ABC: Bayesian Computation Without Likelihoods

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ABC Bayesian Computation Without Likelihoods
David Balding Centre for Biostatistics Imperial
College London (www.icbiostatistics.org.uk)
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Bayesian inference viarejection from prior I

• Generate a posterior random sample for a
  parameter of interest ? by a mechanical version
  of Bayes Theorem
• 1. simulate ? from its prior
• 2. accept/reject, with P(accept) ? likelihood
• 3. if not enough acceptances yet, go to 1.
• Problem if likelihood involves integration over
  many nuisance parameters, hard/slow to compute.
• Solution use simulation to approximate
  likelihood.

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Bayesian inference viarejection from prior II

• Generate an approximate posterior random sample
• 1. simulate parameter vector ? from its prior
• 2. simulate data X given value of ? from 1.
• 2a. if X matches observed data, accept ?
• 3. if not enough acceptances yet, go to 1.
• Problem simulated X hardly ever matches
  observed.
• Solution relax 2a so that ? is accepted when X
  is close to observed data close to is usually
  measured in terms of a vector of summary
  statistics, S.

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Prior p(F)
Marginal likelihood p(S)
Likelihood p(S F)
Posterior density p(F S)
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Approximate Bayesian Computing (ABC)
We simulate to approximate (1) the joint
parameter/ data density then (2) a slice at the
observed data. Few if any simulated points will
lie on this slice so need to assume smoothness
required posterior is approximately the same for
datasets close to that observed. Note (1) we
get approximate likelihood inferences but we
didnt calculate the likelihood (2) different
definitions of close can be tried for the same
set of simulations (3) these can even be retained
and used for different observed datasets.
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? values of these points are treated as random
sample from posterior
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When to use ABC ?

• When likelihood is hard to compute because of
  need for integration over many nuisance
  parameters BUT easy to simulate
• Population genetics nuisance parameters are the
  branching times and topology of the genealogical
  tree underlying the observed DNA sequences/genes.
• Epidemic models nuisance parameters are
  infection times and infectious periods.

ABC implies 3 approximations 1. finite
simulations 2. non-sufficiency of S 3. S need
not match S exactly
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Population genetics example

• Parameters
• N effective population size
• µ mutation rate per generation
• G genealogical tree (topology branch
  lengths) nuisance
• Summary Statistics
• S1 number of distinct alleles/sequences
• S2 number of polymorphic/segregating sites
• Algorithm
• 1. simulate N and µ from joint prior
• 2. simulate G from the standard coalescent model
• 3. simulate mutations on G and calculate S
• 4. accept (N, µ,G) if S S
• This generates a sample from the joint posterior
  of (N, µ,G).
• To make inference about ? 2Nµ, simply ignore G.

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Model comparison via ABC
Can also use ABC for model comparison, as well as
for parameter estimation within models. Ratio of
acceptances
approximates the Bayes Factor. Better fit
(weighted) multinomial regression to predict
model from observed data. Beaumont (2006) used
this to infer the topology of a tree representing
the history of 3 Californian fox populations.
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Problems/limitations

• Rejection-ABC is very inefficient most simulated
  datasets are far from observed and must be
  rejected. No learning.
• How to find/assess good summary statistics?
• Too many summary statistics can make matters
  worse (see later)
• How to choose metric for (high-dimensional) S

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Beaumont, Zhang, and DJB
Approximate Bayesian Computation in Population
Genetics.
Genetics 162 2025-2035, 2002
Use local-linear regression to adjust for the
distance between observed and simulated
datasets. Use a smooth (Epanechnikov) weighting
according to distance. Can now weaken the close
criterion (i.e. increase the tolerance) and
utilize many more points.
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Parameter
Summary Statistic
1
0
Weight
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1
0
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Estimation of scaled mutation rate q 2Nm

• Full data-
• 445 Y chromosomes each typed at 8
  microsatellite loci
• i.e. 3560 numbers

Standard Rejection
Relative mean square error

• Summary statistics-
• mean variance in length
• mean heterozygosity
• number of haplotypes
• i.e. 3 numbers

With regression adjustment
MCMC
Tolerance
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Population growth

• Population constant size NA until t generations
  ago, then exponentially rate r per gen. growth to
  NC. 4 model params, but only 3 identifiable. We
  choose
• Data same as above, except smaller sample size n
  200 (because of time taken for MCMC to
  converge).

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ABC applications in population genetics
Standard rejection method Estoup et al. (2002,
Genetics) Demographic history of invasion of
islands by cane toads. 10 microsatellite loci, 22
allozyme loci. 4/3 summary statistics, 6
demographic parameters. Estoup and Clegg (2003,
Molecular Ecology) Demographic history of
colonisation of islands by silvereyes. With
regression adjustment Tallmon et al (2004,
Genetics) Estimating effective population size
by temporal method. One main parameter of
interest (Ne), 4 summary statistics. Estoup et
al. (2004, Evolution) Demographic history of
invasion of Australia by cane toads. 75/63
summary statistics, model comparison, up to 5
demographic parameters.
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More sophisticated regressions?

• Although global linear regression usually gives
  a poor fit to joint ?/S density, Calabrese (USC,
  unpublished) uses projection pursuit regression
• to fit a large feature set of summary
  statistics. Iterate to improve fit within
  vicinity of S. Application to estimate human
  recombination hotspots.
• Could also consider quantile regression to adapt
  adjustment to different parts of the distribution.

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Do ABC within MCMC

• Marjoram et al. (2003). Two accept/reject
  steps
• Simulate a dataset at the current parameter
  values if it isnt close to observed data, start
  again.
• If it is close, accept or reject according to
  prior ratio times Hastings ratio (no likelihood
  ratio)
• Note now close must be defined in advance
  also cannot reuse simulations for different
  observed datasets. Can apply regression-adjustmen
  t to MCMC outputs.
• Problems
• proposals in tree space
• few acceptances in tail of target distribution -
  stickiness

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Importance sampling within MCMC

• In fact, the Marjoram et al. MCMC approach can be
  viewed as a special case of a more general
  approach developed by Beaumont (2003).
• Instead of simulating a new dataset
  forward-in-time, Beaumont used a backward-in-time
  IS approach to approximate the likelihood.
• His proof of the validity of the algorithm is
  readily extended to forwards-in-time approaches
  based on one or multiple datasets (cf ONeill et
  al. 2000). Could also use a regression
  adjustment.

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ABC within Sequential MCSisson et al at UNSW,
Sydney

• Sample initial generation of ? particles from
  prior.
• Sample ? from previous generation, propose new
  value and generate dataset calculate S.
• Repeat until S S BUT tolerance reduces each
  gen.
• Calculate prior ratio times Hastings ratio use
  as weight W for sampling the next generation.
• If variance of W is large, resample with
  replacement according to W and set all W1/N.
• Application to estimate parameters of TB
  infection.

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Adaptive simulation algorithm(Molitor and Welch,
in progress)

• simulate N values of ? from prior
• calculate corresponding datasets and use
  similarity of S with S to generate a density
• resample from density, replace value with lowest
  similarity of S and S.
• use final density as importance sampling weights
  for a conventional ABC.
• idea is to use preliminary pseudo-posterior
  based on weights to choose something better than
  prior as basis for ABC

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• "number of data generation steps for rejection
  ABC"
• 1 35064 2 27877
• "number of data generation steps for SMC ABC"
• 1 14730 2 12629
• "number of data generation steps for Johns ABC"
• 1 10314 2 6130

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ABC to rescue poor estimators(inspired by DJ
Wilson, Lancaster)

• evaluate estimator based on simplistic model at
  many datasets simulated under more sophisticated
  model.
• for observed dataset, use as estimator regression
  predictor of simplistic estimator at the observed
  data value.
• for example, many population genetics estimators
  assume no recombination, and infinite sites
  mutation model
• use this estimator and simulations to correct for
  recombination and finite-sites mutation

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Acknowledgments

• David Welch and John Molitor, both of Imperial
  College.
• David has just started on an EPSRC grant to
  further develop ABC ideas and apply particularly
  in population genomics.