Commentary on Chris Genovese’s “Nonparametric inference and the Dark Energy equation of state” Eric Feigelson (Penn State) SCMA IV

Nonparametrics today …. … is far more than the Kolmogorov-Smirnov test & Kendall’s . More than the 2-point correlation function, the Kaplan-Meier estimator, etc. includes “density estimation” techniques: histograms, smoothers, splines, lowess, kriging includes “nonparametric regression” techniques: modeling continuous behavior from discrete data with variance & derivative estimation. Computationally efficient.

Question When should we use parametric models vs. nonparametric methods in astronomy? Note to statisticians: The models I address here are not your familiar heuristic models: linear, polynomial, exponential, Weibull. These Are physical models based on the physical laws of nature: gravity, electromagnetism, quantum mechanics  fluid flows, stellar structure, plasma physics, nuclear astrophysics, concordance models of particle physics & cosmology, etc. Our job as astronomers is to establish the conditions (`parameters’) in which these physical processes are actualized in planets, stars, galaxies and the Universe as a whole.

Historical example #1 Eclipsing binary stars HD : `hot Jupiter’ binary system Periodic brightness variation Periodic radial velocity variation Interesting parameters: a orb, M p, R p Charbonneau et al. 2000

A more complicated case: V505 Sgr Triple, partial eclipsing, tidally distorted, asynchronous rotation, reflection ~36 parameters, least-squares fit Lazaro et al. 2006

Although one can debate the statistics (chisq?), computational procedures (least squares? MCMC?), and model selection criteria (chisq? BIC?), there is no debate regarding the astrophysical model involved in binary star orbits (orbits following Newtonian gravity). There are many problems in astronomy where the link to astrophysical models is clear, and parametric methods are appropriate.

Historical example #2 Elliptical galaxy structure W. Keel, WWW M32, HST

Radial profile of starlight in the elliptical M 32 with King model fit King 1962

A long history of incompatible parametric models of elliptical galaxy radial profiles (These five papers have 3,776 citations)

Hubble’s and King’s models are based on simple physical Interpretation (truncated isothermal sphere). Hernquist & NFW models have more complicated physical interpretation. The de Vaucouleurs model makes no physical sense. But the entire issue of elliptical galaxy structure models was rendered moot by several insights since the 1980s: the observed star distribution does not reflect the distribution of the dominant Dark Matter many ellipticals formed from multiple collisions of spiral galaxies their resulting structure is triaxial and can not be represented by any analytical formula.

I suggest that the study of elliptical galaxy structure was confused by the belief that any interpretation of data must be based on a parametric model, however heuristic or implausible. Much fruitless debate might be been avoided had simple density estimation techniques, or preferably the new nonparametric regression methods described by Prof. Genovese, been applied.

Conclusions Astronomers should use parametric models when the underlying physical processes and astrophysical situation is clear (e.g. binary stars/planets). When the astrophysics is not well-founded (e.g. elliptical galaxy structure), nonparametric approaches may be preferable to heuristic parametric modeling. For cosmology, one must decide whether the concordance  CDM model with Dark Energy is “clear” or whether alternatives (quintessence? Bianchi universes?) are viable.