A Gaussian Process Model of Quasar Spectral Energy Distributions
We propose a method for combining two sources of astronomical data, spectroscopy and photometry, that carry information about sources of light (e.g., stars, galaxies, and quasars) at extremely different spectral resolutions. Our model treats the spectral energy distribution (SED) of the radiation from a source as a latent variable that jointly explains both photometric and spectroscopic observations. We place a flexible, nonparametric prior over the SED of a light source that admits a physically interpretable decomposition, and allows us to tractably perform inference. We use our model to predict the distribution of the redshift of a quasar from five-band (low spectral resolution) photometric data, the so called "photoz" problem. Our method shows that tools from machine learning and Bayesian statistics allow us to leverage multiple resolutions of information to make accurate predictions with well-characterized uncertainties.
The paper presents a latent variable model for modeling spectral energy distribution of quasars, given stereoscopic and photometric observations. The joint modeling of stereoscopic and photometric measurements allows the model to make inferences about stereoscopic properties of quasars leveraging the more broadly available photometric data. Clarity: The paper for the most part is well written and easy to follow. I have some minor complaints about the exposition, see detailed comments below. Originality: Moderate. The authors develop a well motivated, non trivial latent variable model for capturing the salient properties of distributions of noisy quasar measurements. The use of parallel tempering in the inference procedure is interesting as well.  Significance: Although an application paper, the work is interesting and could spur further interest among the NIPS community, in tackling these important computational problems in astrophysics. Detailed comments:Model: 1) It would be useful to be explicit about the domains and the priors of the various random variables.      a) Is red shift variable z_n, constrained to take any real value in the domain [0, 7]. Is the prior on z_n, a uniform distribution in this interval?      b) What is the domain of the apparent brightness variable m_n, the positive real line? More importantly what prior was placed on m_n? Line 217 alludes to an empirically derived prior, but no further details are provided.    c) T he Dirichlet distribution is a standard prior on a collection of non negative variables that sum to one.What is the more complicated mixture of log-Gaussian priors over w buying us here? A stronger motivation for this non standard, more complicated choice of prior will be helpful. Inference: I found some of the details of the inference procedure confusing.1) Basis estimation: It appears that the point estimate of the bases is being arrived at by finding a MAP estimate of an auxiliary model, one that ignores the photometric data. Why is the true model posterior which incorporates the photometric likelihood term p (y | w, z, m, B) in equation 10, not optimized instead? The MAP estimates of the two models are obviously different. When inferring w, z, and m, why does it make sense to condition on the MAP basis estimates of the auxiliary model? 2) It is interesting that the authors resort to parallel tempering. It is certainly plausible that a naive MCMC sampler will mix poorly. However, it would be good to clearly demonstrate the need for parallel tempering and show how much worse a single slice sampler chain (with T_c = 1) really is, in terms of red shift predictions. Perhaps an extra row could be added to Figure 5 and an additional line plot in Figure 6 to depict prediction results with and without parallel tempering.Experiments: I like the two data splits with different train and test brightness and redshifts, mimicking real world scenarios.The results could be better explained.1) In figure (5) does the y-axis correspond to shifts inferred by the model? The rightmost scatter plot seems weird. 2) I found figure (6) very confusing.It is never referenced in the text and it isn't clear what is being depicted in it. Are we looking at predicted SED (conditioning on photometric observations) for two held out quasars? Is the noisy line plot in light blue the observed SED for these quasars? What is the mysterious PCA-based model displayed in black, which seems to be fitting the data much better. Is it better because it is "cheating" and has been fit to all available SED's including the two SEDs displayed here?