Tutorial on ABC Algorithms - Dr Christopher Drovandi · Approximate Bayesian Computation in...

Intro ABC Rejection MCMC ABC SMC ABC Closing Refs

Tutorial on ABC Algorithms

Dr Chris DrovandiQueensland University of Technology, Australia

[email protected]

July 3, 2014

Chris Drovandi ABC in Sydney 2014


Ingredients for Implementing ABC Algorithm

Code to simulate data from model

Code to compute summary statistics

Code to compute discrepancy function

Combine within rejection, MCMC or SMC algorithm (not toohard)



Importance Sampling

Define importance distribution g(θ)

Draw M iid samples from g , θiiid∼ g(θ), i = 1, . . . ,M

Weight the samples

wi =f (y |θi )π(θi )

g(θi )

Normalise the weights to obtain Wi , i = 1, . . . ,M

{θi ,Wi}Mi=1 represents weighted sample from π(θ|y )

Effective sample size:

ESS =1

∑Mi=1(Wi )2



ABC Importance Sampling

Require proposal on joint space of θ and xLet g(θ, x) = g(θ)f (x |θ)Generate sample {θi , x i}

Mi=1

iid∼ g(θ, x)

Weight

wi =Kǫ(ρ(y , x i))��

��

f (x i |θi)π(θi )

g(θi)��

f (x i |θi )

If we choose g(θ) = π(θ) and letKǫ(ρ(y , x)) = 1(ρ(y , x) ≤ ǫ) then wi is either 0 or 1



ABC Rejection Algorithm

ABC rejection (post-determination of ǫ)

1: Draw θi ∼ π(·) and simulate x i ∼ f (·|θi) for i = 1, . . . ,M.Generates collection {θi , x i}

Mi=1.

2: Compute discrepancy ρi = ρ(y , x i). Obtain particles{θi , ρi}

Mi=1

3: Sort {θi , ρi}Mi=1 based on ρ

4: Keep N = α×M of θi with lowest discrepancy (this definesthe ǫ)

Choice of α trade off between accuracy and Monte Carlo error(Fearnhead and Prangle (2012))



ABC Rejection Algorithm (Pritchard et al 1999)

ABC Rejection (pre-specification of ǫ)

1: Draw θ ∼ π(·)2: Simulate x ∼ f (·|θ)3: If ρ(y , x) ≤ ǫ then accept θ4: Repeat lines 1:, 2: and 3: until N samples are drawn

Avoids storage requirements but choosing ǫ difficult



ABC Rejection Example

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

5

10

15

20

25

30

τ

Den

sity

ε

1 = 0.24 (20%)

ε2 = 0.13 (10%)

ε3 = 0.013 (1%)

ε4 = 0.0014 (0.1%)

Figure: Plot courtesy of Brenda Nho Vo. ABC posterior for different α(leading to different ǫ).



ABC Rejection Algorithm

Advantages

Simplicity

Useful for testing different sets of summary statistics (Nunesand Balding 2010) or analysing multiple datasets from samemodel (see Drovandi and Pettitt 2013 for Bayesianexperimental design example)

Disadvantages

Storage requirements (depending on implementation chosen)

Highly inefficient if posterior different to prior (too high ǫrequired to obtain a reasonable size sample from ABCposterior)



Markov chain Monte Carlo

Motivation: Keep proposals within non-negligible posteriorregions

Construct a Markov chain whose limiting distribution is π(θ|y )Assume current value of chain is θ. Propose next value ofchain θ

∗ ∼ q(·|θ). Accept θ∗ as next value of chain withprobability min(1,A) where

A =f (y |θ∗)π(θ∗)q(θ|θ∗)

f (y |θ)π(θ)q(θ∗|θ),

otherwise set θ as the next value of chain.



MCMC ABC (Marjoram et al 2003, Sisson and Fan 2011)

1: Obtain θ0, x0 using a burnin or from one sample of rejection ABC

2: Compute ψ0 = Kǫ(ρ(y , x0))3: for i = 1 to N do

4: Draw θ∗ ∼ q(·|θi−1)

5: Simulate x∗ ∼ f (·|θ∗)6: Compute ψ∗ = Kǫ(ρ(y , x∗))

7: Compute r = π(θ∗)ψ∗q(θi−1|θ∗)

π(θi−1)ψi−1q(θ∗|θi−1)

8: if uniform(0, 1) < r then

9: θi = θ

∗, ψi = ψ∗

10: else

11: θi = θ

i−1, ψi = ψi−1

12: end if

13: end for



MCMC ABC

Advantages

Still pretty simple

Tends to be more efficient than ABC rejection for single dataanalysis

Disadvantages

Slightly more difficult to implement than ABC rejection

Requires tuning of proposal distribution q

Can get ‘stuck’ in low probability regions (thus long runs areoften required)

Must be re-run and tuned for each new dataset or summarystatistic selection



MCMC ABC Example of “stickiness”

Figure: Plot taken from Sisson and Fan 2011. From bottom to top ǫ is4.5, 4, 3.5, 3.



Variants on MCMC ABC

Bortot et al 2007 - Include ǫ as a ‘parameter’ (every now andthen propose larger value of ǫ)

Baragatti et al 2013 - Population MCMC (parallel tempering,swapping chains with different ǫ)

Picchini et al 2013 - Early rejection (when using uniformkernel, may reject proposal without simulating data)

Aandahl et al 2014 - Multiple try MCMC ABC



Sequential Monte Carlo (SMC) ABC

SMC for sampling from a sequence of target distribution

For ABC natural to define sequence in terms of non-increasingset of tolerances ǫ1 ≥ ǫ2 ≥ ǫT

πt(θ, x |y , ǫ) ∝ f (x |θ)π(θ)1(ρ(x , y ) ≤ ǫt), for t = 1, . . . ,T ,

Traverse set of N ‘particles’ through sequence of targets byiteratively applying re-weighting (importance sampling),re-sampling and mutation steps

Ultimately obtain set of weighted samples from πT

Nice accessible reference to SMC is Chopin (2002) (see alsoDel Moral et al (2006))



SMC ABC Algorithm of a few papers

1: Initialise ǫ1, . . . , ǫT and specify the initial importance sampling distributionπ0(·)

2: for t = 1 to T do

3: for i = 1 to N do

4: If t = 1 sample θ∗∗ from π0(·). If t > 1 sample θ

∗ from the previouspopulation {θt−1i ,W t−1

i }Ni=1 and perturb the particle θ∗∗ ∼ Kt(·|θ

∗).5: Generate a dataset x∗∗ ∼ f (·|θ∗∗).6: If ρ(y , x∗∗) > ǫt then go back to step 4.7: Set θit = θ

∗∗ and re-weight

wit =

π(θit)π0(θ

it ) if t = 1

π(θit)∑

Nj=1

Wjt−1Kt (θ

it |θjt−1)

if t > 1.

8: end for

9: Normalise the weights W it = w i

t/∑N

j=1 wjt for i = 1, . . . ,N.

10: Update the tuning parameters of Kt+1 using the set of particles{θit ,W i

t }Ni=1.

11: end for



SMC ABC

Advantages

Advantages of SMC over MCMC (deals better withmulti-modal posteriors, easy adaptation of proposal)

Tends to be more efficient than ABC rejection and MCMCABC (from adaptive proposal)

Disadvantages

Not as simple to implement as ABC rejection and MCMCABC

Must be re-run and tuned for each new dataset or summarystatistic selection

Requires specification of sequence of tolerances



SMC ABC of Drovandi et al 2011

1: Set Na as the integer part of αN2: Perform the rejection sampling algorithm with ǫ1. This produces a set of

particles {θi , ρi}Ni=1

3: Sort the particle set by ρ, so that ρ1 ≤ ρ2 ≤ · · · ≤ ρN , and set ǫt = ρN−Na

and ǫmax = ρN . If ǫmax ≤ ǫT then finish, otherwise go to 4.4: Compute the tuning parameters of the MCMC kernel qt(·|·) using the

particle set {θi}N−Na

i=1

5: for j = N − Na + 1 to N do

6: Resample θj from {θi}N−Na

i=1

7: Perform Rt iterations of MCMC ABC using qt and ǫt to update θj

8: end for

9: Compute Rt based on the overall MCMC acceptance rate of the previousiteration and go to 3.

See also Del Moral et al 2012 for similar algorithm



Movie

It’s movie time!



SMC ABC of Drovandi et al 2011

Advantages

Same advantages as previous but also do not require tospecify sequence of tolerances, removes weight calculation andhas stopping rule

Disadvantages

Same disadvantages as previous but do not require sequenceof tolerances but uses MCMC ABC so not all particles getmoved (some duplication inevitable)



Other Variants on SMC ABC

Silk et al (2013) - different way to determine adaptivesequence of tolerances

Filippi et al (2013) - determine optimal proposal distributionfor SMC ABC

Probably a few other papers out there...



ABC Software Packages

R Packages (ABC, EasyABC)

DIYABC, PopABC (ABC for population genetics models)

ABC-SysBio (Python package with GPU support(?))

ABCtoolbox

See Wiki pagehttp://en.wikipedia.org/wiki/Approximate_Bayesian_computation


http://en.wikipedia.org/wiki/Approximate_Bayesian_computation


Closing Remarks on ABC Algorithms

ABC Algorithms generally easy to implement

MCMC and SMC versions tend to be more efficient but ABCrejection still has its charm

Generally still quite computationally intensive

Regression adjustment (see Beaumont et al 2002) can beapplied to output of any ABC algorithm

Many other Likelihood-free algorithms out there...



References

Beaumont et al (2002). Approximate Bayesian Computation inPopulation Genetics. Genetics.Marjoram et al (2003). Markov chain Monte Carlo withoutlikelihoods. PNAS.Sisson et al (2007). Sequential Monte Carlo without likelihoods.PNAS.Drovandi and Pettitt (2011). Estimation of parameters formacroparasite population evolution using approximate Bayesiancomputation. Biometrics.Del Moral et al (2012). An adaptive sequential Monte Carlomethod for approximate Bayesian computation. Statistics andComputing.Drovandi and Pettitt (2013). Bayesian Experimental Design forModels with Intractable Likelihoods. Biometrics.


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