One can see from this new arXival that errors-in-variables models have not yet become widely known within astronomy yet, though astronomers are trying to find ways to deal with this sort of modelling scenario. The errors-in-variables regression problem occurs when you want to regress against , which would usually be described as with a precisely measured covariate and some kind of model taking input —e.g. linear regression, , or Gaussian process regression, —but unfortunately is now observed with error, so our model now features an extra layer describing the relationship between the true but latent (hidden) and the available, noisily-measured , e.g. . If the error term is substantial then ignoring it (and fitting the base model) leaves us exposed to model misspecification errors. A simple Bayesian solution to this problem is to introduce a further layer describing the population distribution of latent ‘s (one that features hyper-parameters allowing shrinkage is a good choice) and then integrate out all the latent variables via posterior simulation (e.g. MCMC).

The astronomers’ approach here is in fact not to bother adding a model: they simply spread the uncertainty in the out by drawing mock for each data point independently and then take a finely-binned non-parametric estimator. There are some nice advantages of modelling in this context, even if a semi-parametric functional prior (like a Gaussian process) is decided to be used. One of these advantages is that you get a ‘structural shrinkage’ of the noisy ‘s towards values that ‘make sense’ given their corresponding ‘s and the assumed functional form. There are some challenges to fitting such a model in the case of a Gaussian process EIV regression: without a nugget term there are multiple ‘crossing points’ that a sampler moving the ‘s must negotiate at which the covariance matrix becomes non-invertible (i.e., when ‘s tie). A nice solution to this is to use a random Fourier feature representation of the GP.

P.s. One of the canonical examples of EIV regression in astronomy is Kelly et al.; that model can be made fancier in a few fun ways: one is to replace the finite mixture of Normals for the population distribution with an infinite mixture model.

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