Transcript Commentary & discussion of nonparametric inference

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 t. 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
Periodic
brightness
variation
Periodic
radial
velocity
variation
Charbonneau
et al. 2000
HD 209458:
`hot Jupiter’
binary system
Interesting parameters:
aorb, Mp, Rp
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 LCDM model with Dark Energy is “clear”
or whether alternatives (quintessence? Bianchi
universes?) are viable.