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Information field theory for cosmological perturbation reconstruction and non-linear signal analysis

Torsten A. Ensslin, Mona Frommert, Francisco S. Kitaura

arXiv:0806.3474v3astro-phcs.IThep-thmath.ITphysics.data-anstat.CO

TL;DR

The paper develops information field theory (IFT) for Bayesian inference on spatially distributed signals. It derives a field-theoretic framework and applies it to galaxy-survey reconstruction and CMB non-Gaussianity estimation, reporting reconstruction through response renormalization and an optimal estimator to linear order in the non-linearity parameter.

Abstract

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We develop information field theory (IFT) as a means of Bayesian inference on spatially distributed signals, the information fields. A didactical approach is attempted. Starting from general considerations on the nature of measurements, signals, noise, and their relation to a physical reality, we derive the information Hamiltonian, the source field, propagator, and interaction terms. Free IFT reproduces the well known Wiener-filter theory. Interacting IFT can be diagrammatically expanded, for which we provide the Feynman rules in position-, Fourier-, and spherical harmonics space, and the Boltzmann-Shannon information measure. The theory should be applicable in many fields. However, here, two cosmological signal recovery problems are discussed in their IFT-formulation. 1) Reconstruction of the cosmic large-scale structure matter distribution from discrete galaxy counts in incomplete galaxy surveys within a simple model of galaxy formation. We show that a Gaussian signal, which should resemble the initial density perturbations of the Universe, observed with a strongly non-linear, incomplete and Poissonian-noise affected response, as the processes of structure and galaxy formation and observations provide, can be reconstructed thanks to the virtue of a response-renormalization flow equation. 2) We design a filter to detect local non-linearities in the cosmic microwave background, which are predicted from some Early-Universe inflationary scenarios, and expected due to measurement imperfections. This filter is the optimal Bayes' estimator up to linear order in the non-linearity parameter and can be used even to construct sky maps of non-linearities in the data.

I. INTRODUCTION

The introduction frames incomplete, noisy observations as Bayesian inference problems and presents information field theory (IFT) as a field-theoretic framework for optimal signal recovery. It situates the framework among prior work and motivates it through cosmological applications to large-scale structure and the CMB.

A. Motivation

  • IFT treats spatially distributed signal recovery as a statistical field theory, using diagrammatic perturbation theory and renormalization flows to construct algorithms and calculate uncertainty moments.The introduction contrasts this with earlier work that mostly treated such problems at a classical field level.Source: A. Motivation · p. 1

B. Overview of the work

  • The paper develops IFT concepts and formalism before presenting interacting-field theory and two cosmological applications: galaxy-survey reconstruction and CMB non-Gaussianity estimation.The applications are proposed as blueprints for similar inference problems.Source: B. Overview of the work · p. 2

3. Statistical and Bayesian field theory

  • The IFT framework unifies information theory and statistical field theory to derive optimal tools for cosmological signal analysis and other disciplines.The introduction notes that the framework is intended to help classify existing signal-extraction methods as well as derive new ones.Source: A. Motivation · p. 1, C. Previous works · p. 2

4. Cosmological large-scale structure

  • Galaxy surveys trace large-scale structure, but non-linear evolution, galaxy formation, and redshift effects complicate reconstruction of the initial density field.The introduction describes Wiener-filter applications to extrapolating matter into the zone of avoidance.Source: §4 · pp. 3–4

5. Cosmic Microwave Background

  • CMB temperature fluctuations encode cosmological information, while inflationary scenarios predict potentially discriminating non-Gaussian features whose detection has remained challenging.Earlier three-point-function tests were mostly negative and insufficiently sensitive to seriously constrain the possible inflationary parameter space [202, 203, 204, 205, 206].Source: §5 · pp. 4–5

A. Motivation

1. Information theory and Bayesian inference; 2. Image reconstruction in astronomy and elsewhere

  • Bayesian inference represents incomplete knowledge probabilistically, while image reconstruction methods such as the Wiener filter and Richardson-Lucy algorithm use assumed data models.The Wiener filter is presented as full Bayesian inference for Gaussian signal and noise statistics.Source: §2 · pp. 2–3, A. Information on physical fields · p. 5

B. Signal response and noise

The section defines response and noise through the data’s linear correlation with a chosen signal, then explains how transformations and signal design affect reconstruction. It identifies the posterior mean as the optimal map under the stated quadratic-error criterion.

B. Signal response and noise

  • The response contains the data component linearly correlated with the chosen signal, while noise is the remaining, linearly uncorrelated component.Noise may still depend on the signal, as in Poisson processes, and retain higher-order correlations.Source: B. Signal response and noise · p. 6
  • Data transformations can change response and noise, so information theory can guide choices that increase signal response and reduce noise.The definitions of response and noise depend on the data-space coordinates and are not unique.Source: B. Signal response and noise · p. 6
  • The posterior mean is the optimal map when minimizing the stated expected quadratic signal uncertainty.The trace of the uncertainty is the expected squared L2 distance between the reconstruction and underlying signal.Source: B. Signal response and noise · p. 6
  • Uncertainty depends on the observed data in non-linear inference, whereas joint-probability averaging characterizes an estimator’s overall performance.The passage distinguishes uncertainty for a given dataset from a frequentist uncertainty estimate averaged before observing data.Source: B. Signal response and noise · pp. 6–7

C. Signal design

  • Signal design favors simple forms correlated with the data and practical enough to support tractable likelihood and prior approximations.In the galaxy example, a low-pass-filtered initial density field suppresses small-scale modes that discrete galaxy positions may not constrain.Source: C. Signal design · p. 7

D. Signal moment calculation

The posterior contains the signal information from which moments can be computed, but non-Gaussian posteriors are often analytically difficult. IFT organizes signal inference through discretized fields and an information Hamiltonian, and uses diagrammatic perturbation to extend Gaussian methods.

D. Signal moment calculation

  • Signal moments follow from averages over the posterior, but non-Gaussian posteriors are often analytically intractable; the paper expands them perturbatively around Gaussian posteriors.For a multivariate Gaussian posterior, moments are known and in principle computable.Source: D. Signal moment calculation · p. 7
  • Diagrammatic expressions encode the minimal computational information flow and can use existing map-making codes to compute perturbation series.The paper reformulates inference in language close to statistical and quantum field theory.Source: D. Signal moment calculation · p. 8

1. Discretisation and continuous limit

  • A well-defined continuous limit means reconstructions on discretizations finer than the signal structures should agree up to differences that vanish with resolution.The authors note that practical analyses rarely need to take the limit explicitly.Source: §1 · p. 8

2. Parameter spaces

  • IFT can also reconstruct an incompletely known parameter-knowledge function P(p|d), including uncertainty induced by an unknown nuisance parameter.When the parameter itself has spatial variations, this is an information field theory over the parameter.Source: §2 · pp. 8–9

A. Information Hamiltonian

  • The Hamiltonian’s linear, quadratic, and higher-order coefficients define the information source, propagator, and interactions between free-theory modes.The source is usually directly and linearly related to the data; the propagator carries information between locations.Source: A. Information Hamiltonian · p. 9

B. Free theory

For a Gaussian signal prior and a linear response with signal-independent Gaussian noise, the free IFT posterior is tractable. Its mean is the generalized Wiener-filter map, and its residual covariance is given by the propagator.

1. Gaussian data model

  • The free Gaussian model assumes a Gaussian signal prior, linear response, and signal-independent Gaussian noise.The response may include window and selection functions, blurring, or a Fourier transformation.Source: §1 · p. 10
  • The posterior mean map is the data after application of a generalized Wiener filter.The free-theory propagator describes how information in the data at one location propagates to another.Source: §1 · p. 10
  • The posterior is Gaussian around the mean map, with residual covariance given by the propagator and all higher connected correlations equal to zero.The connected autocorrelation is the propagator itself.Source: §1 · p. 10

2. Free classical theory

The free classical extremum reproduces the field-theoretical mean, while its Gaussian curvature gives the exact residual covariance in the pure Gaussian model. The section then specifies how non-Gaussian or non-linear features enter IFT and how diagrams generate moments.

2. Free classical theory

  • For the free theory, extremizing the Hamiltonian gives the same map as the field-theoretical result, and the Gaussian approximation to the maximum has the exact residual covariance.The extremum is equivalent to a maximum a posteriori estimator; classical and field-theoretical results may diverge when fluctuations probe more complex posterior regions away from the maximum.Source: §2 · p. 11

1. General Form

  • Non-Gaussian signal or noise, non-linear response, or signal-dependent noise create anharmonic interactions between free-Hamiltonian modes.Perturbations constant, linear, or quadratic in the signal can also represent certain non-ideal effects.Source: §1 · p. 11

2. Feynman rules

  • Connected diagrams built from propagator lines and interaction vertices generate the partition-sum expansion and signal moments.Taking derivatives of log Z[J] generates moments; open diagram ends correspond to their external coordinates.Source: §2 · p. 12

3. Local interactions and Fourier space rules

For local interactions, IFT’s diagrammatic rules simplify and can be expressed in Fourier space. A translationally invariant toy model with local interactions further reduces the rules to power-spectrum quantities and constant interaction coefficients.

3. Local interactions and Fourier space rules

  • For purely local interactions, each vertex with n connected lines represents the local coefficient −λ_n(x′), simplifying the real-space Feynman rules.Fourier-space rules can be obtained by inserting the Fourier identity between scalar products.Source: §3 · p. 13

2. A line connecting momentum k with momentum k′

  • In Fourier space, propagators connect momentum labels, vertices enforce momentum conservation, and internal momenta are integrated with the diagram’s symmetry factor.Momentum directions determine the signs in the momentum-conservation delta function.Source: §2 · p. 13

4. Simplistic interaction Hamiltonians

  • The analytic toy case assumes translational invariance, local interactions, power-spectrum covariances, and signal processing by convolution with an instrumental beam.Under these assumptions the propagator is described by a power spectrum and the interaction coefficients are constant.Source: §4 · p. 13

5. Feynman rules on the sphere

IFT supplies Feynman rules on the sphere for CMB analysis and develops interacting-field tools for inference. Its convergence arguments are conditional rather than a general proof of normalisability.

5. Feynman rules on the sphere

  • The spherical Feynman rules are provided for CMB reconstruction and analysis, and are expected to be useful for terrestrial applications as well.They are given for fields on S2.Source: §5 · p. 13

B. Normalisability of the theory

  • Loop diagrams converge for bounded signal variance; choosing a suitably filtered information field can also ensure convergence, but the paper does not prove normalisability in general.For unbounded variance, convergence requires the stated integral condition to be finite.Source: B. Normalisability of the theory · p. 14

2. Local interactions

  • The classical solution contains tree diagrams, while loop diagrams from fluctuations around it provide uncertainty corrections; local interactions make this expansion explicit.When the classical solution is exact, the fluctuation theory has no one-line internal vertices.Source: §2 · pp. 14–15

1. Helmholtz free energy

  • The Boltzmann-Shannon negative-entropy measure quantifies signal information through the phase-space volume available to signal uncertainty.The corresponding Helmholtz free energy is related to the information measure through the thermodynamic relation F = E − T S_B, with T set to 1.Source: §1 · p. 15
  • The Helmholtz free energy generates connected signal correlations, and diagram order is determined by vertices minus propagators minus one.Tree diagrams are order β^0, one-loop diagrams β^-1, and two-loop diagrams β^-2; only loop terms affect the information content.Source: §1 · pp. 15–16

2. Free theory

The free theory characterizes data-induced information gain by the signal-response-to-noise ratio. The section also outlines an IFT workflow and a Poissonian galaxy-count model for nonlinear reconstruction.

2. Free theory

  • In the free theory, information gain rises linearly with signal-response-to-noise ratio Q when Q ≪ 1 and logarithmically when Q ≫ 1.For the free theory, information gain does not depend on the particular data realization.Source: §2 · p. 16

E. IFT Recipe

  • The IFT recipe specifies a signal prior and data likelihood, expands the Hamiltonian into source, propagator, and interaction terms, then computes quantities with diagrams.If interactions prevent a finite diagram set from performing well, the recipe calls for resummation, including through a saddle-point or renormalization-flow analysis.Source: E. IFT Recipe · pp. 16–17

A. Poissonian data model and Hamiltonian

  • For galaxy reconstruction, the Poissonian data model uses a log-density Gaussian signal and an exponential response, allowing spatially varying bias and zero-response parameters assumed known.This model accommodates unrestricted negative signal fluctuations, but is presented as an illustrative, exchangeable choice.Source: A. Poissonian data model and Hamiltonian · p. 17
  • In the Poisson model, effective inverse noise decreases as mean galaxy number and bias increase, while the information source vanishes at zero response or when counts match their expected mean.The interaction terms are local in position space and vanish with decreasing bias and zero-response parameter.Source: A. Poissonian data model and Hamiltonian · p. 18

B. Galaxy types and bias variations

The paper extends Poissonian galaxy reconstruction to variable galaxy types and nonlinear response corrections. Renormalization-based and classical reconstructions can look distinct, yet their statistical difference is reported as marginal.

B. Galaxy types and bias variations

  • A single bias value can give reasonably accurate matter reconstruction because numerous smaller galaxies dominate the total information source, noise reduction, and most low-order interactions.Galaxy-type dependence matters for unity-variance signal fluctuations, but incorporating types does not increase reconstruction complexity beyond added setup bookkeeping.Source: B. Galaxy types and bias variations · p. 18

C. Non-linear map making

  • The nonlinear map corrects the linear Wiener map for asymmetric data fluctuations and signal-response curvature; at b ≈ 1, first-order corrections can overcorrect and require resummation.At b = 0.5, the corrections mostly point in the right direction, and the classical solution is close to the first-order map.Source: C. Non-linear map making · p. 19

D. Classical solution

  • The classical solution is more conservative than naive response inversion because a damping term limits the influence of overly large data points.The cited inversion is considered suitable only in regimes of large observed counts.Source: D. Classical solution · p. 19
  • In the strongly nonlinear, masked example, the classical estimate can be improved in high-uncertainty regions because missing loop corrections encode posterior structure away from the MAP.The figure compares the classical, intermediate, and renormalization-based reconstructions and their deviations from the signal.Source: D. Classical solution · p. 20, Figure 2 · p. 21

E. Uncertainty-loop corrections

Uncertainty-loop corrections refine the classical field solution and its density estimator, while response renormalization continuously updates a Gaussian approximation as information and interactions are introduced. In the reconstruction example, the intermediate T = 1/2 solution performs better than the classical and fully renormalized endpoints.

E. Uncertainty-loop corrections

  • Resumming loop diagrams into effective interaction vertices absorbs uncertainty fluctuations into the interaction coefficients.The fluctuations increase expected response to a reference field because the Hamiltonian’s exponential term is asymmetric.Source: E. Uncertainty-loop corrections · p. 22
  • The uncertainty-field Hamiltonian and a self-consistently determined propagator provide loop corrections to the classical solution, confirmed by a renormalization calculation within an approximation.The resulting loop-corrected density and propagator support estimators for the dark matter density itself.Source: E. Uncertainty-loop corrections · pp. 21–22
  • Uncertainty-loop corrections shift the mean density estimator to account for the nonlinear transformation from log-density to density.The corrected estimator incorporates the propagator-based uncertainty term.Source: E. Uncertainty-loop corrections · p. 22

F. Response renormalization

  • Response renormalization treats information acquisition as pseudo-time, feeding the information source and interactions into the knowledge state at the same rate.At each step, the knowledge state is approximated by a Gaussian with updated mean and dispersion.Source: F. Response renormalization · pp. 22–23
  • The response-renormalization flow converges to the previously proposed fixed point for the uncertainty-loop-enhanced classical equation.The flow updates the propagator from the connected two-point function; the fixed point and classical equations correspond to T = 1 and T = 0, respectively.Source: F. Response renormalization · p. 24
  • T = 1/2 reconstructions outperform the T = 0 classical and T = 1 renormalization solutions, although their statistical difference is at most marginal.The comparison uses 1000 signal and data realizations; the authors attribute the differing estimates to Gaussian approximations probing a non-Gaussian posterior differently.Source: F. Response renormalization · p. 24

G. Uncertainty structure

The renormalized propagator gives a data-informed approximation to posterior uncertainty, rather than the prior-and-mask structure alone. Its spatial variation reflects the nonlinear response and expected galaxy counts, with implications for how additional observations affect the reconstruction.

G. Uncertainty structure

  • The renormalized propagator better approximates posterior uncertainty around the mean map than the original propagator, and its structure reflects the data.Uncertainty is smaller where galaxy counts are high, and the density estimator benefits from this uncertainty structure.Source: G. Uncertainty structure · p. 24
  • Propagator height and width encode response strength and information-propagation distance, while the original propagator reflects the prior and survey mask.Where the mask is transparent, expected information is richer, so response per source and propagation width are lower.Source: G. Uncertainty structure · p. 24
  • With the nonlinear response, high-count regions have narrower propagators, so an additional galaxy there has less impact on the map than one in a low-density region.More galaxies may nevertheless be observed per unit time in high-density regions, potentially compensating for the lower information per galaxy.Source: G. Uncertainty structure · p. 24

H. Information gain

In nonlinear inference, information gain can depend on the realized data as well as observational sensitivity, unlike in a free theory. The paper develops information-gain calculations for galaxy reconstruction and describes a local-non-Gaussianity model for CMB fluctuations, assuming Gaussian noise while noting that non-Gaussian noise must also be addressed for optimal reconstruction.

H. Information gain

  • In a free theory, information depends on the experimental setup and prior but is independent of obtained data; in a nonlinear IFT, it can strongly depend on the actual data.Source: H. Information gain · p. 25
  • Expected information gain is especially large where uncertainty, expected galaxy number density, and nonlinearities are high, with an inverse term capping the available gain.The examples show that observation priorities depend on coverage and inferred density: filling gaps can be preferred, while high-density regions can sometimes warrant further observations over poorly observed low-density regions.Source: H. Information gain · pp. 26–27

A. Data model

  • For local CMB non-Gaussianity, the model treats the primordial potential as a slightly nonlinear transformation of a Gaussian field, with f controlling the level and sign of non-Gaussianity.The measurement response maps the primordial gravitational field to temperature and polarization data.Source: A. Data model · p. 28
  • The CMB data model assumes Gaussian noise, but non-Gaussian noise components are expected and would need inclusion for an optimal f reconstruction.The stated scope is to outline the principles and show how f-maps could help identify some such noise sources through comparison with foreground structures.Source: A. Data model · p. 28

B. Spectrum, bispectrum, and trispectrum

The paper uses IFT to derive moments of the nonlinearly transformed primordial potential and formulate a CMB Hamiltonian. After marginalizing over primordial fluctuations, it obtains a lowest-order estimator for f and describes extending inference to spatial maps that can be compared with foreground sources.

B. Spectrum, bispectrum, and trispectrum

  • Because the nonlinear relation between the hidden Gaussian field and gravitational potential imprints non-Gaussianity, extracting f requires statistics up to the trispectrum.A moment-generating function and its derivatives provide the mean, covariance, and connected three-point function; the bispectrum is given through O(f^3).Source: B. Spectrum, bispectrum, and trispectrum · p. 28

C. CMB-Hamiltonian

D. fnl-evidence and map making

  • After marginalizing over primordial fluctuations, the paper derives the optimal f estimator to lowest order from the linear and quadratic coefficients of its Hamiltonian.The derivation expands the log-evidence to quadratic order in f.Source: D. fnl-evidence and map making · pp. 29–30
  • The method can be extended to spatially varying f maps, which can be compared with known foreground sources to assess non-Gaussian contamination.The paper describes f-map construction in spherical harmonics space.Source: D. fnl-evidence and map making · p. 30

E. Comparison to traditional estimator

The IFT estimator differs from the traditional CMB non-Gaussianity estimator in its treatment of inhomogeneity, exposure coupling, and normalization. The paper also summarizes IFT’s formal results, cosmological applications, and the need for more realistic cosmological data models.

E. Comparison to traditional estimator

  • The IFT filter contains terms up to fourth order in the data, compared with the traditional estimator’s bispectrum-based third-order form.The authors expect the higher-order terms may improve accuracy, while noting that trispectrum information is more subtle because it is insensitive to the sign of f_nl.Source: E. Comparison to traditional estimator · p. 30
  • The IFT filter handles spatially varying non-linearity, corrects chance correlations between the CMB sky and exposure, and is unbiased with respect to the posterior.The traditional estimator is described as unbiased over signal and data realizations; the IFT estimator’s normalization depends on the observed data.Source: E. Comparison to traditional estimator · pp. 30–31
  • The IFT estimator’s tree-diagram term is identical to the unnormalized traditional estimator, while its loop term corrects chance correlations under inhomogeneous exposure.The loop term vanishes for homogeneous M; the cited comparison notes that a linear correction is needed for inhomogeneous sky coverage.Source: E. Comparison to traditional estimator · p. 31
  • The IFT estimator uses data-dependent normalization, unlike the traditional estimator’s data-independent constant, reflecting its Bayesian rather than frequentist unbiasedness criterion.The paper relates this dependence to differing detectability across data realizations, including cases with identical source ˜j.Source: E. Comparison to traditional estimator · p. 31

VII. SUMMARY AND OUTLOOK

  • The framework derives an information Hamiltonian, recovers Wiener-filter theory in the free case, and supplies Feynman rules and a Boltzmann-Shannon information measure.The measure is based on Helmholtz free energy; the propagator is closely related to the Wiener filter and the information source is usually a noise-weighted version of the data.Source: §VII · p. 31
  • The two cosmological applications are reconstruction of continuous matter fields from incomplete galaxy counts and an optimal method for measuring or constraining local CMB non-linearities.The galaxy-reconstruction algorithm is also described as applicable to low-number photon image reconstruction, including low-dose X-ray imaging.Source: §VII · pp. 31–32
  • More realistic cosmological data models remain to be investigated, including statistical galaxy-formation responses and CMB foreground-subtraction remnants.The proposed galaxy response model would require higher-order correlation coefficients from simulations or semi-analytic descriptions.Source: §VII · p. 32

APPENDIX A: NOTATION

The appendices set out notation for fields and probabilities, give spherical-harmonic Feynman rules, and list terms in the f_nl propagator.

APPENDIX A: NOTATION

  • Functions are treated as vectors in a Hilbert space, while a statistically homogeneous Gaussian field’s covariance is described by its power spectrum.The notation suppresses dependence on model parameters, which are assumed known in this work.Source: Appendix A · pp. 32–33

APPENDIX B: FEYNMAN RULES ON THE SPHERE

  • For an isotropic propagator on the sphere, spherical harmonics diagonalize it; interaction structure is more complicated than in Fourier space, so high-order vertices may be computed in real space.The rules use directed propagator lines and angular-momentum labels; the interaction structure is encoded by coefficients involving Wigner coefficients.Source: Appendix B · pp. 33–34

APPENDIX C: fnl-PROPAGATOR

  • The appendix lists the individual diagrams contributing to the f_nl propagator, each of order O(f^2), for f = 1.Source: Appendix C · p. 34
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