The Living Thing / Notebooks :

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.

Read this stuff:

Arvanitis, S. (2014) A simple example of an indirect estimator with discontinuous limit theory in the MA(1) model. Journal of Time Series Analysis, 35(6), 536–557. DOI.
Bretó, C., He, D., Ionides, E. L., & King, A. A.(2009) Time series analysis via mechanistic models. The Annals of Applied Statistics, 3(1), 319–348. DOI.
Cauchemez, S., & Ferguson, N. M.(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.
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Cook, A. R., Otten, W., Marion, G., Gibson, G. J., & Gilligan, C. A.(2007) Estimation of multiple transmission rates for epidemics in heterogeneous populations. Proceedings of the National Academy of Sciences, 104(51), 20392–20397. DOI.
Cox, D. R., & Kartsonaki, C. (2012) The fitting of complex parametric models. Biometrika, 99(3), 741–747. DOI.
Creel, M., & Kristensen, D. (2012) Estimation of dynamic latent variable models using simulated non-parametric moments. The Econometrics Journal, 15(3), 490–515. DOI.
Creel, M., & Kristensen, D. (2013) Indirect Likelihood Inference (revised) (UFAE and IAE Working Paper No. 93113). . Unitat de Fonaments de l’Anàlisi Econòmica (UAB) and Institut d’Anàlisi Econòmica (CSIC)
Czellar, V., & Ronchetti, E. (2010) Accurate and robust tests for indirect inference. Biometrika, 97(3), 621–630. DOI.
Dridi, R., Guay, A., & Renault, E. (2007) Indirect inference and calibration of dynamic stochastic general equilibrium models. Journal of Econometrics, 136(2), 397–430. DOI.
Efron, B. (2010) The Future of Indirect Evidence. Statistical Science, 25(2), 145–157. DOI.
Fackler, P., & Tastan, H. (2008) A framework for indirect inference. . Working paper
Forneron, J.-J., & Ng, S. (2015) The ABC of Simulation Estimation with Auxiliary Statistics. arXiv:1501.01265 [stat].
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Gallant, A. R., & Tauchen, G. (1996) Which Moments to Match?. Econometric Theory, 12(04), 657–681. DOI.
Gallant, A. R., & Tauchen, G. (1997) Estimation of continuous-time models for stock returns and interest rates. Macroeconomic Dynamics, 1(01), 135–168. DOI.
Genton, M. G., & Ronchetti, E. (2003) Robust Indirect Inference. Journal of the American Statistical Association, 98(461), 67–76. DOI.
Gibson, G. J., & Renshaw, E. (2001) Likelihood estimation for stochastic compartmental models using Markov chain methods. Statistics and Computing, 11(4), 347–358. DOI.
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Gourieroux, C., Monfort, A., & Renault, E. (1993) Indirect Inference. Journal of Applied Econometrics, 8, S85–S118.
He, D., Ionides, E. L., & King, A. A.(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.
Ionides, E. L., Bhadra, A., Atchadé, Y., & King, A. (2011) Iterated filtering. The Annals of Statistics, 39(3), 1776–1802. DOI.
Ionides, E. L., Bretó, C., & King, A. A.(2006) Inference for nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 103(49), 18438–18443. DOI.
Jiang, W., & Turnbull, B. (2004) The Indirect Method: Inference Based on Intermediate Statistics—A Synthesis and Examples. Statistical Science, 19(2), 239–263. DOI.
Kendall, B. E., Ellner, S. P., McCauley, E., Wood, S. N., Briggs, C. J., Murdoch, W. W., & Turchin, P. (2005) Population cycles in the pine looper moth: Dynamical tests of mechanistic hypotheses. Ecological Monographs, 75(2), 259–276.
Nickl, R., & Pötscher, B. M.(2009) Efficient Simulation-Based Minimum Distance Estimation and Indirect Inference. Mathematical Methods of Statistics 19, 327–364.
Roberts, G. O., & Stramer, O. (2001) On inference for partially observed nonlinear diffusion models using the Metropolis–Hastings algorithm. Biometrika, 88(3), 603–621. DOI.
Shrestha, S., King, A. A., & Rohani, P. (2011) Statistical Inference for Multi-Pathogen Systems. PLoS Comput Biol, 7(8), e1002135. DOI.
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Smith, A. A.(2008) Indirect Inference. In The New Palgrave Dictionary of Economics. Palgrave Macmillan
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Zhao, L. (2010) A model of limit-order book dynamics and a consistent estimation procedure. . Citeseer