Information Field Theory Philipp Arras September 20, 2018 Max-Planck Institute for Astrophysics, Garching, Germany Technical University of Munich Teaser: Radio Aperture Synthesis Figure 1: Dirty image Figure 2: Reconstructed sky model Supernova remnant SN1006, public data VLA (24/01/2003), 12.5 MHz bandwidth, single-channel mode, one spectral windows at 1.37 GHz. Reconstructed with RESOLVE, only Stokes I. 1 Acknowledgements • Torsten Enßlin, my supervisor. • Henrik Junklewitz, inventor of RESOLVE. • Philipp Frank, Sebastian Hutschenreuter, Jakob Knollmüller, Reimar Leike, Natalia Porqueres, Daniel Pumpe, Julian Rüstig and many more, fellow sufferers. • Martin Reinecke, numerical genius. • NIFTy, solves all problems (https://gitlab.mpcdf.mpg.de/ift/NIFTy). 2 Inverse Problems Inverse Problems Data d (Ndof < 1) True sky s (Ndof = 1) −! Data arrays 3 −! Fields (e.g. ρ : R ! R≥0) (e.g. vis.shape = (4132094,)) P(sjd) 3 P(djs)P(s) P(sjd) = P(d) 3 Probability Distributions Over All Possible Fields ) d j s ( P Npix dimensions This is a lot! Theoretically, infinite-dimensional. 4 2 Practically, e.g. Npix = 1024 ! 1 Mio. dimensions. Bayesian Reconstruction Algorithms Likelihood Data, instrument description, statistics Prior Posterior Variational Bayes power spectrum 2 10 power power spectrum, Bayes 100 2 10− correlation 4 10− 6 smoothness prior 10− 100 101 102 harmonic mode 5 Information Field Theory Information Field Theory • Information Field Theory := Information theory with fields. • Provides dictionary: Statistical mechanics Bayesian inference & Field theory • Define H(s; d) := − log P(s; d). Then: P(s; d) e−H(s;d) P(sjd) = = : P(d) R Ds e−H(s;d) • Exploits continuity and correlation. • Variational Bayes corresponds to minimizing the Gibbs free energy. 6 Example: Log-normal Poisson Log-normal Poisson Reconstruction • Imagine a light curve. • Data: Point-wise drawn from Poisson distribution: di x Pλi (di): λ: expectation value of poisson process. • Measurement equation: st λi = R(φt) = R(e ); φ: continuous photon flux field. • Prior knowledge: st is a Gaussian process whose kernel is known. 7 Log-normal Poisson Reconstruction • Bayes theorem: P(sjd) /P(djs)P(s): • Bayesian hierarchical model: P(s) τ sjτ djs Information Hamiltonian H 8 Log-normal Poisson Reconstruction 9 Log-normal Poisson Reconstruction 9 Log-normal Poisson Reconstruction 9 Log-normal Poisson Reconstruction 9 Example: Estimating Power Spectra Critical Log-normal Poisson Reconstruction • Infer power spectrum as well: P(sjd) /P(djs)P(sjτ)P(τ): • Bayesian hierarchical model: k a iv im sv sm smoothja; k slopejsm; sv; im; iv τjslope; smooth P(τ) sjτ djs Information Hamiltonian H 10 Critical Log-normal Poisson Reconstruction 11 Critical Log-normal Poisson Reconstruction 11 Critical Log-normal Poisson Reconstruction 11 Critical Log-normal Poisson Reconstruction 11 Example: Multi-Messenger Multi-Messenger with IFT • Observe a physical field s with two different measurement devices: d1 = R1s + n1 d2 = R2s + n2 • Reconstruct s from d = (d1; d2): P(sjd1; d2) /P(d1; d2js)P(s) = P(d1js)P(d2js)P(s) −! One only needs to multiply likelihoods. • Bayesian hierarchical model: d2js Information Hamiltonian H sjτ τ d1js 12 Multi-Messenger with IFT Information source aka dirty image: j = RyN−1d. Figure 4: Information Figure 5: Information Figure 3: Ground truth source interferometer source single dish 13 Multi-Messenger with IFT Ground Truth Single Dish Interferometer Combined 14 ? 14.