InformationInformation fieldfield theorytheory Max-Planck-Institut turningturning datadata intointo imagesimages für Astrophysik Torsten Enßlin Michael Bell, Mona Frommert, Maskim Greiner, Jens Jasche, Henrik Junklewitz, Francisco Kitaura, Niels Oppermann, Georg Robbers, Marco Selig, Cornelius Weig signal fields: ∞ degrees of freedom data sets: finite → additional information needed information: physical laws, symmetries, continuity, statistical homogeneity/isotropy, ... space is continuous → information field theory InformationInformation HamiltonianHamiltonianTheoryTheory s = signal d = data posterior likelihood prior evidence inferenceinference problemproblem asas informationinformationstatisticalstatistical fieldfield theorytheory An information field theory is defined over continuous space … … so that the pixelization is irrelevant if sufficiently fine FreeFree TheoryTheory GaussianGaussian signalsignal && noise,noise, linearlinear responseresponse WienerWienerFreeFree filterfilter TheoryTheory theorytheory GaussianGaussian signalsignalknown && for noise,noise, 60 yearslinearlinear responseresponseinformationinformation sourcesource informationinformation propagatorpropagator signalsignal fieldfield mock signal from known power spectrum datadata == integratedintegrated signalsignal informationinformation sourcesource reconstructionreconstruction InteractingInteracting TheoryTheory non-Gaussiannon-Gaussian signal,signal, noise,noise, oror non-linearnon-linear responseresponse IFTIFT dictionarydictionary Translation: inference problem → statistical field theory Dictionary: log joint PDF = negative Hamiltonian evidence = partition function Z Wiener variance = information propagator noise weighted data → information source inference algorithms ← Feynman diagrams maximum a Posteriori = classical solution uncertainty corrections = loop corrections Shannon information = negative entropy PrimordialPrimordial non-Gaussianitynon-Gaussianity detectiondetection CosmographyCosmography reconstructionreconstruction ofof thethe cosmiccosmic largelarge scalescale structurestructure CosmographyCosmography reconstructionreconstruction ofof thethe cosmiccosmic largelarge scalescale structurestructure ● galaxies trace dark matter density field ● log-normal density field: ● galaxy shot noise due to Poisson statistics ● inhomogeneous observation data = galaxy bias Hamiltonian: galaxy counts Enßlin,Enßlin, Frommert,Frommert, covariance of expected mean Gaussian galaxy counts KitauraKitaura (2009)(2009) log-density field s classicalclassical && renormalizedrenormalized solutionsolution fix point solution for different temperatures T T=0 classical solution, T=1: field theoretical solution 3d3d CosmographyCosmography Jasche, Kitaura, Li, Enßlin (2010) log-density from SDSS DR7 using Hamiltonian sampling main sample galaxies, 0.001< z < 0.4, ¼ sky box : 750 Mpc/h per side, 256 voxels per side → resolution ~ 3 Mpc/h corrected for • selection function • completeness • full shot noise mean + mask sample + mask weakweak gravitationalgravitational lensinglensing grav lensing forecast (Jasche, in prep.) FaradayFaraday skysky Oppermann et al. (arXiv:1111.6186) ConclusionsConclusions Information field theory (IFT) ... is defined over continuous space ● IFT is a mathematical language for field inference ● field reconstruction requires prior = regularization e.g. Gaussianity, symmetries, ... ● field theoretical toolbox: free theory = Wiener filter theory, classical theory = max. a posteriori, Feynman diagrams, renormalisation flow, thermodynamical inference, critical filter methodology ... Cosmology: ● clear assumptions (stat. homogeneity & isotropy, power, noise) → unique, high fidelity IFT algorithm Numerics: ● simulation schemes from information field dynamics ThankThank youyou !! www.mpa-garching.mpg.de/ift Wahrscheinlichkeit = probability Deutsches Wörterbuch von Jacob Grimm und Wilhelm Grimm (1854) … unter wahrscheinlichkeit ist ein fürwahrhalten aus unzureichenden gründen zu verstehen, die aber zu den zureichenden ein gröszeres verhältnisz haben, als die gründe des gegentheils. durch diese erklärung unterscheiden wir die wahrscheinlichkeit (probabilitas) von der blosen scheinbarkeit (verisimilitudo). … Wiktionary (2012) Grad der Möglichkeit = degree of possibility Wahr – = true schein – = appearance lichkeit = [adjective (how much)] Wahrscheinlichkeit = how true something appears to be.