihood
In red: no bootstrapping and M=500 independent datasets;

Stratiﬁed Monte Carlo
Stratiﬁed Monte Carlo is a variance reduction technique.
In full generality: want to approximate
µ =
D
f(x)p(x)dx
over some space D, for some function f and density (or probability
mass) function p.
Now partition D into J “strata” D1, ..., DJ :
• ∪J
j=1Dj = D
• Dj ∩ Dj = ∅, j = j

Samples from bivariate N2(0, I2)
6 concentric rings and 7 strata.
Each stratum has exactly 3 points sampled from within it.
Better to oversample inside the most “important” strata, where the
integrand has higher mass.

Ideally the statistician should decide how many Monte Carlo draws to
sample from each stratum Dj.
• Call this number ñj;
• define ωj := P(X ∈ Dj)
Probabilities ωj should be known.
Then we approximate µ = D
f(x)p(x)dx with
ˆµstrat =
J
j=1
ωj
ñj
x∗∈Dj
f(x∗
) , x∗
∼ p(x|x ∈ Dj)
This is the (unbiased) stratified MC estimator.
Variance reduction compared to vanilla MC estimator can be obtained if
we know how many ñj to sample from each stratum (e.g “proportional
allocation method”5
)
5
Art Owen (2013), Monte Carlo theory, methods and examples.

However in our settings we can’t assume ability to simulate
from within a given stratum; so we can’t decide ñj.
And we can’t assume to know anything about ωj := P(X ∈ Dj).
We use a “post stratification” approach (e.g. Owen 2013)6
• first generate many x∗ ∼ p(x) (i.e. from the model simulator);
• count the number of x∗ ending up in each stratum Dj;
• call these frequencies nj (a random variable);
So these frequencies are known after the simulation is done, not
before.
However we still do not know anything about the ωj = P(X ∈ Dj).
We are going to address this soon within an ABC framework.
6
Art Owen: “Monte Carlo theory, methods and examples” 2013.

Deﬁne strata for ABC
Suppose we have an ns-dimensional summary, i.e. ns = dim(sobs)
and consider the Gaussian kernel
Kδ(s∗
, sobs) =
1
δns
exp −
1
2δ2
(s∗
− sobs) (s∗
− sobs) .
In ABC the µ to approximate via stratiﬁed MC is the ABC
likelihood
D
Kδ(s∗
, sobs)p(s∗
|θ)ds∗
So lets partition D...

Deﬁne strata for ABC
Example to partition D using three strata:
• D1 = {s∗ s.t. s∗ − sobs < δ/2}
• D2 = {s∗ s.t. s∗ − sobs < δ}D1
• D3 = D{D1 ∪ D2}
And more explicitly:
• D1 = {s∗ s.t. (s∗ − sobs) (s∗ − sobs) ∈ (0, δ/2]}
• D2 = {s∗ s.t. (s∗ − sobs) (s∗ − sobs) ∈ (δ/2, δ]}
• D3 = {s∗ s.t. (s∗ − sobs) (s∗ − sobs) ∈ (δ, ∞)}.
Because of our resampling approach, for every θ we have R 1
simulated summaries, say R = 500.
We just need to count how many summaries fall into D1
instead of D2 instead of D3.
This give us n1, n2 and n3 = R − (n1 + n2).

n1n1n1
ssdsdddsobs
n1=2
n2=13
n3=10
R = 25 simulated s∗ distributed across three strata.

How about the strata probabilities?
We still need to estimate the strata probabilities ωj = P(s∗ ∈ Dj).
This is easy because ωj = Dj
p(s∗|θ)ds∗ which we estimate by
another MC simulation.
So
1 simulate once from the model x∗ ∼ p(x|θ)
2 resample R times from x∗ to obtain x∗1, ..., x∗R
3 compute summaries s∗1, ..., s∗R
4 obtain distances dr := (s∗r − sobs) (s∗r − sobs)
ˆω1 :=
1
R
R
r=1
I{dr≤δ/2}, ˆω2 :=
1
R
R
r=1
I{δ/2<dr≤δ},
ˆω3 := 1 −
2
j=1
ˆωj.

We ﬁnally have a (biased) estimator of the ABC likelihood using J
strata:
ˆˆµstrat =
J
j=1
ˆωj
nj
r∈Dj
Kδ(s∗r
, sobs) ,
Bias due both to resampling and using estimated ωj.
All in all we simulated twice from the model at given θ, once to
obtain nj and once to obtain ˆωj.

Notice the above is not quite ok. What if some nj = 0?
(neglected stratum)
In our ABC-MCMC we reject proposal θ∗ as soon as nj = 0, so
the actual estimator is
ˆˆµstrat =
J
j=1
ˆωj
nj
r∈Dj
Kδ(s∗r
, sobs) I{nj>0,∀j}
This is both a curse and a blessing. We’ll see why...

Stratiﬁed MC within ABC-MCMC
As usual, we accept a proposal using a MH step:
propose θ∗ ∼ q(θ|θ#) and accept with probability
1 ∧
ˆˆµstrat(θ∗)
ˆˆµstrat(θ#)
·
π(θ∗)
π(θ#)
·
q(θ#|θ∗)
q(θ∗|θ#)
if we accept, set: θ# := θ∗ and ˆˆµstrat(θ#) := ˆˆµstrat(θ∗).
Repeat a few thousands of times,
with
ˆˆµstrat =
J
j=1
ˆωj
nj
r∈Dj
Kδ(s∗r

Reprising the Gaussian example
Data: 1000 iid observations ∼ N(θ = 0, 1). Gaussian prior →
exact posterior
Red: exact posterior.
Blue: different types of ABC-MCMC posteriors.
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
0
2
4
6
8
10
12
14
pseudomarginal ABC
Exact Bayes
(a) pseudomarginal ABC,
M = 500, small δ
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
0
2
4
6
8
10
12
14
ABC with resampling
Exact Bayes
(b) M = 1 and R = 500
resamples, small δ
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
0
2
4
6
8
10
12
14
resampling + stratification
Exact Bayes
(c) M = 1, R = 500 and
stratification, large δ
With stratification and only M = 1 we get results as good as with
M = 500 (compare left and right).
...and I used a 10x larger δ with the stratified approach!

Basically, if the model is simple enough, we can aﬀord using a very
small δ if we have the computational power to run very many
iterations.
Even if the ABC likelihood is badly approximated (highly variable)
and hence chain-mixing is poor, we just need to run many MCMC
iterations.
But what if computational power is limited? If the model is
complex, we can use a larger δ, and employ stratiﬁcation.

Basically when δ is “very small”, ABC likelihood is very
“concentrated” around sobs → most simulated summaries end
up in low probability regions → we often only obtain a poor
ABC-likelihood approximation with high variance.
ssd
sdd
d
sobs
𝛿

With a larger δ we get more samples in higher probability
regions, and in each stratum, hence a better approximated
(though wider) ABC likelihood.
If we only accept θ∗ when ALL strata are represented, a good
approximation of a “wider ABC likelihood” (larger δ) works better
than a poorly approximated “narrow ABC likelihood” (small δ).

Did stratiﬁcation work to reduce the variance of the likelihood?
Here M = 1 and R = 500 bootstrapped datasets:
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
-25
-20
-15
-10
-5
0
5
And here we have standard pseudomarginal ABC with M = 500:
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
105

The very reason why it works so well is that we necessarily need
at least one sample in each stratum, since
ˆˆµstrat =
J
j=1
ˆωj
nj
r∈Dj
Kδ(s∗r
which crucially implies that the most internal stratum needs a
sample (hence relatively close to sobs).
This is why we need to use a larger δ when implementing
stratiﬁcation.

Further variance reduction
But we can do more:
at each θ∗
• (1) we have simulated once from p(x|θ∗), resampled annd
computed the strata frequencies nj;
• (2) we have simulated once more from p(x|θ∗), resampled and
estimated the strata probabilities ωj;
At the same θ∗, swap the roles of the two simulated datasets:
• reuse simulated data from (1) to estimate the ωj instead;
• reuse simulated data from (2) to compute the nj instead;
We have now obtained a second likelihood at nearly zero cost: call
this ˆˆµ
(2)
strat.
Use ¯µstrat =
(ˆˆµstrat+ˆˆµ
(2)
strat)
2 into the MCMC

Averaging the two likelihoods reduces the likelihood variance
further.
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
20
40
60
80
100
120
140
160
Variances for ABC loglikelihoods
What you see above implies a 50% reduction in the estimated
variance compared to the already good stratiﬁed estimator ˆˆµstrat.
More details in the paper...

An important point to keep in mind:
• we cannot create as many strata as we want;
• if one stratum is “neglected”, i.e. nj = 0 for some j, then we
have ˆˆµstrat = 0 since
ˆˆµstrat =
J
j=1
ˆωj
nj
r∈Dj
Kδ(s∗r
• this forces us to keep the number of strata “small” (we used 3
strata) to reduce the chance of nj = 0.
• enlarge δ. This is ok if you use stratiﬁcation with the
estimator above.

Further questions
Using a larger δ increases acceptance rate...however
• resampling gives some computational overhead. Worth it?
Let’s see in next example....

Time series
Our methodology is not restricted to iid data.
Next example, uses “block bootstrap”7
, where we resample blocks of
observations for stationary time-series.
B =



(1 : B)
block1
, (B + 1 : 2B)
block2
, ..., (nobs − B + 1 : nobs)
block
nobs
B



.
• blocks are chosen to be suﬃciently large such that they retain the
short range dependence structure of the data;
• so that a resampled time series, constructed by concatenating
resampled blocks, has similar statistical properties to real data
At each θ we resample blocks of indeces of simulated data.
7
Kunsch, H. R. (1989). The jackknife and the bootstrap for general
stationary observations. The Annals of Statistics, 1217-1241.

2D Lotka-Volterra time series
A predator-prey model with an intractable likelihood (Markov
jump process).
Two interacting species: X1 (# predators) and X2 (# prey).
Populations evolve according to three interactions:
• A prey may be born, with rate θ1X2, increasing X2 by one.
• The predator-prey interaction in which X1 increases by one
and X2 decreases by one, with rate θ2X1X2.
• A predator may die, with rate θ3X1, decreasing X1 by one.
Its solution may be simulated exactly using the “Gillespie
algorithm”8.
8
Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical
reactions. The journal of physical chemistry, 81(25), 2340-2361.

We have 32 observations for each species simulated via Gillespie’s
algorithm.
At each θ we simulate and resample 4 blocks each having size
B = 8.
We want inference for reaction rates (θ1, θ2, θ3).
Set vague priors log θj ∼ U(−6, 2)

We run several experiments:
• pseudomarginal ABC-MCMC with M = 2 indep. datasets for each
θ;
• ABC-MCMC with M = 2, R = 500 resampled datasets and
allocated across three strata;
D1 = {s∗
s.t. distance ∈ (0, δ/2)}
D2 = {s∗
s.t. distance ∈ (δ/2, δ]}
D2 = {s∗
s.t. distance ∈ (δ, ∞)}
• ABC sequential Monte Carlo (ABC-SMC), with 1,000 particles.
Very expensive, 12 hrs of computation.
We used the very general ABC-SMC with automatically determined
sequence of δ’s from Fan-Sisson9
, generalizing Del Moral et al 2012.
9
Y. Fan and S. Sisson. ABC samplers, in Handbook of approximate
Bayesian computation. Chapman and Hall/CRC, 2018.

We start with ABC-SMC: slow, but safe gold-standard procedure.
After 12 hrs determines that δ = 0.205 is ok (acceptance rate ≈
1%).
θ1 θ2 θ3 ESS accept. rate runtime ESS/min efficiency
(%) (min) increase
true parameters 1 0.005 0.6
ABC-SMC (δ = 0.205) 0.966 [0.749,1.173] 0.0048 [0.0036,0.0058] 0.587 [0.485,0.711] – 1 – –
pmABC (δ = 0.230) 1.085 [0.881,1.401] 0.0052 [0.0043,0.0063] 0.598 [0.475,0.720] 22.4 1.5 16.2 1.38 1
strat. ABC (δ = 1.380) 0.943 [0.742,1.177] 0.0048 [0.0037,0.0060] 0.613 [0.482,0.757] 145.8 7.5 28.7 5.08 3.7
Figure: Lotka-Volterra: Posterior mean and 95% intervals for θ.
For stratified ABC we used a δ 6-times larger than for standard
ABC!
With stratification: inference similar to ABC-SMC, but acceptance
rate way higher than pmABC and better tails exploration.

10,000 MCMC iterations is a very small number when targeting a 1%
acceptance rate.
High autocorrelation, potentially bad tails exploration.
Here is an example from parameter θ1.
Without stratiﬁcation: (δ = 0.23), M = 2
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
With stratiﬁcation: (δ = 1.34), M = 2, R = 500
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
1
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4

Low acceptance rate means that typically we are unable to explore
the tails of the posterior adequately.
→ high IAT (integrated autocorrelation time)
→ Low ESS (eﬀective sample size).
Remember, we are assuming an expensive model. So we cannot
just increase the number of iterations ad libitum.

Conclusions
• stratified Monte Carlo is effective in reducing resampling bias;
• Allows for precise ABC while using a larger δ;
• smaller variance ABC likelihood → better mixing MCMC;
• Downside: neglected strata may increase rejection rate;
• to reduce chances for neglected strata, initialize the simulation
with bootstrapping without stratification. Run initial iterations
this way to cheaply approach high posterior mass.
• Downside: requires a bootstrapping strategy that makes sense
for the given data;
• more research needed for constructing optimal strata.

• Ongoing work: comments most welcome!
• P and Everitt (2019). Stratiﬁed sampling and resampling for
approximate Bayesian computation,
https://arxiv.org/abs/1806.05982
picchini@chalmers.se
@uPicchini

PhD opportunity
PhD position with me. Deadline 29 May.
Simulation-based Bayesian inference and deep learning for
stochastic modelling
https://tinyurl.com/ru5j9co

Stratified Monte Carlo and bootstrapping for approximate Bayesian computation

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