Bayesian modeling of gravitational microlensing events 2nd Year PhD Assessment Talks Fran Bartolić University of St Andrews fbartolic 

2 What is gravitational microlensing? Source ● Star(s) Lens ● Star(s) ● Star + planet ● Black hole ● Brown Dwarf Observer ● Photometry using a network of ground based telescopes ● Space telescopes 

Credit: Y. Tsapras 

Credit: A. Udalski 

5 Why are people interested in microlensing? Credit: Matthew Penny Distance from star [AU] Planet mass [Earth masses] 

6 What does the data look like? 

7 The inverse problem 

8 The inverse problem Model parameters Data Posterior Likelihood Prior 

9 What problems am I trying to solve? Mróz et. al. 2016 Correlated noise in the data Mróz et. al. 2019 Dominik et. al. 2019 Highly correlated non-linear parameter space Population inference 

10 What problems am I trying to solve? Mróz et. al. 2016 Correlated noise in the data Mróz et. al. 2019 Dominik et. al. 2019 Highly correlated non-linear parameter space Population inference 

11 What problems am I trying to solve? Mróz et. al. 2016 Correlated noise in the data Mróz et. al. 2019 Dominik et. al. 2019 Highly correlated non-linear parameter space Population inference 

12 The inverse problem Posterior Likelihood Prior 

13 A generative model for the data + = Observed data Deterministic physical model Probabilistic noise model 

14 A deterministic forward model Trajectory Magnification Flux 

15 A probabilistic noise model Deterministic physical model Probabilistic noise model Multivariate Gaussian White noise Correlated noise Covariance matrix 

16 The inverse problem Posterior Likelihood Prior How to efficiently sample the posterior? 

17 Sampling the posterior with MCMC ● Metropolis Hastings MCMC is inefficient at exploring complex posteriors ● It doesn’t scale to more than ~20 dimensions (parameters) ● Often fails silently Metropolis Hastings Credit: https://github.com/chi-feng/mcmc-demo Posterior 

18 Sampling the posterior with Hamiltonian MCMC Hamiltonian Monte Carlo Credit: https://github.com/chi-feng/mcmc-demo Potential energy Hamiltonian Hamilton’s equations Posterior 

19 Example results Data space Parameter space 

20 Example results Covariance matrix 

21 Take home messages ● Microlensing enables discovery of cold exoplanets and objects such as Brown Dwarfs and Black Holes ● Fitting models is hard because the physics of interest maps poorly onto the observed data ● Correlated noise matters ● Hamiltonian Monte Carlo efficiently samples posteriors using information about the geometry of the posterior probability density