# Indirect inference

A.k.a the “auxiliary method”. Also possibly a.k.a Approximate Bayesian Computation? (Should work out if that is so.)

Here be economists and ecologists.

Maybe this will solve my current weird intractable model issues?

There is an R package for at least some versions of it: pomp

Quoting Cosma:

your model is too complicated for you to appeal to any of the usual estimation methods of statistics. there is no way to even calculate the likelihood of a given data set $$x_1,x_2,…x_t\equiv x_t$$ under parameters $$\theta$$ in closed form, which would rule out even numerical likelihood maximization, to say nothing of Bayesian methods Yet you can simulate; it seems like there should be some way of saying whether the simulations look like the data.

This is where indirect inference comes in Introduce a new model, called the “auxiliary model”, which is mis-specified and typically not even generative, but is easily fit to the data, and to the data alone. (By that last I mean that you don’t have to impute values for latent variables, etc., etc., even though you might know those variables exist and are causally important.) The auxiliary model has its own parameter vector $$\beta$$, with an estimator $$\hat{\beta}$$. These parameters describe aspects of the distribution of observables, and the idea of indirect inference is that we can estimate the generative parameters $$\theta$$ by trying to match those aspects of observations, by trying to match the auxiliary parameters.

Surely those conditions in themselves don’t necessarily rule out Bayesian methods? But anyway that’s the drift. I clearly have more to learn here.

Aaron King’s lab at UMichigan does a lot of this, although they do even more recursive estimation, which seems less insane for my purposes.

Babtie, Ann C., Paul Kirk, and Michael P. H. Stumpf. 2014. “Topological Sensitivity Analysis for Systems Biology.” Proceedings of the National Academy of Sciences 111 (52): 18507–12. https://doi.org/10.1073/pnas.1414026112.

Batz, Philipp, Andreas Ruttor, and Manfred Opper. 2017. “Approximate Bayes Learning of Stochastic Differential Equations,” February. http://arxiv.org/abs/1702.05390.

Bretó, Carles, Daihai He, Edward L. Ionides, and Aaron A. King. 2009. “Time Series Analysis via Mechanistic Models.” The Annals of Applied Statistics 3 (1): 319–48. https://doi.org/10.1214/08-AOAS201.

Cauchemez, Simon, and Neil M. Ferguson. 2008. “Likelihood-Based Estimation of Continuous-Time Epidemic Models from Time-Series Data: Application to Measles Transmission in London.” Journal of the Royal Society Interface 5 (25): 885–97. https://doi.org/10.1098/rsif.2007.1292.

Clark, James S., and Ottar N. Bjørnstad. 2004. “Population Time Series: Process Variability, Observation Errors, Missing Values, Lags, and Hidden States.” Ecology 85 (11): 3140–50. https://doi.org/10.1890/03-0520.

Commandeur, Jacques J. F., Siem Jan Koopman, and Marius Ooms. 2011. “Statistical Software for State Space Methods.” Journal of Statistical Software 41 (1). https://doi.org/10.18637/jss.v041.i01.

Cook, Alex R., Wilfred Otten, Glenn Marion, Gavin J. Gibson, and Christopher A. Gilligan. 2007. “Estimation of Multiple Transmission Rates for Epidemics in Heterogeneous Populations.” Proceedings of the National Academy of Sciences 104 (51): 20392–7. https://doi.org/10.1073/pnas.0706461104.

Cox, D. R., and Christiana Kartsonaki. 2012. “The Fitting of Complex Parametric Models.” Biometrika 99 (3): 741–47. https://doi.org/10.1093/biomet/ass030.

Creel, Michael, and Dennis Kristensen. 2012. “Estimation of Dynamic Latent Variable Models Using Simulated Non-Parametric Moments.” The Econometrics Journal 15 (3): 490–515. https://doi.org/10.1111/j.1368-423X.2012.00387.x.

———. 2013. “Indirect Likelihood Inference (Revised).” UFAE and IAE Working Paper 931.13. Unitat de Fonaments de l’Anàlisi Econòmica (UAB) and Institut d’Anàlisi Econòmica (CSIC). https://ideas.repec.org/p/aub/autbar/931.13.html.

Czellar, Veronika, and Elvezio Ronchetti. 2010. “Accurate and Robust Tests for Indirect Inference.” Biometrika 97 (3): 621–30. https://doi.org/10.1093/biomet/asq040.

Dridi, Ramdan, Alain Guay, and Eric Renault. 2007. “Indirect Inference and Calibration of Dynamic Stochastic General Equilibrium Models.” Journal of Econometrics, The interface between econometrics and economic theory, 136 (2): 397–430. https://doi.org/10.1016/j.jeconom.2005.11.003.

Efron, Bradley. 2010. “The Future of Indirect Evidence.” Statistical Science 25 (2): 145–57. https://doi.org/10.1214/09-STS308.

Forneron, Jean-Jacques, and Serena Ng. 2015. “The ABC of Simulation Estimation with Auxiliary Statistics,” January. http://arxiv.org/abs/1501.01265.

Gallant, A. Ronald, David Hsieh, and George Tauchen. 1997. “Estimation of Stochastic Volatility Models with Diagnostics.” Journal of Econometrics 81 (1): 159–92. https://doi.org/10.1016/S0304-4076(97)00039-0.

Gallant, A. Ronald, and George Tauchen. 1996. “Which Moments to Match?” Econometric Theory 12 (04): 657–81. https://doi.org/10.1017/S0266466600006976.

———. 1997. “Estimation of Continuous-Time Models for Stock Returns and Interest Rates.” Macroeconomic Dynamics 1 (01): 135–68. https://doi.org/10.1017/S1365100597002058.

Genton, Marc G, and Elvezio Ronchetti. 2003. “Robust Indirect Inference.” Journal of the American Statistical Association 98 (461): 67–76. https://doi.org/10.1198/016214503388619102.

Gourieroux, Christian, and Alain Monfort. 1993. “Simulation-Based Inference: A Survey with Special Reference to Panel Data Models.” Journal of Econometrics 59 (1–2): 5–33. https://doi.org/10.1016/0304-4076(93)90037-6.

Gourieroux, C., A. Monfort, and E. Renault. 1993. “Indirect Inference.” Journal of Applied Econometrics 8 (December): S85–S118.

He, Daihai, Edward L. Ionides, and Aaron A. King. 2010. “Plug-and-Play Inference for Disease Dynamics: Measles in Large and Small Populations as a Case Study.” Journal of the Royal Society Interface 7 (43): 271–83. https://doi.org/10.1098/rsif.2009.0151.

Ionides, Edward L., Anindya Bhadra, Yves Atchadé, and Aaron King. 2011. “Iterated Filtering.” The Annals of Statistics 39 (3): 1776–1802. https://doi.org/10.1214/11-AOS886.

Ionides, E. L., C. Bretó, and A. A. King. 2006. “Inference for Nonlinear Dynamical Systems.” Proceedings of the National Academy of Sciences 103 (49): 18438–43. https://doi.org/10.1073/pnas.0603181103.

Jiang, Wenxin, and Bruce Turnbull. 2004. “The Indirect Method: Inference Based on Intermediate Statistics—A Synthesis and Examples.” Statistical Science 19 (2): 239–63. https://doi.org/10.1214/088342304000000152.

Kendall, Bruce E., Stephen P. Ellner, Edward McCauley, Simon N. Wood, Cheryl J. Briggs, William W. Murdoch, and Peter Turchin. 2005. “Population Cycles in the Pine Looper Moth: Dynamical Tests of Mechanistic Hypotheses.” Ecological Monographs 75 (2): 259–76. http://www.sysecol2.ethz.ch/Refs/EntClim/K/Ke169.pdf.

Nickl, Richard, and Benedikt M. Pötscher. 2009. “Efficient Simulation-Based Minimum Distance Estimation and Indirect Inference.” Mathematical Methods of Statistics 19, August, 327–64. http://arxiv.org/abs/0908.0433.

Roberts, G. O., and O. Stramer. 2001. “On Inference for Partially Observed Nonlinear Diffusion Models Using the Metropolis–Hastings Algorithm.” Biometrika 88 (3): 603–21. https://doi.org/10.1093/biomet/88.3.603.

Smith, A. A. 1993. “Estimating Nonlinear Time-Series Models Using Simulated Vector Autoregressions.” Journal of Applied Econometrics 8 (S1): S63–S84. https://doi.org/10.1002/jae.3950080506.

Smith, A A. 2008. “Indirect Inference.” In The New Palgrave Dictionary of Economics. Palgrave Macmillan. http://www.econ.yale.edu/smith/palgrave7.pdf.

Wood, Simon N. 2010. “Statistical Inference for Noisy Nonlinear Ecological Dynamic Systems.” Nature 466 (7310): 1102–4. https://doi.org/10.1038/nature09319.