Reconstructing an X-ray luminosity function from noisy data …

Last week I visited Basel (again!) to give an overview talk, entitled ‘a quantitative history of malaria in maps‘, at the STPH Winter Symposium.  The subsequent speakers were all very interesting and I learnt a lot about new vector control methods (e.g. eave traps) and the search for new anti-malarial compounds by brute force testing of 20,000 organic compounds from the Novartis ‘library’.  Back in Oxford now with a spare five minutes to mention a paper I read a couple of weeks ago …

In this arXival by Aird et al. the authors use a Bayesian model to reconstruct the X-ray luminosity functions of various populations of star formation galaxies.  The challenge is that the observed X-ray emission from the pointing to each galaxy is a mix of emission from a background of uncertain intensity and perhaps some photons from the target itself.  The galaxy X-ray luminosity function the authors aim to estimate thus appears as a latent function intermediate in their hierarchical model below the top level of the observed likelihood by which it is in places only weakly constrained.  The model adopted for this latent function is similar to that used by Warren et al. in their solar corona study (which I’ve blogged about here recently): namely, a mixture of regular functions (here gamma densities) spaced evenly on a grid.  (Similar also to the NPMLE mixture model set up of Feng & Dicker 2016.)  The prior on their weights is taken to be something like a moving average model with sum to one constraint.

Importantly, the authors recognise that the success of this type of Bayesian inversion problem is ultimately heavily dependent on the suitability of the chosen prior and model structure for the situation at hand, which they therefore test extensively with mock datasets drawn from a range of plausible luminosity functions outside their prior class of functions.  While this raises some theoretical questions as to how one can have a prior that doesn’t encompass all of one’s prior belief about the possible set of luminosity functions one might find in the wild, in practice this is a very common situation in that one often has to choose a model that can be readily mathematically described and fit to the data and simply remember going forwards that it is indeed only a model.  Hence, I’m very much in favour of such model testing procedures.

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