Since I’ve been a bit distracted by travel lately I haven’t had time to blog about a large number of recent papers that have been hanging around as open tabs on my browser; so to clear the decks I’ve decided to just make a few short comments about each.
“Bayesian inference on the sphere beyond statistical isotropy” by Das et al. describes a Hamiltonian MCMC sampler for exploring the posterior of the CMB power spectrum with marginalisation over departures from statistical isotropy. This is achieved by representing the covariance matrix of the underlying CMB signal with the so-called “BipoSH” representation, summing spherical harmonics to a given order set by the informative limit of the pixelisation scheme. This sort of ‘non-parametric’ (my words) approach to inferring the covariance matrix of a Gaussian process makes an interesting contrast to the parametric approach more commonly used outside of cosmology (and perhaps medical imaging) in which optimisation or marginalisation is with respect to the parameters of e.g. a Matern covariance function. For our purposes at MAP this approach would be unsuitable because our target of interest is strictly pixels on the continental land mass (i.e. dealing with ocean pixels runs into memory limitations) and the observations are taken at irregular points rather than a regular array of pixels. However, even if it were suitable I would worry about two aspects: (i) the limitations for estimating posterior functionals that might depend on small scale fluctuations in the field (i.e., without a parametric covariance function we could not make any useful inferences below the scale of pixelization; and (ii) the lack of intuitive control on the regularisation (i.e., it is much more difficult to imagine the impact a given choice of priors [in the Das et al. study: uniform / improper] on the coefficients in the ‘non-parametric’ case will have on prior distribution of random fields [and hence the regularisation in functional space] than for familiar parametric covariance functions). For the data-rich CMB reconstruction task these issues don’t seem to be problematic though.
I noticed the following confusion presumably inspired by the quantiles equivalent to 1-, 2-, and 3-sigma limits (not mentioned in the text) in Piedipalumbo et al.: “the confidence levels are estimated from the final sample (the merged chain): “the 15.87-th and 84.13-th quantiles define the 68% confidence interval; the 2.28-th and 97.72-th quantiles define the 95% confidence interval; and the 0.13-th and 99.87-th quantiles define the 99% confidence interval.” More like 99.74 …
I was disappointed to see unnecessary binning (of a 44 object sample) and the smearing of (rather than deconvolution of) redshift uncertainties in Strolger et al.’s paper on “The rate of core collapse supernovae to redshift 2.5 from the CANDELS and CLASH supernova surveys”. Although it’s a common way of dealing with the uncertainties in the outputs from photometric redshift routines, it’s clearly sub-optimal to set a single estimate of the rate in a given redshift bin from the sum of fractional contributions to that bin from each SNe. Introducing almost any kind of model (whether parametric or non-parametric) imposing a degree of regularisation on the variation of the rate as a function of redshift will bring the fits back towards deconvolution.
I see that semi-parametric schemes for errors-in-variables regression are finally coming to astronomy as I’d hoped; in “A Gibbs Sampler for Multivariate Linear Regression” Adam Mantz describes an R package for posterior simulation of a Normal linear regression with uncertainties in the covariates which treats the latent covariate distribution with a Dirichlet process mixture model a la West et al. 1994. In principle, the same Gibbs sampling framework could be expanded to various generalised linear models (like probit regression; we used a DP-based probit regression between two binomially-distributed variables in Bonnie’s recently submitted paper [only poster abstract available at present]), although ultimately it would make sense for more astronomers to be able to code up this kind of hierarchical model in JAGS or similar so as to have the greatest possible flexibility to deal with the quirks of their data. One other comment I have is that one shouldn’t get too caught up with the number of unique components assigned to the observed data with these models, since in any case the number of components in the DP is infinite and the number of assigned components is not a consistent estimator.