# Indirect inference

A.k.a the auxiliary method. AFAICT the same thing as simplified synthetic likelihoods. Maybe the same thing as simulation-based inference.

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.

Surely those conditions in themselves don’t necessarily rule out Bayesian methods? Is that not what Approximate Bayes does? Anyway,

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.

One wonders whether the optimal observations can be learned from the data. This is what learning summary statistics aims to do.

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