Bayesian Statistics as a New* Tool for Spectral Analysis …

*warning your understanding of the word ‘new’ and that of the authors may differ!

I noticed this paper on astro ph today by J-M. Mugnes & C. Robert, the latter I initially thought might be Xian, but indeed it rapidly became apparent on reading the text that these two Bayesians could not be one and the same. I have to say it’s always surprising to me how each tiny subfield of astronomy can support multiple ‘first’ applications of Bayesian techniques, each time with a non-trivial ‘introduction to Bayesian statistics’ section. In describing their ‘new’ approach Mugnes & Robert ignore all past work on Bayesian fitting techniques for spectral lines for general astronomical purposes, noting only a single Bayesian study in the specific field of ‘stellar astronomy with high resolution spectra’ about which they write:

“More recently, Schoenrich & Bergemann (2014) developed a method which allow the combined use of photometric, spectroscopic, and astrometry informations for the determination of stellar parameters and metallicities. Nevertheless, none of these studies were applied on massive stars, nor did they performed a spectral analysis involving more than three parameters at the same time (the spectroscopic part of the work of Schoenrich & Bergemann 2014 focused only on the effective temperature, surface gravity, and global metallicity).”

So, how many parameters do you think are now analysed by Mugnes & Robert? Yeah, you guessed it, a grand total of four parameters! And in fact although the target of the spectroscopic likelihood in Shoenrich & Bergemann is three parameters, they do include additional photometric data with its own parameters and nuisance parameters, so the full dimensionality of their posterior is also greater than three.

Accepting that many astronomers will probably never feel the need to read literature outside of their tiny sub-field, no matter how relevant, I suppose readers of the blog would prefer some more interesting insights. Well, there are some here: it turn out that both Mugnes & Robert and Schoenrich & Bergemann use quite unusual likelihood functions which are better described as ad hoc goodness-of-fit criteria than representations of a sampling distribution approximating the observational process.

In the former the reduced, background subtracted spectral product is assumed to have iid Gaussian noise in each pixel, with a specific group of pixels a priori ascribed to each specral line. In addition to the assumed observational noise there’s also assumed to be an iid Gaussian ‘model noise’ to account for model imperfections which is, bizarrely for a Bayesian analysis, not assigned a prior and marginalized but is set empirically from the RMS under the best-fitting template. Another non-standard feature of Mugnes & Robert’s fit is that since they “want to treat all available lines equally” they interpolate each line (both broad and narrow) to the same number of bins. So ‘bin’ here shouldn’t even really be thought of as representing a pixel or group of pixels.

Schoenrich & Bergemann, on the other hand, introduce a type of ad hockery that I’ve not seen before: limiting the allowed precision of the posterior itself!

“To avoid over-confident estimates, we demand that either the temperature uncertainty σTeff > 80 K or the metallicity uncertainty σ[Fe/H] > 0.08 dex, and otherwise flatten the PDF by multiplying the χ2 distribution with a fixed factor until the condition is met.”

And that after criticising two earlier studies (Burnett & Binney 2010; Casagrande et al. 2011) for “simplifications of the observational likelihoods” …

I wonder if — no, not really, I know that — all of this could be rolled up into one principled approach with a single, simple idea: hierarchical Bayesian modelling.

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4 Responses to Bayesian Statistics as a New* Tool for Spectral Analysis …

  1. hogghogg says:

    Love the word “ad hockery” and use it myself. Am I right that this is a Jaynes-ism?

  2. I learned the noun ad hockery from Jaynes as well.

    FWIW, I’d happily self-identify as a Jaynesian, although I’m more comfortable with openly subjective Bayes than he was. Check out http://www.github.com/eggplantbren/StatisticalCompass

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