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I have set out to implement a simple ABC rejection sampling algorithm in order to approximate the posterior distribution of parameters for Lotka-Volterra system and I have a few questions:

1) What kind of prior would one impose on the parameters of LV model? Is a uniform distribution of parameters a reasonable choice? I understand that i'd need something a bit more powerful than rejection sampling approach (e.g. ABC SMC).

2) What would be a good choice of summary statistic and a distance measure for acceptance decision? I can imagine that KL distance could be a candidate but, from experience, what works best?

3) How do we "sample" from the model? The first thing that comes to mind is using Gillespie algorithm but i can't be sure.

Thank you!

ambushed
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1 Answers1

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  1. Prior choice for ABC is not different than for standard Bayesian analysis - if your parameters act strongly nonlinear on the likelihood, uniform may not be a sensible choice. See, e.g. Kass, R. E. & Wasserman, L. (1996) The selection of prior distributions by formal rules. J. Am. Stat. Assoc., American Statistical Association, 91, 1343-1370. A sensitivity analysis may be useful.

  2. Impossible to say if you don't say what data you have - in any case, you typically want to have several summary statistics. You could have a look at the summary statistics tested in Wood, S. N. (2010) Statistical inference for noisy nonlinear ecological dynamic systems. Nature, 466, 1102-1104, this is a fairly similar setup.

  3. You need a stochastic model - standard LV is not stochastic, you will need to introduce some stochasticity in the equations.

Florian Hartig
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  • Thank you for your input! I will check out the references. As for your answer to question 3: true, this is where i am planning to use a stochastic system of rate equations with corresponding propensities and use Gillespie to draw samples inspired by R. Erban, et al "A practical guide to stochastic simulation of reaction diffusion processes". I am just not 100% sure it is the way to go. – ambushed Apr 03 '16 at 21:36
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    You might also want to have a look at this paper: http://arxiv.org/abs/1403.6886. They use ABC to fit LV. As for summaries, you could probably use those of Wood (2010) (they are implemented by the synlik R package if you are interested), but if both prey and predator densities are observed you will need to add statistics measuring their dependence (for instance vector autoregressive coefficients?). – Matteo Fasiolo Apr 03 '16 at 21:58
  • As for the data, i will fake it for now. Choose a parameter set, sample with it and try to see how close I can get to the posterior with ABC. Nothing fancy. – ambushed Apr 03 '16 at 21:59
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    @ambushed / regarding 3 - if you just want to play around I see no runtime issue with simply doing the LV as a stochastic difference equation, as in Np (t+1) = f(Np, Npre) + stochasticity ... and so on. – Florian Hartig Apr 04 '16 at 16:27
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    I would suggest looking at [Toni et al. (2009) "Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems" J Roy Soc Inferface. 6. (doi: 10.1098/rsif.2008.0172)](http://rsif.royalsocietypublishing.org/content/6/31/187) as they illustrate this problem (choosing parameters for LV based upon noisey data) and compare ABC-rejection, ABC-MCMC, and ABC-SMC. In response to your questions: 1) they use a uniform prior for parameters; 2) they use SSE; 3) they first solve deterministic LV with known parameters and add Gaussian noise. – p-robot Oct 14 '16 at 11:33