`Information Field Theory <http://www.mpa-garching.mpg.de/ift/>`_ [1]_ (IFT) is information theory, the logic of reasoning under uncertainty, applied to fields.

`Information Field Theory <https://www.mpa-garching.mpg.de/ift/>`_ [1]_ (IFT) is information theory, the logic of reasoning under uncertainty, applied to fields.

A field can be any quantity defined over some space, e.g. 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.

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@@ -22,89 +21,18 @@ NIFTy comes with reimplemented MAP and VI estimators.

.. tip:: *In-a-nutshell introductions to information field theory* can be found in [2]_, [3]_, [4]_, and [5]_, with the latter probably being the most didactical.

.. [1] T.A. Enßlin et al. (2009), "Information field theory for cosmological perturbation reconstruction and nonlinear signal analysis", PhysRevD.80.105005, 09/2009; `[arXiv:0806.3474] <http://www.arxiv.org/abs/0806.3474>`_

.. [1] T.A. Enßlin et al. (2009), "Information field theory for cosmological perturbation reconstruction and nonlinear signal analysis", PhysRevD.80.105005, 09/2009; `[arXiv:0806.3474] <https://www.arxiv.org/abs/0806.3474>`_

.. [2] T.A. Enßlin (2013), "Information field theory", proceedings of MaxEnt 2012 -- the 32nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering; AIP Conference Proceedings, Volume 1553, Issue 1, p.184; `[arXiv:1301.2556] <http://arxiv.org/abs/1301.2556>`_

.. [2] T.A. Enßlin (2013), "Information field theory", proceedings of MaxEnt 2012 -- the 32nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering; AIP Conference Proceedings, Volume 1553, Issue 1, p.184; `[arXiv:1301.2556] <https://arxiv.org/abs/1301.2556>`_

.. [3] T.A. Enßlin (2014), "Astrophysical data analysis with information field theory", AIP Conference Proceedings, Volume 1636, Issue 1, p.49; `[arXiv:1405.7701] <http://arxiv.org/abs/1405.7701>`_

.. [3] T.A. Enßlin (2014), "Astrophysical data analysis with information field theory", AIP Conference Proceedings, Volume 1636, Issue 1, p.49; `[arXiv:1405.7701] <https://arxiv.org/abs/1405.7701>`_

.. [4] Wikipedia contributors (2018), `"Information field theory" <https://en.wikipedia.org/w/index.php?title=Information_field_theory&oldid=876731720>`_, Wikipedia, The Free Encyclopedia.

.. [5] T.A. Enßlin (2019), "Information theory for fields", accepted by Annalen der Physik; `[DOI] <https://doi.org/10.1002/andp.201800127>`_, `[arXiv:1804.03350] <http://arxiv.org/abs/1804.03350>`_

Discretized continuum

---------------------

The representation of fields that are mathematically defined on a continuous space in a finite computer environment is a common necessity.

The goal hereby is to preserve the continuum limit in the calculus in order to ensure a resolution independent discretization.

Any partition of the continuous position space :math:`\Omega` (with volume :math:`V`) into a set of :math:`Q` disjoint, proper subsets :math:`\Omega_q` (with volumes :math:`V_q`) defines a pixelization,

Here the number :math:`Q` characterizes the resolution of the pixelization and the continuum limit is described by :math:`Q \rightarrow \infty` and :math:`V_q \rightarrow 0` for all :math:`q \in \{1,\dots,Q\}` simultaneously.

Moreover, the above equation defines a discretization of continuous integrals, :math:`\int_\Omega \mathrm{d}x \mapsto \sum_q V_q`.

Any valid discretization scheme for a field :math:`{s}` can be described by a mapping,

if the weighting function :math:`w_q(x)` is chosen appropriately.

In order for the discretized version of the field to converge to the actual field in the continuum limit, the weighting functions need to be normalized in each subset; i.e., :math:`\forall q: \int_{\Omega_q} \mathrm{d}x \; w_q(x) = 1`.

Choosing such a weighting function that is constant with respect to :math:`x` yields

.. [5] T.A. Enßlin (2019), "Information theory for fields", accepted by Annalen der Physik; `[DOI] <https://doi.org/10.1002/andp.201800127>`_, `[arXiv:1804.03350] <https://arxiv.org/abs/1804.03350>`_

which corresponds to a discretization of the field by spatial averaging.

Another common and equally valid choice is :math:`w_q(x) = \delta(x-x_q)`, which distinguishes some position :math:`x_q \in \Omega_q`, and evaluates the continuous field at this position,

In practice, one often makes use of the spatially averaged pixel position, :math:`x_q = \left< x \right>_{\Omega_q}`.

If the resolution is high enough to resolve all features of the signal field :math:`{s}`, both of these discretization schemes approximate each other, :math:`\left< s(x) \right>_{\Omega_q} \approx s(\left< x \right>_{\Omega_q})`, since they approximate the continuum limit by construction.

(The approximation of :math:`\left< s(x) \right>_{\Omega_q} \approx s(x_q \in \Omega_q)` marks a resolution threshold beyond which further refinement of the discretization reveals no new features; i.e., no new information content of the field :math:`{s}`.)

All operations involving position integrals can be normalized in accordance with the above definitions.

For example, the scalar product between two fields :math:`{s}` and :math:`{u}` is defined as

where :math:`\dagger` denotes adjunction and :math:`*` complex conjugation.

Since the above approximation becomes an equality in the continuum limit, the scalar product is independent of the pixelization scheme and resolution, if the latter is sufficiently high.

The above line of argumentation analogously applies to the discretization of operators.

For a linear operator :math:`{A}` acting on some field :math:`{s}` as :math:`{A} {s} = \int_\Omega \mathrm{d}y \; A(x,y) \; s(y)`, a matrix representation discretized with constant weighting functions is given by

The proper discretization of spaces, fields, and operators, as well as the normalization of position integrals, is essential for the conservation of the continuum limit.

Their consistent implementation in NIFTy allows a pixelization independent coding of algorithms.

Free Theory & Implicit Operators

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@@ -205,11 +133,11 @@ NIFTy takes advantage of this formulation in several ways:

3) The response can be non-linear, e.g. :math:`{R'(s)=R \exp(A\,\xi)}`, see `demos/getting_started_2.py`.

4) The amplitude operator may dependent on further parameters, e.g. :math:`A=A(\tau)= F\, \widehat{e^\tau}` represents an amplitude operator with a positive definite, unknown spectrum defined in the Fourier domain.

4) The amplitude operator may depend on further parameters, e.g. :math:`A=A(\tau)= F\, \widehat{e^\tau}` represents an amplitude operator with a positive definite, unknown spectrum defined in the Fourier domain.

The amplitude field :math:`{\tau}` would get its own amplitude operator, with a cepstrum (spectrum of a log spectrum) defined in quefrency space (harmonic space of a logarithmically binned harmonic space) to regularize its degrees of freedom by imposing some (user-defined degree of) spectral smoothness.

5) NIFTy calculates the gradient of the information Hamiltonian and the Fisher information metric with respect to all unknown parameters, here :math:`{\xi}` and :math:`{\tau}`, by automatic differentiation.

The gradients are used for MAP and HMCF estimates, and the Fisher matrix is required in addition to the gradient by Metric Gaussian Variational Inference (MGVI), which is available in NIFTy as well.

The gradients are used for MAP estimates, and the Fisher matrix is required in addition to the gradient by Metric Gaussian Variational Inference (MGVI), which is available in NIFTy as well.

MGVI is an implicit operator extension of Automatic Differentiation Variational Inference (ADVI).

The reconstruction of a non-Gaussian signal with unknown covariance from a non-trivial (tomographic) response is demonstrated in `demos/getting_started_3.py`.

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@@ -296,7 +224,7 @@ Thus, only the gradient of the KL is needed with respect to this, which can be e

We stochastically estimate the KL-divergence and gradients with a set of samples drawn from the approximate posterior distribution.

The particular structure of the covariance allows us to draw independent samples solving a certain system of equations.

This KL-divergence for MGVI is implemented in the class MetricGaussianKL within NIFTy5.

This KL-divergence for MGVI is implemented in the class :class:`~minimization.metric_gaussian_kl.MetricGaussianKL` within NIFTy5.

The demo `getting_started_3.py` for example not only infers a field this way, but also the power spectrum of the process that has generated the field.

**NIFTy** [1]_, [2]_, "\ **N**\umerical **I**\nformation **F**\ield **T**\heor\ **y**\ ", is a versatile library designed to enable the development of signal inference algorithms that are independent of the underlying spatial grid and its resolution.

**NIFTy** [1]_, [2]_, "\ **N**\umerical **I**\nformation **F**\ield **T**\heor\ **y**\ ", is a versatile library designed to enable the development of signal inference algorithms that are independent of the underlying grids (spatial, spectral, temporal, …) and their resolutions.

Its object-oriented framework is written in Python, although it accesses libraries written in C++ and C for efficiency.