Information Field Theory
(IFT)
Information field theory (IFT) is information theory, logic under uncertainty, applied to fields. A field can be any quantity defined over some space, such as the air temperature over Europe, the magnetic field strength in the Milky Way, or the matter density in the Universe. IFT describes how data and knowledge can be used to infer field properties. Mathematically it is a statistical field theory and exploits many of the tools developed for such. Practically, it is a framework for signal processing and image reconstruction. IFT is fully Bayesian. How else can infinitely many field degrees of freedom be constrained by finite data? It can be used without the knowledge of Feynman diagrams. There is a full toolbox of methods. It reproduces many known well working algorithms. This should be reassuring. And, there were certainly previous works in a similar spirit, like Bayesian Field Theory (BFT). See below for IFT & BFT publications and previous works. Anyhow, in many cases IFT provides novel rigorous ways to extract information from data.
Highlights
Black hole as video: M87* in time, space and frequency
In April 2017 the Event Horizon Telescope (EHT) observed the supermassive black hole M87* and provided a first image of its shadow that went around the world. Researchers at the Max Planck Institute for Astrophysics have now reconstructed a video of the immediate surroundings of a black hole from the same underlying data. This not only confirms previous findings, video of the immediate surroundings of a black hole from the same underlying data. This not only confirms previous findings, it also hints at new structures and dynamics in the gas disk around the black hole.
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Watch stars move around the Milky Way’s supermassive black hole in deepest images yet
The European Southern Observatory’s Very Large Telescope Interferometer (ESO’s VLTI) has obtained the deepest and sharpest images to date of the region around the supermassive black hole at the centre of our galaxy. The new images zoom in 20 times more than what was possible before the VLTI and have helped astronomers find a neverbeforeseen star close to the black hole. By tracking the orbits of stars at the centre of our Milky Way, the team has made the most precise measurement yet of the black hole’s mass.
Algorithmic improvements for radio interferometry
Radio telescopes observe the sky in a very indirect fashion. Sky images in the radio frequency range therefore have to be computed using sophisticated algorithms. Scientists at the MPI for Astrophysics have developed a series of improvements for these algorithms, which help to improve the telescopes' resolution considerably.
Artificial intelligence combined
Artificial intelligence expands into all areas of the daily life, including research. Neural networks learn to solve complex tasks by training them on the basis of enormous amounts of examples. Researchers at the Max Planck Institute for Astrophysics in Garching have now succeeded in combining several networks, each one specializing in a different task, to jointly solve tasks using Bayesian logic in areas none was originally trained on. This enables the recycling of expensively trained networks and is an important step towards universally deductive artificial intelligence.
Next generation imaging
The Information Field Theory Group at the Max Planck Institute for Astrophysics has released a new version of the NIFTy software for scientific imaging. NIFTy5 generates an optimal imaging algorithm from the complex probability model of a measured signal. Such algorithms have already proven themselves in a number of astronomical applications and can now be used in other areas as well.
The embarrassment of false predictions 
How to best communicate probabilities?
Complex predictions such as election forecasts or the weather reports often have to be simplified before communication. But how should one best simplify these predictions without facing embarrassment? In astronomical data analysis, researchers are also confronted with the problem of simplifying probabilities. Two researchers at the Max Planck Institute for Astrophysics now show that there is only one mathematically correct way to measure how embarrassing a simplified prediction can be. According to this, the recipient of a prediction should be deprived of the smallest possible amount of information.
Galactic anatomy with gamma rays
The anatomy of the Milky Way as seen in gamma light is full of mysteries. For example, there are gigantic bubbles of unknown origin above and below the center of the Milky Way that emit a lot of this highenergy radiation. A new method for imaging, developed at the Max Planck Institute for Astrophysics, now divided the Galactic gammaradiation into three fundamental components: radiation from point sources, radiation from reactions of energetic protons with dense cold gas clouds, and radiation from electrons scattering light in the thin, hot, Galactic gas. The anatomic insights gained unravel some Galactic mysteries. Thus, it appears that the gammaray bubbles are simply outflows of ordinary, hot gas from the central region of the Milky Way.
New allsky map shows the magnetic fields of the Milky Way with the highest precision
With a unique new allsky map, scientists at MPA have made significant progress toward measuring the magnetic field structure of the Milky Way in unprecedented detail. Specifically, the map is of a quantity known as Faraday depth, which among other things, depends strongly on the magnetic fields along a particular line of sight. To produce the map, data were combined from more than 41,000 individual measurements using a novel image reconstruction technique. The work was a collaboration between scientists at the Max Planck Institute for Astrophysics (MPA), who are specialists in the new discipline of information field theory, and a large international team of radio astronomers. The new map not only reveals the structure of the galactic magnetic field on large scales, but also smallscale features that provide information about urbulence in the galactic gas.
D3PO: Denoising, Deconvolving, and Decomposing Photon Observations
A common problem for scientists analysing astronomical images is the separation of diffuse and pointlike components. This analysis has now become easier: scientists at the Max Planck Institute for Astrophysics have recently published the D3PO algorithm, which removes noise effects and instrumental artefacts from the observed images, while simultaneously separating diffuse and pointlike contributions.
Resolving the radio sky
Radio astronomers obtain extremely high resolution sky images by using interferometers, instruments where several single radio telescopes are linked together. However, optimal data analysis procedures for such an instrument are significantly more involved than for a single telescope. Scientists from the Max Planck Institute for Astrophysics have now developed the algorithm RESOLVE which solves a number of outstanding problems in radio imaging.
Data analysis and steam engines
As astronomical telescopes become more and more sensitive, the analysis techniques become ever more sophisticated. But do we need a new theoretical approach for a modern image reconstruction method? Not necessarily, a wellknown theory, originally developed for a better understanding of steam engines, does the trick: thermodynamics. Two researchers at the Max Planck Institute for Astrophysics have now shown that the so called Gibbs energy in thermodynamics, known for more than a century, can be applied to the development of new, optimal imaging techniques.
Mathematics of digital senses: Information Field Theory for signal recognition
The correct interpretation of signals through our senses is not only an essential problem of living creatures, but also of fundamental scientific relevance. Scientists at the MaxPlanckInstitute for Astrophysics have shown that mathematical methods from particle physics can be used for developing image reconstruction techniques. These yield optimal results even for incomplete, defective, and distorted data. Information Field Theory, which is used to develop such image reconstruction techniques, provides us with algorithms, i.e. mathematical instructions, for computing complicated perception processes in engineering and science, such as in cosmology.
IFT Introduction

Information Theory & Information Field Theory
University lectures by Torsten A. Enßlin (LMU Munich, 20.4.2020  ), script, handouts, exercises, ... 
Information Field Theory: from astrophysical imaging to artificial intelligence
Talk by Torsten A. Enßlin at Joint Astronomical Colloquium (ESO Garching, 13.2.2020)  Information theory for fields
Torsten A. Enßlin, Annalen der Physik special issue on "Physics of Information" online article" arXiv:1804.03350  Information field theory
Wikipedia  Why information field theory *is* a field theory
Torsten A. Enßlin, May 2015. link  Information field theory, an introduction in a nutshell
Torsten A. Enßlin, in MaxEnt 2012, the 32nd Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, arXiv:1301.2556  Astrophysical data analysis with information field theory, advanced recipes, also in a nutshell
Torsten A. Enßlin, accepted chapter to the conference proceedings for MaxEnt 2013, to be published by AIP, arXiv:1405.7701  Bayesian Field Theory: Nonparametric Approaches to Density Estimation, Regression, Classification, and Inverse Quantum Problems
J. C. Lemm, arxiv:physics/9912005 and more of his works  Bayesian Field Theory applied to scattered data interpolation and inverse problems.
C. Farmer. Algorithms for Approximation, pages 147166, 2007. eBook  A pathintegral approach to Bayesian inference for inverse problems using the semiclassical approximation
Joshua C Chang, Van Savage, Tom Chou, J Stat Phys (2014) 157: 582 arXiv:1312.2974  Information field theory for cosmological perturbation reconstruction and nonlinear signal analysis
Torsten A. Enßlin, Mona Frommert, Francisco S. Kitaura 2009, Phys. Rev. D 80, 105005 arXiv:0806.3474  IFT lecture notes and crash course notes
Theory
 Information Field Theory and Artificial Intelligence
Torsten A. Enßlin, Entropy 2022, 24, 374, arXiv:2105.10470  Geometric Variational Inference
Philipp Frank, Reimar Leike, Torsten A. Enßlin, Entropy 2021, 23, 853, arXiv:2105.10470  Towards Bayesian Data Compression
Johannes HarthKitzerow, Reimar Leike, Philipp Arras, Torsten A. Enßlin, Annalen der Physik 2021, Volume 533, 2000508, arXiv:2010.10375  Bayesian Reasoning with DeepLearned Knowledge
Jakob Knollmüller, Torsten A. Enßlin, 2021, Entropy 23, 693 arXiv:2001.11031  Metric Gaussian Variational Inference
Jakob Knollmüller, Torsten A. Enßlin, submitted arXiv:1901.11033  A Bayesian Model for Bivariate Causal Inference
Maximilian Kurthen, Torsten A. Enßlin, Entropy 2020, 22, 46; doi:10.3390/e22010046 arXiv:1812.09895  Bayesian parameter estimation of missspecified models
Johannes Oberpriller, Torsten A. Enßlin, submitted arXiv:1812.08194  Encoding prior knowledge in the structure of the likelihood
Jakob Knollmüller, Torsten A. Enßlin, submitted arXiv:1812.04403  Renormalization group computation of likelihood functions for cosmological data sets
Patrick McDonald, submitted arXiv:1810.08454  Separating diffuse from pointlike sources  a Bayesian approach
Jakob Knollmüller, Philipp Frank, Torsten A. Enßlin, submitted arXiv:1804.05591  Inference of signals with unknown correlation structure from nonlinear measurements
Jakob Knollmüller, Theo Steininger, Torsten A. Enßlin, submitted arXiv:1711.02955  Towards information optimal simulation of partial differential equations
Reimar H. Leike, Torsten A. Enßlin, Physical Review E, Volume 97, Issue 3, id.033314 ePrint, arXiv:1709.02859  Noisy independent component analysis of autocorrelated components
Jakob Knollmüller, Torsten A. Enßlin, Phys. Rev. E 96, 042114 (2017) arXiv:1705.02344  Correlated signal inference by free energy exploration
Torsten A. Enßlin, Jakob Knollmüller, submitted arXiv:1612.08406  Optimal Belief Approximation
Reimar Leike, Torsten A. Enßlin, Entropy 2017, 19, 402; doi:10.3390/e19080402 arxiv:1610.09018  Operator Calculus for Information Field Theory
Reimar Leike, Torsten A. Enßlin, Physical Review E, Volume 94, Issue 5, id.053306 (2016) arxiv:1605.00660  Signal inference with unknown response: calibration uncertainty renormalized estimator
Sebastian Dorn, Torsten A. Enßlin, Maksim Greiner, Marco Selig, Vanessa Boehm, PRE 91, 013311 (2015) arXiv:1410.6289  Improving selfcalibration
Torsten A. Enßlin, Henrik Junklewitz, Lars Winderling, Maksim Greiner, Marco Selig, 2014, PRE 90, id.043301 arXiv:1312.1349  Reconstruction of Gaussian and lognormal fields with spectral smoothness
Niels Oppermann, Marco Selig, Michael R. Bell, Torsten A. Enßlin, 2013, Phys. Rev. E 87, 032136 arXiv:1210.6866  Reconstructing signals from noisy data with unknown signal and noise covariances
Niels Oppermann, Georg Robbers, Torsten A. Enßlin, 2011, Physical Review E 84, 041118 arXiv:1107.2384  Reconstruction of signals with unknown spectra in information field theory with parameter uncertainty
Torsten A. Enßlin, Mona Frommert 2011, Physical Review D 83, 105014 arXiv:1002.2928  Inference with minimal Gibbs free energy in information field theory
Torsten A. Enßlin, Cornelius Weig 2010, Physical Review E 82, 051112 arXiv:1004.2868 and Comment on Paper and Reply to Comment  and more related papers
IFT Applications
 Variable structures in M87* from space, time and frequency resolved interferometry
Philipp Arras, Philipp Frank, Philipp Haim, Jakob Knollmüller, Reimar Leike, Martin Reinecke, Torsten A. Enßlin (2022), Nature Astronomy, https://doi.org/10.1038/s41550021015480 arXiv:2002.05218  Deep Images of the Galactic Center with GRAVITY
Julia Stadler et al. (2021), A&A, Forthcoming article arXiv:2112.07477  Studying Bioluminescence Flashes with the ANTARES Deep Sea Neutrino Telescope
Nico Reeb et al. (2021), arXiv:2107.08063  Nonparametric Bayesian Causal Modeling of the SARSCoV2 Viral Load Distribution vs. Patient's Age
Matteo Gurdiani et al. (2021), arXiv:2105.13483  The Galactic Faraday rotation sky 2020
Sebastian Hutschenreuter et al. (2021), arXiv:2102.01709  Reconstructing nonrepeating radio pulses with Information Field Theory
Christoph Welling, Philipp Frank, Torsten A. Enßlin, Anna Nelles (2021), arXiv:2102.00258  Comparison of classical and Bayesian imaging in radio interferometry
Philipp Arras, Richard A. Perley, Hertzog L. Bester, Reimar Leike, Oleg Smirnov, Rüdiger Westermann, Torsten A. Enßlin (2020), arXiv:2008.11435  Resolving nearby dust clouds
Reimar H. Leike, Martin Glatze, Torsten A. Enßlin (2020), Astronomy & Astrophysics, Volume 639, A138, 10.1051/00046361/202038169, arXiv:2004.06732  Unified Radio Interferometric Calibration and Imaging with Joint Uncertainty Quantification
Philipp Arras, Philipp Frank, Reimar Leike, Rüdiger Westermann, Torsten A. Enßlin (2019), Astronomy & Astrophysics, Volume 627, id.A134, 12 pp. 10.1051/00046361/201935555; arXiv:1903.11169  The Galactic Faraday depth sky revisited
Sebastian Hutschenreuter, Torsten A. Enßlin, Astronomy & Astrophysics, Volume 633, id.A150, 16 pp.d eJournal, arXiv:1903.06735  Charting nearby dust clouds using Gaia data only
Reimar Leike, Torsten A. Enßlin, A&A 631, A32 (2019); doi:10.1051/00046361/201935093; arXiv:1901.05971  The primordial magnetic field in our cosmic backyard
Sebastian Hutschenreuter, Sebastian Dorn, Jens Jasche, Franco Vazza, Daniela Paoletti, Guilhem Lavaux, Torsten A. Enßlin, Classical and Quantum Gravity, Volume 35, Issue 15, article id. 154001 (2018) DOI: 10.1088/13616382/aacde0 arXiv:1803.02629  Radio Imaging With Information Field Theory
Philipp Arras, Jakob Knollmüller, Henrik Junklewitz, Torsten A. Enßlin, submitted arXiv:1803.02174  Denoising, Deconvolving and Decomposing multiDimensional Photon Observations The D4PO Algorithm
Daniel Pumpe, Martin Reinecke, Torsten A. Enßlin, Astronomy & Astrophysics 619 , 119 (2018) arXiv:1802.02013  Inferring Galactic magnetic field model parameters using IMAGINE  An Interstellar MAGnetic field INference Engine
Theo Steininger, Torsten A. Enßlin, Maksim Greiner, Tess Jaffe, Ellert van der Velden, Jiaxin Wang, Marijke Haverkorn, Jörg R. Hörandel, Jens Jasche, Jörg P. Rachen, submitted arXiv:1801.04341  Bayesian weak lensing tomography: Reconstructing the 3D largescale distribution of matter with a lognormal prior
Vanessa Böhm, Stefan Hilbert, Maksim Greiner, Torsten A. Enßlin, Physical Review D, Volume 96, Issue 12, id.123510 arXiv:1701.01886 DOI  Search for quasiperiodic signals in magnetar giant flares
Daniel Pumpe, Michael Gabler, Theo Steininger, Torsten A. Enßlin, Astronomy & Astrophysics, Volume 610, id.A61, 12 pp. eJournal, arXiv:1708.05702  Information Field Theory with INTEGRAL/SPI data
Mahsa Ghaempanah, Dissertation, LMU München: Faculty of Physics (2017) thesis  Cosmic expansion history from SN Ia data via information field theory
Natalia Porqueres, Torsten A. Enßlin, Maksim Greiner, Vanessa Boehm, Sebastian Dorn, Pilar RuizLapuente, Alberto Manrique, submitted arXiv:1608.04007  Tomography of the Galactic free electron density with the Square Kilometer Array
Maksim Greiner, Dominic Schnitzeler, Torsten A. Enßlin, A&A 590, A59 (2016) arXiv:1512.03480  Stochastic determination of matrix determinants
Sebastian Dorn, Torsten A. Enßlin, Phys. Rev. E 92, 013302 (2015) arXiv:1504.02661  Using rotation measure grids to detect cosmological magnetic fields  a Bayesian approach
V. Vacca, N. Oppermann, T. Enßlin, J. Jasche, M. Selig, M. Greiner, H. Junklewitz, M. Reinecke, M. Brueggen, E. Carretti, L. Feretti, C. Ferrari, C. A. Hales, C. Horellou, S. Ideguchi, M. JohnstonHollitt, R. F. Pizzo, H. Roettgering, T. W. Shimwell, K. Takahashi, A&A 591, A13 (2016) arXiv:1509.00747  Estimating extragalactic Faraday rotation
Niels Oppermann, Henrik Junklewitz, Maksim Greiner, Torsten A. Enßlin, Takuya Akahori, Ettore Carretti, Bryan M. Gaensler, Ariel Goobar, Lisa HarveySmith, Melanie JohnstonHollitt, Luke Pratley, Dominic H. F. M. Schnitzeler, Jeroen M. Stil, Valentina Vacca, (2015) A&A 575, id.A118, 25 arXiv:1404.3701  A Bayesian method for pulsar template generation
M. Imgrund, D.J. Champion, M. Kramer, H. Lesch, MNRAS (June 01, 2015) 449 (4): 4162 arXiv:1501.03497  Allsky reconstruction of the primordial scalar potential from WMAP temperature data
Sebastian Dorn, Maksim Greiner, Torsten A. Enßlin, JCAP02 (2015) 041 arXiv:1412.8315 data  The Denoised, Deconvolved, and Decomposed Fermi gammaray Sky  An Application of the D3PO Algorithm
Marco Selig, Valentina Vacca, Niels Oppermann, Torsten A. Enßlin, A&A 581, A126 (2015) arXiv:1410.4562 data  Logtransforming the matter power spectrum
Maksim Greiner, Torsten A. Enßlin, A&A 574, A86 (2015) arXiv:1312.1354  Estimating extragalactic Faraday rotation
Niels Oppermann, Henrik Junklewitz, Maksim Greiner, Torsten A. Enßlin, Takuya Akahori, Ettore Carretti, Bryan M. Gaensler, Ariel Goobar, Lisa HarveySmith, Melanie JohnstonHollitt, Luke Pratley, Dominic H. F. M. Schnitzeler, Jeroen M. Stil, Valentina Vacca, accepted ba A&A, arXiv:1404.3701 arXiv:1404.3701 data  A new approach to multifrequency synthesis in radio interferometry
Henrik Junklewitz, Michael Bell, Marco Selig, Torsten A. Enßlin, Astronomy & Astrophysics, Volume 581, id.A5 (2015) arXiv:1401.4711  RESOLVE: A new algorithm for aperture synthesis imaging of extended emission in radio astronomy
Henrik Junklewitz, Michael Bell, Marco Selig, Torsten A. Enßlin, submitted, arXiv:1311.5282 arXiv:1311.5282  D3PO  Denoising, Deconvolving, and Decomposing Photon Observations
Marco Selig, Torsten A. Enßlin, accepted by Physical Review E, arXiv:1311.1888 arXiv:1311.1888  A fast and precise way to calculate the posterior for the local nonGaussianity parameter f_nl from Cosmic Microwave Background observations
Sebastian Dorn, Niels Oppermann, Rishi Khatri, Marco Selig, Torsten A. Enßlin, Phys. Rev. D 88, 103516 (2013) arXiv:1307.3884  The XENON100 exclusion limit without considering Leff as a nuisance parameter
Jonathan H. Davis, Celine Boehm, Niels Oppermann, Torsten A. Enßlin, Thomas Lacroix, Physical Review D, vol. 86, Issue 1, id. 015027, 2012 arXiv:1203.6823  An improved map of the Galactic Faraday sky
Niels Oppermann, et al., 2012, Astronomy & Astrophysics, Volume 542, id.A93 (2012) arXiv:1111.6186  Improving stochastic estimates with inference methods: calculating matrix diagonals
Marco Selig, Niels Oppermann, Torsten A. Enßlin, Phys. Rev. E 85, 021134 (2012) arXiv:1108.0600  Probing Magnetic Helicity with Synchrotron Radiation and Faraday Rotation
Niels Oppermann, Henrik Junklewitz, Georg Robbers, Torsten A. Enßlin 2011, Astronomy and Astrophysics, 530, id.A89 arXiv:1008.1246  Bayesian analysis of spatially distorted cosmic signals from Poissonian data
Cornelius Weig, Torsten A. Enßlin 2010, MNRAS 409, 1393 arXiv:1003.1311  Bayesian nonlinear large scale structure inference of the Sloan Digital Sky Survey data release 7
Jens Jasche, Francisco S. Kitaura, Cheng Li, Torsten A. Enßlin 2010, MNRAS 409, 355 arXiv:0911.2498  Fast Hamiltonian sampling for largescale structure inference
Jens Jasche, Francisco S. Kitaura 2010, MNRAS 407, 29 arXiv:0911.2496  Bayesian powerspectrum inference for Large Scale Structure data
Jens Jasche, Francisco S. Kitaura, Benjamin D. Wandelt, Torsten A. Enßlin 2010, MNRAS 406, 60 arXiv:0911.2493  Cosmic Cartography of the LargeScale Structure with Sloan Digital Sky Survey Data Release 6
Francisco S. Kitaura, Jens Jasche, Cheng Li, Torsten A. Enßlin, R.Benton Metcalf, Benjamin D. Wandelt, Gerard Lemson, Simon D.M. White 2009, MNRAS 400, 183 arXiv:0906.3978
Dynamics
 Dynamical field inference and supersymmetry
Margret Westerkamp, Igor Ovchinnikov, Torsten A. Enßlin, Entropy 2021, 23(12), 1652;
arXiv:2010.15414
 Probabilistic simulation of partial differential equations
Philipp Frank, Torsten A. Enßlin, 2020, submitted, arXiv:2010.06583  Field dynamics inference for local and causal interactions
Philipp Frank, Reimar Leike, Torsten A. Enßlin, 2021, Annalen der Physik, 2000486, arXiv:1902.02624  Towards informationoptimal simulation of partial differential equations
Reimar Leike, Torsten A. Enßlin, 2018, Physical Review E, 97, 033314; journal article, arXiv:1709.02859  Consistency and convergence of simulation schemes in Information field dynamics
Martin Dupont, Torsten A. Enßlin, 2018, Physical Review E, Vol. 98, No. 4, DOI: 10.1103/PhysRevE.98.043307, arXiv:1802.00971  Field dynamics inference via spectral density estimation
Philipp Frank, Theo Steininger, Torsten A. Enßlin, Phys. Rev. E 96, 052104 (2017) arXiv:1708.05250  Supersymmetric theory of stochastic ABC model
Igor Ovchinikov, Yuquan Sun, Torsten A. Enßlin and Kang L. Wang, J. Phys. Commun. (2018) doi.org/10.1088/23996528/aac94a  Dynamic system classifier
Daniel Pumpe, Maksim Greiner, Ewald Mueller, Torsten A. Enßlin, Physical Review E (Vol.94, No.1) DOI: 10.1103/PhysRevE.94.012132 arXiv:1601.07901  Kinematic dynamo, supersymmetry breaking, and chaos
Igor Ovchinikov, Torsten A. Enßlin, Physical Review D, Volume 93, Issue 8, id.085023 (2016) arXiv:1512.01651  Mathematical foundation of Information Field Dynamics
Christian Muench, master thesis, Technical University Munich arXiv:1412.1226  Simulation of stochastic network dynamics via entropic matching
Tiago Ramalho, Marco Selig, Ulrich Gerland, Torsten A. Enßlin, Phys. Rev. E 87, 022719 (2013) arXiv:1209.3700  Information field dynamics for simulation scheme construction
Torsten A. Enßlin, 2013, Phys. Rev. E 87, 013308 arXiv:1206.4229
IFT Tools
Bayesian Density Estimation for Poisson Data
(open source code) .
NIFTy5: Numerical Information Field Theory v5
NIFTy (Numerical Information Field Theory) facilitates the construction of Bayesian field reconstruction algorithms for fields being defined over multidimensional domains. A NIFTy algorithm can be developed for 1D field inference and then be used in 2D or 3D, on the sphere, or on product spaces thereof. NIFTy5 is a complete redesign of the previous framework (ascl:1302.013), and requires only the specification of a probabilistic generative model for all involved fields and the data in order to be able to recover the former from the latter. This is achieved via Metric Gaussian Variational Inference, which also provides posterior samples for all unknown quantities jointly. (open source code) .
HMCF  Hamiltonian Monte Carlo Sampling for Fields  A Python framework for HMC sampling with NIFTy
HMCF "Hamiltonian Monte Carlo for Fields", is a software addon for the NIFTy "Numerical Information Field Theory" framework implementing Hamilton Monte Carlo (HMC) sampling in Python. HMCF as well as NIFTy are designed to address field in ference problems especially in  but not limited to  astrophysics. They are optimized to deal with the typically high number of degrees of freedom as well as their correlation structure. HMCF adds an HMC sampler to NIFTy that automatically adjusts the many free pa rameters steering the HMC sampling machinery such as integration step size and the mass matrix according to the requirements of field inference. Furthermore, different convergence measures are available to check whether the burnin phase has finished. Multiprocessing in the sense of running individual Markov chains (MC) on several cores is possible as well. A primary application of HMCF is to provide samples from the full field posterior and to verify conveniently approximate algorithms implemented in NIFTy. (open source code) .
NIFTy 3  Numerical Information Field Theory  A Python framework for multicomponent signal inference on HPC clusters
NIFTy, "Numerical Information Field Theory", is a software framework designed to ease the development and implementation of field inference algorithms. Field equations are formulated independently of the underlying spatial geometry allowing the user to focus on the algorithmic design. Under the hood, NIFTy ensures that the discretization of the implemented equations is consistent. This enables the user to prototype an algorithm rapidly in 1D and then apply it to highdimensional realworld problems. This paper introduces NIFTy 3, a major upgrade to the original NIFTy framework. NIFTy 3 allows the user to run inference algorithms on massively parallel high performance computing clusters without changing the implementation of the field equations. It supports ndimensional Cartesian spaces, spherical spaces, power spaces, and product spaces as well as transforms to their harmonic counterparts. Furthermore, NIFTy 3 is able to treat nonscalar fields. The functionality and performance of the software package is demonstrated with example code, which implements a real inference algorithm from the realm of information field theory. NIFTy 3 is opensource software available under the GNU General Public License v3 (GPL3) at https://gitlab.mpcdf.mpg.de/ift/NIFTy/ .
NIFTY, Numerical Information Field Theory,
is a versatile library designed to enable the development of signal inference algorithms that operate regardless of the underlying spatial grid and its resolution. Its objectoriented framework is written in Python, although it accesses libraries written in Cython, C++, and C for efficiency.NIFTY offers a toolkit that abstracts discretized representations of continuous spaces, fields in these spaces, and operators acting on fields into classes. Thereby, the correct normalization of operations on fields is taken care of automatically without concerning the user. This allows for an abstract formulation and programming of inference algorithms, including those derived within information field theory. Thus, NIFTY permits its user to rapidly prototype algorithms in 1D and then apply the developed code in higherdimensional settings of real world problems. The set of spaces on which NIFTY operates comprises point sets, ndimensional regular grids, spherical spaces, their harmonic counterparts, and product spaces constructed as combinations of those.
 NIFTy 1 A versatile Python library for signal inference
 NIFTy 3 (supporting D20)
 D3PO Denoising, Deconvolving, and Decomposing Photon Observations
 Resolve aperture synthesis imaging in radio interferomerty
 Charm cosmic history agnostic reconstruction method
 D20 a distributed data object for parallel highperformance computing in Python PAPER
 Keepers
Further literature
 DIP  Diagnostics for Insufficiencies of Posterior calculations in Bayesian signal inference
Sebastian Dorn, Niels Oppermann, Torsten A. Enßlin, Phy. Rev. E 88 arXiv:1307.3889  Lectures on Probability, Entropy, and Statistical Physics
A. Caticha arXiv:0808.0012  MAGIC: Exact Bayesian Covariance Estimation and Signal Reconstruction for Gaussian Random Fields
B. Wandelt arXiv:ph/0401623