### Strategies for exploring parameter space for planetary microlensing events:
### Lessons from the RV and TTV community
Eric Ford
Penn State
23rd International Microlensing Conference
Center for Computational Astrophysics
28-30 January, 2019
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## Questions posted
- How to efficiently search a complex (non-Gaussian) multi-variate parameter space?
- How to weight degenerate solutions?
- Efficient computational methods for solving the triple lens equation.
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## Challenge: Multi-modal likelihood
- Multiple local maxima
- Widely spaced
- Separated by deep valleys
- Sharp features
- High-dimensional
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## Who else has these challenges?
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## Who else has these challenges?
- Radial velocity
- Astrometry
- Transit search
- Transit timing variations
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## Similarities
- Non-linear likelihood
- Likelihood highly multi-modal in orbital period
- 5-7 parameters per planet
- Unknown number of planets
- Keplerian orbits
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## Explore
vs
## Exploit
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## Explore
1. Physical intuition
2. Computational power
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## Explore: Physical intution
Full model: N-body
Surrogate models:
- Keplerian orbits
- Epicycle approximation
- Circular orbits
- Linear/quadratic trajectory?
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## Explore: Physical intution
Hierarchy of parameters
- 1-3 key multi-modal parameters
- Secondary non-linear parameters
- (Nearly) linear parameters
- Nuisance parameters
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### Key multi-modal parameters
Brute force search
- Parallelizes efficiently
- GPUs offer >100x speedup (if high compute per memory load)
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### Linear (or nearly linear) parameters
- Optimize analytically w/ linear algebra
- Marginalize w/ Laplace approximation
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### Other non-linear parameters
- Optimize itteratively
- Marginalize via quadrature
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## Explore: Physical intution
Hierarchy of parameters
- 1-3 key parameters:
- Brute force
- Non-linear parameters:
- Optimize itteratively, or
- Integration by quadrature
- (Nearly) linear parameters:
- Optimize analytically, or
- Integration via Laplace approximation
- Nuisance parameters
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## Example: RV (Keplerian Orbits)
Hierarchy of parameters
- Key parameters:
- Orbital Period(s)
- Linear parameters:
- RV amplitiudes (2/planet)
- Instrumental offsets
- Non-linear parameters:
- Eccentricities
- One angle per planet
- Noise model
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## Example: RV (Epicycle Approximation)
Hierarchy of parameters
- Key parameters:
- Orbital Period(s)
- Linear parameters:
- RV amplitiudes (4/planet)
- Instrumental offsets
- Non-linear parameters:
- Noise model
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## Example: Transit Photometry
Hierarchy of parameters
- Key parameters:
- Orbital period
- Orbital phase
- Transit duration
- Linear parameters:
- Transit depth
- Coefficients for detrending
- Non-linear parameters:
- Noise model
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## Example: Transit Timing Variations
Hierarchy of parameters
- Key parameters:
- Orbital periods
- Orbital phases
- Non-linear parameters:
- Masses
- Eccentricity vectors
- Nuisance parameters:
- Inclinations
- Nodes
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## Explore: Physical Intuition
Fast model evaluation
- Density of sampling (e.g., in periods)
- One Physical model... many viewing geometries
- Small eccenricity approximation
- Small mutual inclination approximation
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## Exploit
What is your goal?
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## Exploit
- Find optima: Itterative optimizer
- Draw posterior sample (near optimum): MCMC
- Marginalize around optimum: Importance Sampling
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## Exploit: Optimization
Optimization algorithms using gradients
- Differentiable form of Kepler (Pal 2009)
- Derivatives of Kepler equation by hand
- Autodifferentiation
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## Exploit: Posterior Sampling
Artisinal MCMC proposals
- Use physically motivated proposals for correlated parameters (Ford 2005)
- Useful when nearly degeneracies
- Multiple parameterizations for multiple regimes.
- Enable jumps between few known modes (Hu+ 2014)
- Be extra careful to use well-motivated priors
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## Exploit: Posterior Sampling
Workhorse algorithms
- Ensemble samplers (D ~ 10's)
- Differential Evolution MCMC (ter Braak 2006; Nelson+ 2013)
- Affine invariant ensemble sampler (Goodman & Weare 2010)
Often "good enough" and saves expert from spending time to design artisinal MCMC proposals
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## Exploit: Posterior Sampling
Workhorse algorithms
- Geometric samplers (large D)
- HMC + No U-Turn Sampler (NUTS; Hoffman & Gelman 2014)
- Geometric Adaptive Monte Carlo (Tuchow+ 2019)
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## Exploit: Posterior Sampling
Combine:
- Artisinal proposals (e.g., between modes)
- Ensemble samplers (most important parameters)
- Geometric samplers (e.g., nuisance parameters)
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## Exploit: Posterior Sampling
Weighting of degenerate solutions comes out naturally
Warnings:
- Posterior width depends on measurement uncertainties
- Correlated noise also affects weighting and location of posterior modes
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## Exploit: Posterior Sampling
Suggestions:
- Report what you measure well (even if it's not what physicists want)
- Particularly important for summmary statistics
- Can avoid some biases
- e & omega vs e sin(omega) & e cos(omega)
- Similar for inclinations
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## How many planets?
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## How many planets does the data provide strong evidence for?
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## How many planets?
From Astronomer intuition to Bayesian model comparison...
- Requires accurate noise model
- Better to marginalize over noise model parameters than guess wrong
- "Jitter parameter"
- Strength of correlated noise
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## Bayesian Model Comparison
- In general, computationally very expensive
- AIC or BIC are merely heuristics
- Unlikely asymptotics apply to your data
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## Bayesian Model Comparison
- Laplace approximation
- Integrated Nested Laplace Approximation (INLA; Rue+ 2009)
- Importance Sampling
- Mixture of Gaussians (or Student t-distributions)
- Product of marginals (Perrakis+ 2014)
- Ratio estimator (Nelson+ 2014)
- Diffusive Nested Sampling (DNest4; Brewer & Foreman-Mackey 2018)
- Thermodynamic integration
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## Extremely Precise RV Evidence Challenge
Lessons learned
- Validate codes
- Do not trust internal errors estimates
- Plan for uncertainty in evidence estimates
- Perform sensitivity tests
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## Extremely Precise RV Evidence Challenge
Dispersion in evidence estimates
- 0 planets: ~3x
- 1 planet: ~10x
- 2 planets: 100-1000x
- 3 planets: >10,000x
Nelson+ 2019
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## What's the real goal?
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## Characterizing Planet Populations
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## Hierarchical Bayesian Models
- High-dimensional
- Priors can be surprisingly important
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## Hierarchical Bayesian Models
- Probabilistic programming languages good for prototyping
- JAGS (probably too restrictive for microlensing models)
- STAN (flexible, heavily templatized C++)
- Turing.jl (flexible, easier to implement thanks to Julia)
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## Hierarchical Bayesian Models
- Survey may be homogeneous
- Astrophysical complexity
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## Hierarchical Bayesian Models
- Incorporating complex survey details can be difficult
- Approximate Bayesian Computing
- Allows for complex physics, survey practicalities
- Can gradually build model complexity
- Enables testingn sensitivity to unmodeled complexities
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## Towards Reproducibile Science
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## Towards Reproducibile Science
- Report what you measure well
- Report more than summary statistics
- If degeneracies, report mixture model as approximation to posterior
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## Towards Reproducibile Science
- Share posterior distributions
- Ideally labeled with log prior & log likelihood (perhaps multiple terms)
- Plan for both data & codes to be shared
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## Towards Reproducibile Science
Experimental design matters
- Simulations should inform key decissions
- Use algorithmic observing strategies (most of the time)
- Document decisions (esp. deviations)
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# Questions?