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Hierarchical inference for exoplanet populations

Hierarchical inference for exoplanet populations

My slides for #iau2015

Dan Foreman-Mackey

August 03, 2015
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  1. summary ▶︎ hierarchical inference and probabilistic modeling provide a consistent

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

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

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

    of exoplanets based on the existing set of large, heterogeneous datasets?
  5. 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.?
  6. k = 1, · · · , K ✓ wk

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

    }, { wk } | ✓ ) d{ wk } = Z p({ xk } | { wk }) p({ wk } | ✓ ) d{ wk } the Big Integral™
  8. 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
  9. an example: Kepler ▶︎ what can we say about the

    joint period– radius distribution based on the Kepler dataset?
  10. 101 102 orbital period [days] 100 101 planet radius [R

    ] Data from: The Exoplanet Archive typical error bar
  11. ▶︎ 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)
  12. inhomogeneous Poisson process p ( {wn } | ✓ )

    = exp ✓ Z ˆ✓( w ) d w ◆ N Y n=1 ˆ✓( wn)
  13. 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)
  14. 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)
  15. 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)
  16. 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)
  17. aside: inverse detection efficiency ✓j = Nj R j Q(w)

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

    dw bin height survey completeness or detection efficiency number of points in bin
  19. 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
  20. 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)
  21. attempt #1: intuition DO NOT TRY THIS AT HOME! ignoring

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

    uncertainties truth intuitive resampling w p(w) BAD IDEA
  23. 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
  24. 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
  25. 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)
  26. when does all this matter? ▶︎ when you want precise

    measurements with realistic uncertainty estimates ▶︎ near detection limit (esp. extrapolation!) ▶︎ missing data ▶︎ ...
  27. 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