The likelihood function and its discontents

Talk given at Spergelfest, 2021 October.

Transcript

1. David W. Hogg — likelihood function and its discontents The likelihood function and its discontents David W. Hogg NYU — MPIA — Flatiron
2. David W. Hogg — likelihood function and its discontents What I’m not going to talk about 2
3. David W. Hogg — likelihood function and its discontents Spergel changed my life • my appointment at NYU • my role at CCA • saving me from myself? ◦ (example: Gaia Sprints; see Price-Whelan talk) 3
4. David W. Hogg — likelihood function and its discontents What I am going to talk about 4
5. David W. Hogg — likelihood function and its discontents The likelihood function • p(data | model) 5
6. David W. Hogg — likelihood function and its discontents Everyone needs this function • Frequentists: Optimize this to get a Cramér-Rao-bound-saturating estimator! • Bayesians: The only way to update your posterior pdf is with this function! 6
7. David W. Hogg — likelihood function and its discontents Ideal case • data = (mean expectation) + noise • mean expectation is a deterministic function of model parameters • noise is from a known function (God help us if it isn’t Gaussian) • p(data | parameters) 7
8. David W. Hogg — likelihood function and its discontents Ideal case: marginalization • p(data | nuisances, pars) • p(nuisances) • p(data | pars) = ∫ p(data | nuisances, pars) p(nuisances) d(nuisances) ◦ see arXiv:1205.4446 8
9. David W. Hogg — likelihood function and its discontents Aside: The way I give talks is the way MK hears talks • Kamionkowski: All I hear is “blah blah blah blah blah Marc Kamionkowski blah blah blah blah.” • Hogg: All I say is “blah blah blah blah blah Hogg paper blah blah blah blah.” 9
10. David W. Hogg — likelihood function and its discontents You don’t have to be Bayesian: profiling • p profile (data | pars) = max nuisances p(data | nuisances, pars) • In the Gaussian likelihood case, this is identical to marginalization for some generic choice of p(nuisances)! 10
11. David W. Hogg — likelihood function and its discontents How do you share data and results? • If someone else wants to combine your data with theirs, they need your likelihood function! • So when you share your data, either share your data… • ...or share something from which you can reconstruct the LF. 11
12. David W. Hogg — likelihood function and its discontents Aside: The Lutz-Kelker correction and H 0 • This is absurdly Inside Baseball, but • the Lutz-Kelker corrections for parallaxes deliver maximum-posterior estimators under a particular prior. • If you use them in a data analysis, you are including a prior pdf you don’t control. • They will bias your results. 12
13. David W. Hogg — likelihood function and its discontents Aside: The Gaia Collaboration is awesome • ESA Gaia isn’t releasing all of its raw data (yet). • But it is releasing parallaxes (say) and proper motions (say). • These are the parameters of a likelihood function! ◦ see, for example, arXiv:1804.07766 ◦ connection to negative parallaxes. 13
14. David W. Hogg — likelihood function and its discontents Aside: NASA WMAP and the Lambda Archive • The NASA Lambda Archive is about sharing likelihoods. 14
15. David W. Hogg — likelihood function and its discontents Gaussian likelihoods are almost always approximations • Hennawi to Philcox: What do you do when the density field isn’t Gaussian, because then your likelihood function isn’t Gaussian? • Philcox to Hennawi: Actually even in the Gaussian-random-field case, the likelihood isn’t Gaussian! • CMB likelihood functions are close to Gaussian because of the central limit theorem, not because the field is Gaussian! 15
16. David W. Hogg — likelihood function and its discontents Aside: CMB and LSS projects use surrogate likelihoods • Do lots of simulations, compute mean and variance (tensor) on observables. • Use those as the mean and variance (tensor) of a likelihood function? • That’s a surrogate. ◦ It’s clever, wrong in detail, and fine in high signal-to-noise circumstances. 16
17. David W. Hogg — likelihood function and its discontents Non-Gaussian likelihood: Ideal case • Likelihood function is non-Gaussian but computationally tractable. • Frequentists: Optimize (the hard way). • Bayesians: Use MCMC. 17
18. David W. Hogg — likelihood function and its discontents MCMC is a method of last resort • You must choose a prior pdf over all parameters. ◦ No, you can’t “sample your likelihood function”. ◦ No, flat priors are not equivalent to “no priors”. ◦ No, you can’t undo your choice of priors later. • You must evaluate your LF (and priors) at many useless points. ◦ My most highly cited paper (arXiv:1202.3665) is on a method almost no-one should ever use! 18
19. David W. Hogg — likelihood function and its discontents The point of science is not to figure out your posterior! • You might care about your posterior pdf. • But I don’t care about your posterior pdf. • I care about the likelihood function, evaluated at your data! ◦ (Because I have different knowledge...) ◦ (...and because I will get new data in the future.) 19
20. David W. Hogg — likelihood function and its discontents Non-Gaussian likelihood: Non-ideal case • The likelihood function is non-Gaussian and it is computationally intractable. • Example: In LSS, the LF for the galaxy positions ◦ (You would have to marginalize out all phases and all of galaxy formation.) 20
21. David W. Hogg — likelihood function and its discontents Non-Gaussian likelihood: Non-ideal case • Frequentist: Make sensibly and cleverly defined estimators ◦ (like the Landy-Szalay estimator for the correlation function) ◦ (and see arXiv:2011.01836) • Bayesian: Likelihood-free inference 21
22. David W. Hogg — likelihood function and its discontents Likelihood-free inference • Simulate the crap out of the Universe. • Choose the simulations that look really, really like the data. • This makes MCMC seem like a good idea. ◦ It inherits everything that’s bad about MCMC. ◦ It is (usually) way, way more expensive. 22
23. David W. Hogg — likelihood function and its discontents Surrogate statistics • What does it mean for a simulation to look really, really like the data? • You have to choose some set of statistics (and a metric). • Choose them by intuition and cleverness! ◦ or... 23
24. David W. Hogg — likelihood function and its discontents Machine learning • Here’s a responsible use of machine learning in astrophysics! • Use the machine to find the statistics of the data that are most discriminatory about theory. • Lots of approaches and recent results. ◦ (see, eg, Zoltan Haiman’s slides from Friday!) • My take: Impose exact symmetries! ◦ arXiv:2106.06610 24
25. David W. Hogg — likelihood function and its discontents Summary • The likelihood function is important. • If your likelihood function is hard to work with, I pity you. • Machine learning is likely to play a role in the future of data analysis. 25