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Causal structure and machine learning (in astronomy)

David W Hogg
December 13, 2019

Causal structure and machine learning (in astronomy)

Presented on 2019-12-13 in Ringberg Castle at #MLringberg2019.

David W Hogg

December 13, 2019

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  1. Causal structure and
    machine learning
    David W Hogg
    (NYU) (MPIA) (Flatiron)

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  2. Three concepts of causal structure
    ● Generative vs discriminative models.
    ○ (cf S. Villar)
    ● Enforcing symmetries with graph structure.
    ○ The “convolutional” in CNN, or the “recurrent” in RNN, for example.
    ● Building models that represent our strong causal beliefs.
    ○ As in “the image is blurred by the seeing, pixelized, and Poisson-sampled at the device”.
    ○ (cf Lanusse, or Green)

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  4. But unitarity?
    ● If the laws of physics are unitary, then there is no concept of intervention.
    ● Therefore only certain meanings of the word “causal” are appropriate here.
    I think of the causal structure as being the physical dependencies in the
    data-generating process, representable by a directed graph.

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  5. Generative vs discriminative models
    ● Are you asking what function of your labels makes your data?
    ○ x = A(y)
    ○ Labels y generate the data x.
    ● Or are you asking what function of your data makes your labels?
    ○ y = B(x)
    ○ Data x are transformed into labels y.

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  6. Generative vs discriminative models
    ● x = A(y) + noise .
    ● Train with A := argmin_A || x - A(y) || .
    ● Test with y := argmin_y || x - A(y) || .
    ● The test step is like a pseudo-inverse of
    the forward model. Or an inference!
    ● Can deal with missing data and non-trivial
    likelihood functions (heteroskedastic, for
    example). But the test step is an inference,
    ● y = B(x) + prediction error
    ● Train with B := argmin_B || y - B(x) || .
    ○ plus regularization! .
    ● Test with y := B(x) .
    ● There is no inverse, not even a
    ● Test step is generally very fast!

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  7. Example: Linear models
    ● Let the data x be D-dimensional and the labels y be K-dimensional.
    ● Generative: x = A y + noise, where A is D x K
    ○ A has pseudo-inverse (ATA)-1 AT or something like that.
    ○ Training data size N must be N > K.
    ● Discriminative y = B x + prediction error, where B is K x D
    ○ Training requires a regularization if N < D.
    ● Conjecture: The generative model is always more accurate.
    ○ This is even at optimal regularization amplitude.
    ○ I can demonstrate this in a simple sandbox.

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  12. Example: Linear models
    ● Both models (generative and discriminative) do well.
    ● But the generative model does better.
    ○ It saturates some bounds on inference.
    ○ And the discriminative model required a tuned regularization.
    ● Conjecture: No matter what the training-set size, the generative model is
    always more accurate.
    ○ This is even at optimal regularization amplitude.

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  13. Graph structure to enforce symmetries
    ● There are results from math that say that (on graph NNs, anyway), any
    compact symmetry can be enforced on the model.
    ○ Bruna, LeCun, others; see also Charnock: http://bit.ly/NeuralBiasModel
    ● The C in CNN is about translational symmetry.
    ● In many of our problems (cosmology, turbulence, galaxy images), the
    symmetries are exact.
    ○ Would you believe a cosmological parameter inference that depends on how you translate or
    rotate the large-scale structure?
    ○ Would you believe a cosmic shear estimate that isn’t covariant under rotations?
    ● Because these issues are hard, many practitioners resort to data
    ○ But this only enforces the symmetries in the limit. And it’s far away!

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  14. Representing our beliefs about the physics
    ● The stellar spectrum depends on Teff, log g, and element abundances.
    ● The color of the star depends on its temperature and interstellar reddening,
    which in turn depends on its location in the Galaxy.
    ● The galaxy image is sheared by the cosmological gravitational field, blurred
    by the Earth’s atmosphere, and pixelized by the detector.

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  15. Extreme-precision radial velocities
    ● It is now routine to measure stellar Doppler shifts at the m/s level.
    ● Even at resolving power 100,000, this is 1/1000 of a pixel in the spectrograph.
    ● Used to find or confirm many hundreds of extra-solar planets.
    ○ And thousands more coming very soon.
    ● Measured RVs are limited by our ability to model the atmosphere and star.
    ○ (Not everyone would agree with this statement, but it’s a hill I’ll die on.)
    ● The total signal-to-noise in typical data sets is immense.
    ○ 100s of 100,000-pixel observations over many years with SNR of 100s each.
    ● It was awarded the 2019 Nobel Prize.
    ○ Mayor and Queloz

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  17. wobble
    ● Megan Bedell (Flatiron), Dan Foreman-Mackey (Flatiron), Ben Montet
    (Chicago), Rodrigo Luger (Flatiron).
    ● All tested and operating on real HARPS data.
    ● arXiv:1901.00503

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  18. wobble
    ● A spectrum has lines or shape from star, atmosphere, and spectrograph.
    ○ This is causal structure because different spectra are taken at different relative velocities.
    ● These lines have different rest frames.
    ○ Doppler shift is a hard-coded symmetry of the model, because Duh.
    ● And some or all of these components can vary with time.
    ○ And the relative velocities of star, atmosphere, and spectrograph do too.
    ● Linearized model for tractability (convexity).
    ● Justifiable likelihood function to account properly for noise.
    ○ We have a good noise model and the data are heteroskedastic.

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  21. The stellar color-magnitude diagram
    ● In the space of luminosity and temperature, stars lie in an amazingly
    structured and simple distribution.
    ○ See, eg, the last 150 years of astronomy.
    ○ Main sequence, red-giant branch, white dwarfs, horizontal branch, red clump, binary
    sequences, and so on.
    ○ Almost one-dimensional! (with thickness)
    ● Theory does a great job! But small, systematic deviations.
    ○ These lead to biases if you want to use the theory to measure stellar properties, for example.
    ● If we can understand the CMD well, we can infer distances to all the stars!
    ● Gaia changed the world.
    ○ Data now “outweigh” theory in many respects.

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  23. De-noising Gaia
    ● Lauren Anderson (Flatiron), Boris Leistedt (NYU), Axel Widmark (Stockholm),
    Keith Hawkins (Texas), and others.
    ● ESA Gaia DR1 data
    ○ This is out of date now, of course.
    ● arXiv:1706.05055, arXiv:1705.08988, arXiv:1703.08112

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  28. Model structure
    ● Extremely flexible model for the true color-magnitude diagram.
    ○ (The word “true” has many possible meanings here.)
    ○ We could have used a deep-learning model here.
    ● Correct use of the Gaia likelihood function (noise model).
    ○ We didn’t have to cut out noisy or bad objects.
    ○ Every star has its own individual noise properties (heteroskedastic).
    ● The model knows that parallax and brightness both depend on distance!
    ○ Didn’t have to learn that from the data.
    ○ This is causal structure in my sense of this term.
    ○ Technically, it is the embodiment of the symmetries of relativity and electromagnetism.

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  29. Precision
    Our data-driven model was trained on and fit to the Gaia data but produced more
    precise results than the Gaia data. What gives?
    ● Stationarity assumption.
    ○ Stars are similar to one another; related to statistical shrinkage.
    ● Use of high-quality noise model.
    ● Enforcing physical symmetries.
    ○ Lorentz invariance (the symmetries of electromagnetism and spacetime).
    ○ (But no other use of physical models of stars, which we don’t fully believe.)
    ○ There is much more we could do!

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  30. Stellar spectroscopy
    ● Stars (as observed) have only a few first-order parameters.
    ○ Effective temperature, surface gravity, surface abundances (a few dominate).
    ● These parameters reveal themselves in absorption-line strengths.
    ● Stellar interior models and atmosphere models are amazingly detailed.
    ○ Models predict spectra at the few-percent level or even better, depending on stuhh.
    ○ And yet, the data are so incredible that we can see their failures at immense significance.
    ● Spectroscopy at resolving power 20,000 to 100,000 is the standard tool.
    ○ (wavelength over delta-wavelength)

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  33. The Cannon*
    ● Melissa Ness (Columbia), Anna Ho (Caltech), Andy Casey (Monash), Jessica
    Birky (UW), Hans-Walter Rix (MPIA), Soledad Villar (NYU), and many others.
    ● SDSS-III and SDSS-IV APOGEE spectroscopy data.
    ○ These are remarkable projects of which I am very fortunate to be a part.
    ○ We have also run on other kinds of data from other projects.
    ● arXiv:1501.07604, arXiv:1603.03040, arXiv:1609.03195, arXiv:1609.02914,
    arXiv:1602.00303, arXiv:1511.08204 …
    * It’s named after the person, not the weapon!

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  34. The Cannon
    ● Training-set framework.
    ○ Some stars have good labels, from somewhere!
    ○ I’m going to call parameters “labels”.
    ● Every spectral pixel brightness (expectation) is a simple function of labels.
    ● Righteous likelihood function for the spectral pixel brightnesses.
    ○ Fully heteroskedastic.
    ○ Can deal with missing data and low SNR spectra.
    ● Model training is maximum-likelihood.
    ● Model execution on new data is also maximum-likelihood.

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  39. Conclusions
    ● Generative vs discriminative models.
    ○ (cf S. Villar)
    ○ Generative models are more accurate, in at least some settings.
    ○ Generative models better represent our beliefs.
    ● Enforcing symmetries with graph structure.
    ● Building models that represent our strong causal beliefs.
    ○ (cf Lanusse, or Green)
    ○ Creating state-of-the-art radial-velocity measurements for exoplanet discovery.
    ○ De-noising Gaia data through hierarchical inference.
    ○ Labeling stellar spectra more accurately than with physical models.

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