## another instance of ABC?

“These characteristics are (1) likelihood is not available; (2) prior information is available; (3) a portion of the prior information is expressed in terms of functionals of the model that cannot be converted into an analytic prior on model parameters; (4) the model can be simulated. Our approach depends on an assumption that (5) an adequate statistical model for the data are available.”

**A** 2009 JASA paper by Ron Gallant and Rob McCulloch, entitled “On the Determination of General Scientific Models With Application to Asset Pricing”, may have or may not have connection with ABC, to wit the above quote, but I have trouble checking whether or not this is the case.

The true (scientific) model parametrised by θ is replaced with a (statistical) substitute that is available in closed form. And parametrised by g(θ). [If you can get access to the paper, I’d welcome opinions about *Assumption 1* therein which states that the intractable density is equal to a closed-form density.] And the latter is over-parametrised when compared with the scientific model. As in, e.g., a N(θ,θ²) scientific model versus a N(μ,σ²) statistical model. In addition, the prior information is only available on θ. However, this does not seem to matter that much since (a) the Bayesian analysis is operated on θ only and (b) the Metropolis approach adopted by the authors involves simulating a massive number of pseudo-observations, given the current value of the parameter θ *and* the scientific model, so that the transform g(θ) can be estimated by maximum likelihood over the statistical model. The paper suggests using a secondary Markov chain algorithm to find this MLE. Which is claimed to be a simulated annealing resolution (p.121) although I do not see the temperature decreasing. The pseudo-model is then used in a primary MCMC step.

Hence, not truly an ABC algorithm. In the same setting, ABC would use a simulated dataset the same size as the observed dataset, compute the MLEs for both and compare them. Faster if less accurate when Assumption 1 [that the statistical model holds for a restricted parametrisation] does not stand.

Another interesting aspect of the paper is about creating and using a prior distribution around the manifold η=g(θ). This clearly relates to my earlier query about simulating on measure zero sets. The paper does not bring a definitive answer, as it never simulates exactly on the manifold, but this constitutes another entry on this challenging problem…

December 2, 2014 at 4:55 am

We have examined this method in more detail in http://www.e-publications.org/ims/submission/STS/user/submissionFile/19008?confirm=38400261. Actually this approach was proposed earlier in Reeves and Pettitt 2005 (in this proceedings http://www.statmod.org/files/proceedings/iwsm2005_proceedings.pdf).

This approach uses the (tractable) likelihood of an auxiliary parametric model as a replacement to the intractable likelihood. The method requires estimating the mapping function between the scientific and statistical model parameters. A large number of independent replicates of the data, n, (or a long simulation of a stationary time series as is the case in the paper) from the scientific model is used to estimate the mapping for each theta generated in the MCMC. We demonstrate that the (approximate) posterior induced by the method is sensitive to n and ultimately, if the auxiliary model is ‘good’, we want to take n to infinity (as opposed to taking epsilon to 0 in ABC).

You are correct though that this is not ABC in the traditional sense since no data reduction is performed and thus there is no comparison of summary statistics as is typical in ABC. For a very accurate approximation the method relies on a rather strong assumption that the auxiliary likelihood acts as a good replacement likelihood in regions of non-negligible posterior support. But given that ABC can be rather crude, it seems a bit naïve to dismiss the method.

It is interesting to note that the (Bayesian version of the) synthetic likelihood method of Wood (2010) is a special case of this framework but where a multivariate normal auxiliary model is applied to an intractable likelihood of a summary statistic (i.e. some data reduction has been performed).