Forward (& Backward) Thinking

Wei Zhu (祝伟)

2021-12-20

Is one detection statistically significant?

The shortest paper list

One is significant: DM-free galaxy

van Dokkum et al. (2018, 2019), Mancera Piña et al. (2021); see also Trujillo et al. (2019)

One is significant: Direct imaging wide-orbit planets

Forward modeling vs. Backward modeling (or. inversion problem)

$$ P({\rm Model|Data}) = \frac{P({\rm Data|Model}) P({\rm Model})}{P({\rm Data})} \propto P({\rm Data|Model}) $$

John von Neumann: With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.

“Drawing an elephant with four complex parameters” by Jurgen Mayer, Khaled Khairy, and Jonathon Howard,  Am. J. Phys. 78, 648 (2010)

Danger of over-simplification

• Hubble constant from strong lensing time delay.
• Kepler dichotomy.
• ...

\(H_0\) Lenses in COSMOGRAIL's Wellspring (H0LiCOW )

Wong et al. (2019)

• 2.4% error on \(H_0\);
• 5.3\(\sigma\) tension with CMB \(H_0\).

 "Kepler dichotomy"

Ballard & Johnson (2016)

(See also Lissauer et al. 2011, Johansen et al. 2012)

• Half of transiting planets (tranets) are found in single-tranet systems.

• If assuming small (and fixed) mutual inclinations, two populations are needed:
  • Compact systems
  • 1-planet systems (or 2-planet with large mutual inclinations)

The Silent Majority

Weiss et al. (2018)

(See also Lissauer et al. 2011, Ciardi et al. 2013, Millholland et al. 2017)

Radius of the inner planet \(R_\oplus\)

Solar system in the context

See Zhu (2020) and Murchikova & Tremaine (2020).

Figure adapted from Penny et al. (2019)

• Solar system planets have very diverse properties.
• In a (super-)Kepler's view, Solar system planets (Venus & Earth) have very similar properties.

一个物理学家发现，奇数都是质数。他找来两个学生来做实验验证。学生A找来20以内所有的奇数，他发现 9，15不是质数。他和导师在PRL上发了篇文章，声称在10%的误差范围内奇数都是质数。

学生A毕业以后，学生B接过他的工作，继续研究。她找来100以内的质数，发现只有25个。于是她和导师写了篇文章说，奇数都是质数这个结论是正确的，不过在使用的时候，需要乘以一个蒙混因子(fudge factor)。

——来自网络

Brewer et al. (2018): Compact Multi-planet Systems are more Common around Metal-poor Hosts

Winter et al. (2020): Stellar clustering shapes the architecture of planetary systems

Low phase space density

High phase space density

Forward Thinking

• One detection can be significant.
  • Depending on your expectation.
• Advantages of forward modeling
• The danger of forward modeling

  • Over-fitting & over-simplification

• The silent majority.
• Know the physics behind the data.