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Description
Agent-based models (ABMs) are increasingly used to study complex systems in biology, enabled by advances in computing and the growing availability of high-resolution data tracking individual agents. However, fitting ABMs to data remains challenging because their likelihood functions are typically intractable, making standard statistical methods such as maximum likelihood estimation or Markov chain Monte Carlo difficult to apply. Likelihood-free inference approaches, including approximate Bayesian computation and synthetic likelihood \cite{Wood_2010, Price_Drovandi_Lee_Nott_2018}, address this issue by relying on model simulations. Recent work has explored replacing these simulations with learned emulators \cite{Jarvenpaa_Corander_2024, Meeds_Welling_2014}. In this work, we exploit the differentiability of such an emulator within the synthetic likelihood framework, enabling efficient parameter inference using Hamiltonian Monte Carlo with the No-U-Turn Sampler \cite{Hoffman_Gelman_2014}. Using toy models with known likelihoods, we demonstrate the empirical accuracy of our approach. We further show that it improves parameter inference in complex ABMs compared to common likelihood-free inference methods that do not provide a gradient with respect to the parameters, and we investigate the asymptotic accuracy of the proposed emulator.
Bibliography
@article{Hoffman_Gelman_2014, title={The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo}, volume={15}, ISSN={1533-7928}, url={http://jmlr.org/papers/v15/hoffman14a.html}, abstractNote={Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo (MCMC) algorithm that avoids the random walk behavior and sensitivity to correlated parameters that plague many MCMC methods by taking a series of steps informed by first-order gradient information. These features allow it to converge to high-dimensional target distributions much more quickly than simpler methods such as random walk Metropolis or Gibbs sampling. However, HMC’s performance is highly sensitive to two user-specified parameters: a step size $epsilon$ and a desired number of steps $L$. In particular, if $L$ is too small then the algorithm exhibits undesirable random walk behavior, while if $L$ is too large the algorithm wastes computation. We introduce the No-U-Turn Sampler (NUTS), an extension to HMC that eliminates the need to set a number of steps $L$. NUTS uses a recursive algorithm to build a set of likely candidate points that spans a wide swath of the target distribution, stopping automatically when it starts to double back and retrace its steps. Empirically, NUTS performs at least as efficiently as (and sometimes more efficiently than) a well tuned standard HMC method, without requiring user intervention or costly tuning runs. We also derive a method for adapting the step size parameter $epsilon$ on the fly based on primal-dual averaging. NUTS can thus be used with no hand-tuning at all, making it suitable for applications such as BUGS-style automatic inference engines that require efficient samplers.}, number={47}, journal={Journal of Machine Learning Research}, author={Hoffman, Matthew D. and Gelman, Andrew}, year={2014}, pages={1593–1623} }
@article{Jarvenpaa_Corander_2024, title={Approximate Bayesian inference from noisy likelihoods with Gaussian process emulated MCMC}, volume={25}, ISSN={1533-7928}, url={http://jmlr.org/papers/v25/21-0421.html}, abstractNote={We present a framework for approximate Bayesian inference intended for a situation where only a limited number of noisy log-likelihood evaluations can be obtained due to constraints on the available computational budget, which is becoming increasingly common for expensive simulator-based models. We model the log-likelihood function using a Gaussian process (GP) and our main methodological innovation is to apply this model to emulate the progression that an exact Metropolis-Hastings (MH) sampler would take if it was applicable. Informative log-likelihood evaluation locations are selected using a sequential experimental design strategy until the MH accept/reject decisions are performed with sufficient level of accuracy based on a prespecified error tolerance criterion. The resulting approximate sampler is conceptually simple and shown to be sample-efficient. It is also more robust compared with earlier “Bayesian optimisation-like” methods tailored for approximate Bayesian inference, which generally assume a global surrogate model across the parameter space that can be challenging to fit well. We discuss some theoretical aspects and various interpretations of the resulting approximate MH sampler, and demonstrate its benefits in the context of Bayesian and generalised Bayesian likelihood-free inference for simulator-based statistical models.}, number={366}, journal={Journal of Machine Learning Research}, author={Järvenpää, Marko and Corander, Jukka}, year={2024}, pages={1–55} }
@article{Price_Drovandi_Lee_Nott_2018, title={Bayesian Synthetic Likelihood}, volume={27}, ISSN={1061-8600, 1537-2715}, url={https://www.tandfonline.com/doi/full/10.1080/10618600.2017.1302882}, DOI={10.1080/10618600.2017.1302882}, number={1}, journal={Journal of Computational and Graphical Statistics}, author={Price, L. F. and Drovandi, C. C. and Lee, A. and Nott, D. J.}, year={2018}, month=jan, pages={1–11}, language={en} }
@article{Wood_2010, title={Statistical inference for noisy nonlinear ecological dynamic systems}, volume={466}, rights={http://www.springer.com/tdm}, ISSN={0028-0836, 1476-4687}, url={https://www.nature.com/articles/nature09319}, DOI={10.1038/nature09319}, number={7310}, journal={Nature}, author={Wood, Simon N.}, year={2010}, month=aug, pages={1102–1104}, language={en} }
@article{Meeds_Welling_2014, title={GPS-ABC: Gaussian Process Surrogate Approximate Bayesian Computation}, url={http://arxiv.org/abs/1401.2838}, DOI={10.48550/arXiv.1401.2838}, abstractNote={Scientists often express their understanding of the world through a computationally demanding simulation program. Analyzing the posterior distribution of the parameters given observations (the inverse problem) can be extremely challenging. The Approximate Bayesian Computation (ABC) framework is the standard statistical tool to handle these likelihood free problems, but they require a very large number of simulations. In this work we develop two new ABC sampling algorithms that significantly reduce the number of simulations necessary for posterior inference. Both algorithms use confidence estimates for the accept probability in the Metropolis Hastings step to adaptively choose the number of necessary simulations. Our GPS-ABC algorithm stores the information obtained from every simulation in a Gaussian process which acts as a surrogate function for the simulated statistics. Experiments on a challenging realistic biological problem illustrate the potential of these algorithms.}, note={arXiv:1401.2838}, number={arXiv:1401.2838}, publisher={arXiv}, author={Meeds, Edward and Welling, Max}, year={2014}, month=jan }
