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Approximate Bayesian Computation has been suggested as an approach to parameter estimation for computationally intensive simulations, most commonly in population genetics, but also in dynamical systems, for example, Toni 2009 and applied in the corresponding abc-sysbio tool pubished in nat methods.

In a dynamical systems model, the likelihood is expensive to calculate because it requires running a simulation to evaluate the likelihood. However, unlike stochastic population genetics model, one could reasonably obtain a likelihood from a dynamical systems model by imagining a probabilistic representation of the RMSE (e.g. treating errors as normally distributed) and obtain a likelihood from a single simulation.

So in both exact and approximate bayesian computation, each likelihood calculation entails running a single ODE simulation, how is it that ABC is computationally advantageous as compared to exact bayesian computation approaches? Where does the speedup come from?

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    $\begingroup$ If you can compute the likelihood function at the same cost as for the ABC computation, there is no debate as to which method to use: go for the exact Bayesian solution. $\endgroup$ – Xi'an Feb 23 '15 at 16:45
  • $\begingroup$ do you have any comments on the Toni paper? Maybe I'm not understanding something, but the distance functions require ODE integration and it seems to me there's not much of a computational savings in using SSE relative to explicitly evaluating the actual likelihood using a gaussian noise model. Why not apply SMC using the full likelihood? $\endgroup$ – user4733 Feb 23 '15 at 17:12
  • $\begingroup$ Related stats.stackexchange.com/questions/283113/… $\endgroup$ – Tim Jun 19 '17 at 12:42
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It looks like it is similar to HMC in that it uses a type of stochastic gradient instead of a random walk. I didn't see anywhere that they say they could do the estimate based on a single simulation. Instead, it looks like you can simply get good values much faster. It really looks very similar to HMC as it's implemented in Stan. I would actually recommend Stan, since it looks like it has better documentation and there are some books on it currently.

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  • $\begingroup$ i tried stan and i'm generally a fan when it comes to hierarchical models. maybe i did something wrong, but it managed to freeze up trying to solve a single first order equation for an exponential decay. i think the sequential monte carlo aspect of the paper is interesting. i just don't understand the use of ABC when you need to solve the ODE in order to compute the summary statistic. $\endgroup$ – user4733 Feb 26 '15 at 4:41
  • $\begingroup$ Stan is capable of handling differential equation models, though I believe it can be tricky to implement them. You might consider looking around or asking questions on the stan-users mailing list; a lot of the developers hang around there and are often happy to provide answers or support. groups.google.com/forum/#!forum/stan-users $\endgroup$ – C.R. Peterson Dec 3 '15 at 17:06

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