@@ -138,7 +138,7 @@ the information source. The operation in :math:`{d= D\,R^\dagger N^{-1} d}` is a

...

@@ -138,7 +138,7 @@ 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.

Some of these operators are diagonal in harmonic (Fourier) basis, and therefore only require the specification of a (power) spectrum, :math:`{S= F\,\widehat{P_s} F^\dagger}`, where :math:`{F^\dagger= \mathrm{HarmonicTransformOperator}}` and :math:`{\widehat{P_s} = \mathrm{DiagonalOperator}(P_s)}`, and :math:`{P_s(k)}` is the power spectrum of :math:`{s}` as a function of the (absolute value of the) harmonic (Fourier) space koordinate :math:`{k}`. For those, NIFTy can easily also provide inverse operators, as :math:`{S^{-1}= F\,\widehat{\frac{1}{P_s}} F^\dagger}` in case :math:`{F}` is unitary, :math:`{F^\dagger=F^{-1}}`.

Some of these operators are diagonal in harmonic (Fourier) basis, and therefore only require the specification of a (power) spectrum, :math:`{S= F\,\widehat{P_s} F^\dagger}`, where :math:`{F^\dagger= \mathrm{HarmonicTransformOperator}}` and :math:`{\widehat{P_s} = \mathrm{DiagonalOperator}(P_s)}`, and :math:`{P_s(k)}` is the power spectrum of the process that generated :math:`{s}` as a function of the (absolute value of the) harmonic (Fourier) space koordinate :math:`{k}`. For those, NIFTy can easily also provide inverse operators, as :math:`{S^{-1}= F\,\widehat{\frac{1}{P_s}} F^\dagger}` in case :math:`{F}` is unitary, :math:`{F^\dagger=F^{-1}}`.

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

These implicit operators can be combined into new operators, e.g. to :math:`{D^{-1} = S^{-1} + R^\dagger N^{-1} R}`, 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.

...

@@ -151,7 +151,7 @@ The demo codes demos/getting_started_1.py and demos/Wiener_Filter.ipynb illustra

...

@@ -151,7 +151,7 @@ The demo codes demos/getting_started_1.py and demos/Wiener_Filter.ipynb illustra

Generative Models

Generative Models

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

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

For more complex measurement situations, involving non-linear measuremnts, unknown covariances, calibration constants and the like, it is recommended to formulate those as generative models as NIFTy provides powerful inference algorithms for such.

For more complex measurement situations, involving non-linear measuremnts, unknown covariances, calibration constants and the like, it is recommended to formulate those as generative models. NIFTy provides powerful inference algorithms for such generative models.

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

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

...

@@ -172,14 +172,30 @@ The joint information Hamiltonian for the whitened signal field :math:`{\xi}` r

...

@@ -172,14 +172,30 @@ The joint information Hamiltonian for the whitened signal field :math:`{\xi}` r

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 unit variance helping to improve the condition number for the equations to be solved.

1) All prior degrees of freedom have now the same unit variance helping to improve the condition number for the equations to be solved.

2) The amplitude operator can be regarded as part of the response, :math:`{R'=R\,A}`. In general, mre complex responses can be constructed out of the cocatenation of simpler operators.

2) The amplitude operator can be regarded as part of the response, :math:`{R'=R\,A}`. In general, more complex responses can be constructed out of the cocatenation of simpler operators.

3) The response can be made non-linear, e.g. :math:`{R'(s)=R \exp(A\,\xi)}`, see 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)= F\, \widehat{e^\tau}` represents an amplitude model with a positive definite, flexible Fourier spectrum. The amplitude field :math:`{\tau}` gets its own amplitude model, with a cepstrum ( = spectrum of a log spectrum) defined in the quefrency space ( = harmonic space of an harmonic space) to tame its degrees of freedom by imposing some (user defined level of) spectral smoothness.

4) The amplitude operator can be made dependent on unknowns as well, e.g. :math:`A=A(\tau)= F\, \widehat{e^\tau}` represents an amplitude model with a positive definite, flexible Fourier spectrum. The amplitude field :math:`{\tau}` gets its own amplitude model, with a cepstrum ( = spectrum of a log spectrum) defined in the quefrency space ( = harmonic space of an harmonic space) to tame its degrees of freedom by imposing some (user defined level of) spectral smoothness.

5) NIFTy can calculate the gradient of the information Hamiltonian and the Fischer 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 Fischer matrix is required in addition to the gradient by Metric Gaussian Variational Inference (MGVI), which is also available in NIFTY. MGVI is an implicit operator extension of Automatic Differentiation Variational Inference (ADVI).

5) NIFTy can calculate the gradient of the information Hamiltonian and the Fischer 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 Fischer matrix is required in addition to the gradient by Metric Gaussian Variational Inference (MGVI), which is also available in NIFTY. MGVI is an implicit operator extension of Automatic Differentiation Variational Inference (ADVI).

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 provided by the MGVI algorithm.

The reconstruction of 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 the power spectrum of its generating process are probed via posterior samples provided by the MGVI algorithm.