@@ -137,7 +137,8 @@ the posterior covariance operator, and

...

@@ -137,7 +137,8 @@ the posterior covariance operator, and

the information source. The operation in :math:`{d= D\,R^\dagger N^{-1} d}` is also called the generalized Wiener filter.

the information source. The operation in :math:`{d= D\,R^\dagger N^{-1} d}` is also called the generalized Wiener filter.

NIFTy permits to define the involved operators :math:`{R}`, :math:`{R^\dagger}`, :math:`{S}`, and :math:`{N}` implicitely, as coputer 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^{-1} = \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.

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@@ -154,12 +155,15 @@ For more complex measurement situations, involving non-linear measuremnts, unkno

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@@ -154,12 +155,15 @@ For more complex measurement situations, involving non-linear measuremnts, unkno

In a generative model, all known or unknown quantities are described as the results of generative processes, which start with simple probability distributions, like uniform, iid Gaussian, or delta distributions.

In a generative model, all known or unknown quantities are described as the results of generative processes, which start with simple probability distributions, like uniform, iid Gaussian, or delta distributions.

The above free theory case looks as a generative model like the following:

The above free theory case looks as a generative model like the following:

.. math::

.. math::

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, \mathbb{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

@@ -172,7 +176,7 @@ NIFTy takes advantage of this formulation in several ways:

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@@ -172,7 +176,7 @@ NIFTy takes advantage of this formulation in several ways:

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{FourierTransform}\,\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_2.py.

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.