After missing the blood moon eclipse last night, I had a meeting today at the Paris observatory (IAP), where we discussed an ABC proposal made by Tom Charnock, Guilhem Lavaux, and Benjamin Wandelt from this institute.
“We introduce a simulation-based machine learning technique that trains artificial neural networks to find non-linear functionals of data that maximise Fisher information : information maximising neural networks.” T. Charnock et al., 2018
The paper is centred on the determination of “optimal” summary statistics. With the goal of finding “transformation which maps the data to compressed summaries whilst conserving Fisher information [of the original data]”. Which sounds like looking for an efficient summary and hence impossible in non-exponential cases. As seen from the description in (2.1), the assumed distribution of the summary is Normal, with mean μ(θ) and covariance matrix C(θ) that are implicit transforms of the parameter θ. In that respect, the approach looks similar to the synthetic likelihood proposal of Wood (2010). From which an unusual form of Fisher information can be derived, as μ(θ)’C(θ)⁻¹μ(θ)… A neural net is trained to optimise this information criterion at a given (so-called fiducial) value of θ, in terms of a set of summaries of the same dimension as the data. Which means the information contained in the whole data (likelihood) is not necessarily recovered, linking with this comment from Edward Ionides (in a set of lectures at Wharton).
“Even summary statistics derived by careful scientific or statistical reasoning have been found surprisingly uninformative compared to the whole data likelihood in both scientific investigations (Shrestha et al., 2011) and simulation experiments (Fasiolo et al., 2016)” E. Ionides, slides, 2017
The maximal Fisher information obtained in this manner is then used in a subsequent ABC step as the natural metric for the distance between the observed and simulated data. (Begging the question as to why being maximal is necessarily optimal.) Another question is about the choice of the fiducial parameter, which choice should be tested by for instance iterating the algorithm a few steps. But having to run simulations for a single value of the parameter is certainly a great selling point!