@@ -139,7 +139,7 @@ the information source. The operation in :math:`{d= D\,R^\dagger N^{-1} d}` is a

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@@ -139,7 +139,7 @@ the information source. The operation in :math:`{d= D\,R^\dagger N^{-1} d}` is a

NIFTy permits to define the involved operators :math:`{R}`, :math:`{R^\dagger}`, :math:`{S}`, and :math:`{N}` implicitely, as routines that can be applied to vectors, but which do not require the explicit storage of the matrix elements of the operators.

NIFTy permits to define the involved operators :math:`{R}`, :math:`{R^\dagger}`, :math:`{S}`, and :math:`{N}` implicitely, as routines that can be applied to vectors, but which do not require the explicit storage of the matrix elements of the operators.

These implicit operators can be combined into new operators, e.g. to :math:`{D^{-1} = \left( S^{-1} + R^\dagger N^{-1} R\right)^{-1}}`, as well as their inverses, e.g. :math:`{D^{-1} = \left( D^{-1} \right)^{-1}}`.

These implicit operators can be combined into new operators, e.g. to :math:`{D^{-1} = \left( S^{-1} + R^\dagger N^{-1} R\right)^{-1}}`, as well as their inverses, e.g. :math:`{D = \left( D^{-1} \right)^{-1}}`.

The invocation of an inverse operator applied to a vector might trigger the execution of a numerical linear algebra solver.

The invocation of an inverse operator applied to a vector might trigger the execution of a numerical linear algebra solver.

Thus, when NIFTy calculates :math:`{m = D\, j}` it actually solves :math:`{D^{-1} m = j}` for :math:`{m}` behind the scenes.

Thus, when NIFTy calculates :math:`{m = D\, j}` it actually solves :math:`{D^{-1} m = j}` for :math:`{m}` behind the scenes.

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@@ -160,23 +160,23 @@ The above free theory case looks as a generative model like the following:

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@@ -160,23 +160,23 @@ The above free theory case looks as a generative model like the following:

s = A\,\xi

s = A\,\xi

with :math:`{A}` the amplitude operator such that it generates signal field with the correct covariance :math:`{S=A\,A^\dagger}` out of a Gaussian white noise field :math:`{\xi}` with :math:`{\mathcal{P}(\xi)= \mathcal{G}(\xi, \mathbb{1})}`.

with :math:`{A}` the amplitude operator such that it generates signal field with the correct covariance :math:`{S=A\,A^\dagger}` out of a Gaussian white noise field :math:`{\xi}` with :math:`{\mathcal{P}(\xi)= \mathcal{G}(\xi, 1)}`.

The joint information Hamiltonian for the whitened signal field :math:`{\xi}` reads

The joint information Hamiltonian for the whitened signal field :math:`{\xi}` reads

NIFTy takes advantage of this formulation in several ways:

NIFTy takes advantage of this formulation in several ways:

1) all prior degrees of freedom have now the same variance

1) all prior degrees of freedom have now the same variance

2) the amplitude operator can be regarded as part of the response, :math:`{R'=R\,A}`

2) the amplitude operator can be regarded as part of the response, :math:`{R'=R\,A}`

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

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

4) the amplitude operator can be made dependent on unknowns as well, e.g. :math:`{A=A(\tau)=\mathrm{FourierTransform}\,\mathrm{DiagonalOperator}(\exp(\tau))}` represents an amplitude model with a flexible Fourier spectrum

4) the amplitude operator can be made dependent on unknowns as well, e.g. :math:`{A=A(\tau)=\mathrm{HarmonicTransformOperator}\;\mathrm{DiagonalOperator}(\exp(\tau))}` represents an amplitude model with a flexible Fourier spectrum

5) the gradient of the Hamiltonian and the Fischer information metric with respect to all unknown parameters, here :math:`{\xi}` and can be constructed by NIFTy and used for Metric Gaussian Variational Inference.

5) the gradient of the Hamiltonian and the Fischer information metric with respect to all unknown parameters, here :math:`{\xi}` and can be constructed by NIFTy and used for Metric Gaussian Variational Inference.

A demonstration example for reconstructing a non-Gaussian signal with unknown covarinance from a complex (tomographic) response is given by demos/getting_started_3.py. Here, the uncertainty of the field and its power spectra are probed via posterior samples.

The reconstructing a non-Gaussian signal with unknown covarinance from a complex (tomographic) response is performed by demos/getting_started_3.py. Here, the uncertainty of the field and its power spectra are probed via posterior samples.