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 lawsof 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 #1Eclipsing binary stars

HD 209458: `hot Jupiter’ binary system

Periodic brightnessvariation

Periodic radial velocityvariation

Interesting parameters:aorb, Mp, Rp

Charbonneauet 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 #2Elliptical galaxy structure

W. Keel, WWW

M32, HST

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

King 1962

A long history of incompatible parametric modelsof elliptical galaxy radial profiles

(These five papers have 3,776 citations)

Hubble’s and King’s models are based on simple physicalInterpretation (truncated isothermal sphere). Hernquist & NFW models have more complicated physical interpretation. Thede 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.