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.

## Refs

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.

## Refs

- CzRo10: Veronika Czellar, Elvezio Ronchetti (2010) Accurate and robust tests for indirect inference.
*Biometrika*, 97(3), 621–630. DOI - BaRO17: Philipp Batz, Andreas Ruttor, Manfred Opper (2017) Approximate Bayes learning of stochastic differential equations.
*ArXiv:1702.05390 [Physics, Stat]*. - NiPö09: Richard Nickl, Benedikt M. Pötscher (2009) Efficient Simulation-Based Minimum Distance Estimation and Indirect Inference.
*Mathematical Methods of Statistics 19*, 327–364. - Smit93: A. A. Smith (1993) Estimating nonlinear time-series models using simulated vector autoregressions.
*Journal of Applied Econometrics*, 8(S1), S63–S84. DOI - GaTa97: A. Ronald Gallant, George Tauchen (1997) Estimation of continuous-time models for stock returns and interest rates.
*Macroeconomic Dynamics*, 1(01), 135–168. DOI - CrKr12: Michael Creel, Dennis Kristensen (2012) Estimation of dynamic latent variable models using simulated non-parametric moments.
*The Econometrics Journal*, 15(3), 490–515. DOI - COMG07: Alex R. Cook, Wilfred Otten, Glenn Marion, Gavin J. Gibson, Christopher A. Gilligan (2007) Estimation of multiple transmission rates for epidemics in heterogeneous populations.
*Proceedings of the National Academy of Sciences*, 104(51), 20392–20397. DOI - GaHT97: A. Ronald Gallant, David Hsieh, George Tauchen (1997) Estimation of stochastic volatility models with diagnostics.
*Journal of Econometrics*, 81(1), 159–192. DOI - GoMR93: C. Gourieroux, A. Monfort, E. Renault (1993) Indirect Inference.
*Journal of Applied Econometrics*, 8, S85–S118. - Smit08: A A Smith (2008) Indirect Inference. In The New Palgrave Dictionary of Economics. Palgrave Macmillan
- DrGR07: Ramdan Dridi, Alain Guay, Eric Renault (2007) Indirect inference and calibration of dynamic stochastic general equilibrium models.
*Journal of Econometrics*, 136(2), 397–430. DOI - CrKr13: Michael Creel, Dennis Kristensen (2013) Indirect Likelihood Inference (revised) (UFAE and IAE Working Paper No. 931.13). Unitat de Fonaments de l’Anàlisi Econòmica (UAB) and Institut d’Anàlisi Econòmica (CSIC)
- IoBK06: E. L. Ionides, C. Bretó, A. A. King (2006) Inference for nonlinear dynamical systems.
*Proceedings of the National Academy of Sciences*, 103(49), 18438–18443. DOI - IBAK11: Edward L. Ionides, Anindya Bhadra, Yves Atchadé, Aaron King (2011) Iterated filtering.
*The Annals of Statistics*, 39(3), 1776–1802. DOI - CaFe08: Simon Cauchemez, 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), 885–897. DOI - RoSt01: G. O. Roberts, O. Stramer (2001) On inference for partially observed nonlinear diffusion models using the Metropolis–Hastings algorithm.
*Biometrika*, 88(3), 603–621. DOI - HeIK10: Daihai He, Edward L. Ionides, 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), 271–283. DOI - KEMW05: Bruce E. Kendall, Stephen P. Ellner, Edward McCauley, Simon N. Wood, Cheryl J. Briggs, William W. Murdoch, Peter Turchin (2005) Population cycles in the pine looper moth: Dynamical tests of mechanistic hypotheses.
*Ecological Monographs*, 75(2), 259–276. - ClBj04: James S. Clark, Ottar N. Bjørnstad (2004) Population time series: process variability, observation errors, missing values, lags, and hidden states.
*Ecology*, 85(11), 3140–3150. DOI - GeRo03: Marc G Genton, Elvezio Ronchetti (2003) Robust Indirect Inference.
*Journal of the American Statistical Association*, 98(461), 67–76. DOI - GoMo93: Christian Gourieroux, Alain Monfort (1993) Simulation-based inference: A survey with special reference to panel data models.
*Journal of Econometrics*, 59(1–2), 5–33. DOI - Wood10: Simon N. Wood (2010) Statistical inference for noisy nonlinear ecological dynamic systems.
*Nature*, 466(7310), 1102–1104. DOI - CoKO11: Jacques J. F. Commandeur, Siem Jan Koopman, Marius Ooms (2011) Statistical Software for State Space Methods.
*Journal of Statistical Software*, 41(1). DOI - FoNg15: Jean-Jacques Forneron, Serena Ng (2015) The ABC of Simulation Estimation with Auxiliary Statistics.
*ArXiv:1501.01265 [Stat]*. - CoKa12: D. R. Cox, Christiana Kartsonaki (2012) The fitting of complex parametric models.
*Biometrika*, 99(3), 741–747. DOI - Efro10: Bradley Efron (2010) The Future of Indirect Evidence.
*Statistical Science*, 25(2), 145–157. DOI - JiTu04: Wenxin Jiang, Bruce Turnbull (2004) The Indirect Method: Inference Based on Intermediate Statistics—A Synthesis and Examples.
*Statistical Science*, 19(2), 239–263. DOI - BHIK09: Carles Bretó, Daihai He, Edward L. Ionides, Aaron A. King (2009) Time series analysis via mechanistic models.
*The Annals of Applied Statistics*, 3(1), 319–348. DOI - GaTa96: A. Ronald Gallant, George Tauchen (1996) Which Moments to Match?
*Econometric Theory*, 12(04), 657–681. DOI