Approximate Bayesian computation

Approximate Bayesian computation (ABC) is a family of computational techniques in Bayesian statistics. These simulation techniques operate on summary data (such as population mean, or variance) to make broad inferences with less computation than might be required if all available data were analyzed in detail. They are especially useful in situations where evaluation of the likelihood is computationally prohibitive, or whenever suitable likelihoods are not available.

ABC methods originated in population and evolutionary genetics ^[1][2] but have recently also been introduced to the analysis of complex and stochastic dynamical systems ^[3].

In standard Bayesian inference the posterior distribution is given by

${\displaystyle P(\theta |D)\propto P(D|\theta )\pi (\theta )}$

where ${\displaystyle \theta }$ are the parameters of a probability model, ${\displaystyle D}$ are the observed data, and ${\displaystyle \pi (\theta )}$ is the prior distribution of the parameters ${\displaystyle \theta }$. ${\displaystyle P(D|\theta )}$ is the likelihood of ${\displaystyle \theta }$, that is the probability of observing the data ${\displaystyle D}$ given the model with parameter ${\displaystyle \theta }$. The explicit evaluation of the likelihood is avoided in ABC approaches by considering distances between observed and data simulated from a model with parameter ${\displaystyle \theta }$. For sufficiently complex models and large data sets the probability of happening upon a simulation run that yields precisely the same dataset as the one observed will be very small, often unacceptably so. So rather than considering the data we consider a summary statistic of the data, ${\displaystyle S(D)}$, and use a distance ${\displaystyle \Delta (S(D),S(X))}$ between the summary statistics of real and simulated data, ${\displaystyle D}$ and ${\displaystyle X}$, respectively.

The generic ABC approach to infer the posterior probability of a parameter ${\displaystyle \theta }$ is as follows:

1. Sample a candidate parameter vector ${\displaystyle \theta ^{\ast }}$ from some proposal distribution ${\displaystyle \pi (\theta )}$.

2. Simulate a dataset ${\displaystyle X}$ from the model with parameter ${\displaystyle \theta ^{\ast }}$.

3. If ${\displaystyle \Delta (S(D),S(X))<\epsilon }$ then accept ${\displaystyle \theta ^{\ast }}$ as a sample from the posterior.

For ${\displaystyle \epsilon }$ sufficiently small the ABC procedure should deliver a good approximation to the true posterior, in particular if the summary statistic ${\displaystyle S}$ is a sufficient statistic of the probability model. If sufficient statistics do not exist or are hard to come by, setting up a satisfying and efficient ABC approach can be challenging.

The generic procedure outlined above can be computationally inefficient but ABC and likelihood-free inferential procedures can be combined with the standard computational approaches used in Bayesian inference such as Markov chain Monte Carlo ^{[4] [5]} and Sequential Monte Carlo method ^[3]approaches. In these frameworks ABC can be used to tackle otherwise computationally intractable problems.

An increasing number of software implementations of ABC approaches exist ^[6].

See also

Markov chain Monte Carlo

Sequential Monte Carlo Method

References

1. Pritchard, J. K. (1999). "Population Growth of Human Y Chromosomes: A Study of Y Chromosome Microsatellites". Mol. Biol. Evol. 16: 1791–1798. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
2. Beaumont, M. A. (Dec 2002). "Approximate Bayesian Computation in Population Genetics". Genetics. 162: 2025–2035. PMID 12524368. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
3. Toni, T.; Welch, D.; Strelkowa, N.; Ipsen, A.; Stumpf, M.P.H. (2009). "Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems". Journal of the Royal Society Interface. 6 (31): 187–202. {{cite journal}}: Text "doi: 10.1098/​rsif.2008.0172" ignored (help)
4. Marjoram, P. (2003). "Markov chain Monte Carlo without likelihoods". P Natl Acad Sci USA. 100 (26): 15324–15328. doi:10.1073/pnas.0306899100. PMID 14663152. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
5. Plagnol, V. (2004). "Approximate Bayesian computation and MCMC" (PDF). Monte Carlo and Quasi-Monte Carlo Methods 2002. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help) (The link is to a preprint.)
6. Cornuet, J-M. (2008). "Inferring population history with DIY ABC: a user-friendly approach to Approximate Bayesian Computation". Bioinformatics. doi:10.1093/bioinformatics/btn514. PMID 18842597. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

Softwares

DIYABC : "Do it yourself ABC".