# Reconstruction of signals with unknown spectra in information field theory with parameter uncertainty

## Abstract

The optimal reconstruction of cosmic metric perturbations and other signals requires knowledge of their power spectra and other parameters. If these are not known a priori, they have to be measured simultaneously from the same data used for the signal reconstruction. We formulate the general problem of signal inference in the presence of unknown parameters within the framework of information field theory. To solve this, we develop a generic parameter-uncertainty renormalized estimation (PURE) technique. As a concrete application, we address the problem of reconstructing Gaussian signals with unknown power-spectrum with five different approaches: (i) separate maximum-a-posteriori power-spectrum measurement and subsequent reconstruction, (ii) maximum-a-posteriori reconstruction with marginalized power-spectrum, (iii) maximizing the joint posterior of signal and spectrum, (iv) guessing the spectrum from the variance in the Wiener-filter map, and (v) renormalization flow analysis of the field-theoretical problem providing the PURE filter. In all cases, the reconstruction can be described or approximated as Wiener-filter operations with assumed signal spectra derived from the data according to the same recipe, but with differing coefficients. All of these filters, except the renormalized one, exhibit a perception threshold in case of a Jeffreys prior for the unknown spectrum. Data modes with variance below this threshold domore »

- Authors:

- Max-Planck-Institut fuer Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching (Germany)

- Publication Date:

- OSTI Identifier:
- 21502637

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 83; Journal Issue: 10; Other Information: DOI: 10.1103/PhysRevD.83.105014; (c) 2011 American Institute of Physics

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CORRECTIONS; COSMOLOGY; DISTURBANCES; FIELD THEORIES; FILTERS; GALAXIES; INFORMATION THEORY; METRICS; PERTURBATION THEORY; RENORMALIZATION; ROUGHNESS; SMOOTH MANIFOLDS; SPECTRA; MATHEMATICAL MANIFOLDS; SURFACE PROPERTIES

### Citation Formats

```
Ensslin, Torsten A., and Frommert, Mona.
```*Reconstruction of signals with unknown spectra in information field theory with parameter uncertainty*. United States: N. p., 2011.
Web. doi:10.1103/PHYSREVD.83.105014.

```
Ensslin, Torsten A., & Frommert, Mona.
```*Reconstruction of signals with unknown spectra in information field theory with parameter uncertainty*. United States. doi:10.1103/PHYSREVD.83.105014.

```
Ensslin, Torsten A., and Frommert, Mona. Sun .
"Reconstruction of signals with unknown spectra in information field theory with parameter uncertainty". United States. doi:10.1103/PHYSREVD.83.105014.
```

```
@article{osti_21502637,
```

title = {Reconstruction of signals with unknown spectra in information field theory with parameter uncertainty},

author = {Ensslin, Torsten A. and Frommert, Mona},

abstractNote = {The optimal reconstruction of cosmic metric perturbations and other signals requires knowledge of their power spectra and other parameters. If these are not known a priori, they have to be measured simultaneously from the same data used for the signal reconstruction. We formulate the general problem of signal inference in the presence of unknown parameters within the framework of information field theory. To solve this, we develop a generic parameter-uncertainty renormalized estimation (PURE) technique. As a concrete application, we address the problem of reconstructing Gaussian signals with unknown power-spectrum with five different approaches: (i) separate maximum-a-posteriori power-spectrum measurement and subsequent reconstruction, (ii) maximum-a-posteriori reconstruction with marginalized power-spectrum, (iii) maximizing the joint posterior of signal and spectrum, (iv) guessing the spectrum from the variance in the Wiener-filter map, and (v) renormalization flow analysis of the field-theoretical problem providing the PURE filter. In all cases, the reconstruction can be described or approximated as Wiener-filter operations with assumed signal spectra derived from the data according to the same recipe, but with differing coefficients. All of these filters, except the renormalized one, exhibit a perception threshold in case of a Jeffreys prior for the unknown spectrum. Data modes with variance below this threshold do not affect the signal reconstruction at all. Filter (iv) seems to be similar to the so-called Karhune-Loeve and Feldman-Kaiser-Peacock estimators for galaxy power spectra used in cosmology, which therefore should also exhibit a marginal perception threshold if correctly implemented. We present statistical performance tests and show that the PURE filter is superior to the others, especially if the post-Wiener-filter corrections are included or in case an additional scale-independent spectral smoothness prior can be adopted.},

doi = {10.1103/PHYSREVD.83.105014},

journal = {Physical Review. D, Particles Fields},

number = 10,

volume = 83,

place = {United States},

year = {Sun May 15 00:00:00 EDT 2011},

month = {Sun May 15 00:00:00 EDT 2011}

}