Information Field Theory

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

● 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