Dan Foreman-Mackey
August 03, 2015
270

# Hierarchical inference for exoplanet populations

My slides for #iau2015

August 03, 2015

## Transcript

1. ### Hierarchical inference for exoplanet populations Dan Foreman-Mackey (Sagan Fellow, U.

Washington) iau exostats / 2015-08-03
2. ### Probabilistic modeling for exoplanet populations Dan Foreman-Mackey (Sagan Fellow, U.

Washington) iau exostats / 2015-08-03

4. ### "backup Hogg" Dan Foreman-Mackey Sagan Fellow, University of Washington dfm.io

/ @exoplaneteer / github.com/dfm
5. ### summary ▶︎ hierarchical inference and probabilistic modeling provide a consistent

framework for: ▶︎ measurement uncertainties ▶︎ missing data ▶︎ heterogeneous datasets ▶︎ false positives/alarms ▶︎ ...
6. ### summary ▶︎ hierarchical inference and probabilistic modeling provide a consistent

framework for: ▶︎ measurement uncertainties ▶︎ missing data ▶︎ heterogeneous datasets ▶︎ false positives/alarms ▶︎ ... ▶︎ it isn't hard
7. ### population inference ▶︎ population: global distribution and rate of physical

parameters (period, mass, multiplicity, etc.) ▶︎ inference: coming to a conclusion based on evidence
8. ### population inference ▶︎ what can we say about the population

of exoplanets based on the existing set of large, heterogeneous datasets?

11. ### population inference ▶︎ what can we say about the population

of exoplanets based on the full set of photons detected by Kepler, Keck, GPI, [your favorite instrument here], etc.?

13. ### hierarchical inference (hierarchical Bayesian modeling) ▶︎ hierarchical inference: exploit structure

in the problem to make it tractable

18. ### k = 1, · · · , K ✓ wk

xk per-object parameters (period, radius, etc.) per-object observations global population
19. ### p({ xk } | ✓ ) = Z p({ xk

}, { wk } | ✓ ) d{ wk } = Z p({ xk } | { wk }) p({ wk } | ✓ ) d{ wk } the Big Integral™
20. ### solving the Big Integral™ ▶︎ for small problems, use available

tools like JAGS, Stan, PyMC, emcee, etc. ▶︎ for bigger problems, you'll probably need something problem speciﬁc

22. ### an example: Kepler ▶︎ what can we say about the

joint period– radius distribution based on the Kepler dataset?
23. ### 101 102 orbital period [days] 100 101 planet radius [R

] Data from: The Exoplanet Archive typical error bar
24. ### ▶︎ the inverse detection eﬃciency procedure: weighted histogram of the

catalog ▶︎ the inhomogeneous Poisson process: equation for the likelihood of the catalog methods for population inference (occurrence rate calculations)
25. ### inhomogeneous Poisson process p ( {wn } | ✓ )

= exp ✓ Z ˆ✓( w ) d w ◆ N Y n=1 ˆ✓( wn)
26. ### inhomogeneous Poisson process p ( {wn } | ✓ )

= exp ✓ Z ˆ✓( w ) d w ◆ N Y n=1 ˆ✓( wn) "observable" rate density ˆ ✓(w) = ✓(w) Q(w)
27. ### expected number of detections inhomogeneous Poisson process p ( {wn

} | ✓ ) = exp ✓ Z ˆ✓( w ) d w ◆ N Y n=1 ˆ✓( wn) "observable" rate density ˆ ✓(w) = ✓(w) Q(w)
28. ### distribution of detections expected number of detections inhomogeneous Poisson process

p ( {wn } | ✓ ) = exp ✓ Z ˆ✓( w ) d w ◆ N Y n=1 ˆ✓( wn) "observable" rate density ˆ ✓(w) = ✓(w) Q(w)
29. ### observable rate density ▶︎ includes detection eﬃciency ▶︎ "true" rate

density: histogram, power law, physical model, etc. ▶︎ can include false positives/alarms ▶︎ multiple surveys? product of likelihoods ˆ ✓(w) = ✓(w) Q(w)

dw
31. ### aside: inverse detection eﬃciency ✓j = Nj R j Q(w)

dw bin height
32. ### aside: inverse detection eﬃciency ✓j = Nj R j Q(w)

dw bin height number of points in bin
33. ### aside: inverse detection eﬃciency ✓j = Nj R j Q(w)

dw bin height survey completeness or detection eﬃciency number of points in bin
34. ### aside: inverse detection eﬃciency ✓j = Nj R j Q(w)

dw bin height survey completeness or detection eﬃciency bin volume number of points in bin
35. ### aside: inverse detection eﬃciency ✓j = Nj R j Q(w)

dw bin height survey completeness or detection eﬃciency bin volume number of points in bin ✓j = Nj X n=1 1 Q(wn)

w p(w)
39. ### attempt #1: intuition DO NOT TRY THIS AT HOME! ignoring

uncertainties truth w p(w)
40. ### attempt #1: intuition DO NOT TRY THIS AT HOME! ignoring

uncertainties truth intuitive resampling w p(w)
41. ### attempt #1: intuition DO NOT TRY THIS AT HOME! ignoring

uncertainties truth intuitive resampling w p(w) BAD IDEA

44. ### per-object likelihood function Poisson process p({ xk } | ✓

) = Z p({ xk }, { wk } | ✓ ) d{ wk } = Z p({ xk } | { wk }) p({ wk } | ✓ ) d{ wk } the Big Integral™ for our Kepler example
45. ### the "interim" prior aside: what is a catalog? w (n)

k ⇠ p( wk | xk, ↵ ) – 8 – we will reuse the hard work that went into building the ca ach entry in a catalog is a representation of the posterior p p( wk | xk , ↵ ) = p( xk | wk ) p( wk | ↵ ) p( xk | ↵ ) meters wk conditioned on the observations of that object nder that the catalog was produced under a speciﬁc cho ive”— interim prior p( wk | ↵ ). This prior was chosen by the di↵erent from the likelihood p( wk | ✓ ) from Equation (2). e can use these posterior measurements to simplify Equa
46. ### the Big Integral™ for our Kepler example p ( {

xk } | ✓) p ( { xk } | ↵) ⇡ exp ✓ Z ˆ✓(w) dw ◆ K Y k=1 1 Nk Nk X n=1 ˆ✓(w (n) k ) p (w (n) k | ↵) sum over posterior samples product over objects Ref: Foreman-Mackey, Hogg, & Morton (2014)
47. ### when does all this matter? ▶︎ when you want precise

measurements with realistic uncertainty estimates ▶︎ near detection limit (esp. extrapolation!) ▶︎ missing data ▶︎ ...
48. ### summary ▶︎ hierarchical inference and probabilistic modeling provide a consistent

framework for: ▶︎ measurement uncertainties ▶︎ missing data ▶︎ heterogeneous datasets ▶︎ false positives/alarms ▶︎ ... ▶︎ it isn't always hard