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Neel Shah
NIFTy
Commits
6a5cf05c
Commit
6a5cf05c
authored
Jan 16, 2019
by
Philipp Arras
Browse files
Add docs
parent
a4923f0f
Changes
1
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nifty5/operators/energy_operators.py
View file @
6a5cf05c
...
...
@@ 202,9 +202,8 @@ class PoissonianEnergy(EnergyOperator):
class
InverseGammaLikelihood
(
EnergyOperator
):
"""Special class for inverse Gamma distributed covariances.
RL FIXME: To be documented.
"""
FIXME
"""
def
__init__
(
self
,
d
):
...
...
@@ 225,32 +224,34 @@ class InverseGammaLikelihood(EnergyOperator):
class
BernoulliEnergy
(
EnergyOperator
):
"""C
lass for
likelihood

energ
ies
of expected event frequency constrained by
"""C
omputes
likelihood
energ
y
of expected event frequency constrained by
event data.
.. math ::
E(f) = 
\\
log
\\
text{Bernoulli}(df)
= d^
\\
dagger
\\
log f  (1d)^
\\
dagger
\\
log(1f),
where f is a field defined on `d.domain` with the expected
frequencies of events.
Parameters

d : Field
data field with events (=1) or nonevents (=0)
Notes

``E = BernoulliEnergy(d)`` represents
:math:`E(f) = 
\\
log
\\
text{Bernoulli}(df) =
d^
\\
dagger
\\
log f  (1d)^
\\
dagger
\\
log(1f)`,
where f is a field in data space (d.domain) with the expected
frequencies of events.
Data field with events (1) or nonevents (0).
"""
def
__init__
(
self
,
d
):
print
(
d
.
dtype
)
if
not
isinstance
(
d
,
Field
)
or
not
np
.
issubdtype
(
d
.
dtype
,
np
.
integer
):
raise
TypeError
if
not
np
.
all
(
np
.
logical_or
(
d
.
local_data
==
0
,
d
.
local_data
==
1
)):
raise
ValueError
self
.
_d
=
d
self
.
_domain
=
DomainTuple
.
make
(
d
.
domain
)
def
apply
(
self
,
x
):
self
.
_check_input
(
x
)
v
=
x
.
log
().
vdot
(

self
.
_d
)

(
1.

x
).
log
().
vdot
(
1.

self
.
_d
)
v
=

(
x
.
log
().
vdot
(
self
.
_d
)
+
(
1.

x
).
log
().
vdot
(
1.

self
.
_d
)
)
if
not
isinstance
(
x
,
Linearization
):
return
Field
.
scalar
(
v
)
if
not
x
.
want_metric
:
...
...
@@ 261,25 +262,13 @@ class BernoulliEnergy(EnergyOperator):
class
Hamiltonian
(
EnergyOperator
):
"""Class for information Hamiltonians.
"""Computes an information Hamiltonian in its standard form, i.e. with the
prior being a Gaussian with unit covariance.
Parameters

lh : EnergyOperator
a likelihood energy
ic_samp : IterationController
is passed to SamplingEnabler to draw Gaussian distributed samples
with covariance = metric of Hamiltonian
(= Hessian without terms that generate negative eigenvalues)
Notes

``H = Hamiltonian(E_lh)`` represents
Let the likelihood energy be :math:`E_{lh}`. Then this operator computes:
:math:`H(f) = 0.5 f^
\\
dagger f + E_{lh}(f)`
an information Hamiltonian for a field f with a white Gaussian prior
(unit covariance) and the likelihood energy :math:`E_{lh}`.
.. math ::
H(f) = 0.5 f^
\\
dagger f + E_{lh}(f):
Other field priors can be represented via transformations of a white
Gaussian field into a field with the desired prior probability structure.
...
...
@@ 288,9 +277,22 @@ class Hamiltonian(EnergyOperator):
by a generative model, from which NIFTy can draw samples and infer a field
using the Maximum a Posteriori (MAP) or the Variational Bayes (VB) method.
For more details see:
"Encoding prior knowledge in the structure of the likelihood"
Jakob Knollmüller, Torsten A. Ensslin, submitted, arXiv:1812.04403
The metric of this operator can be used as covariance for drawing Gaussian
samples.
Parameters

lh : EnergyOperator
The likelihood energy.
ic_samp : IterationController
Tells an internal :class:`SamplingEnabler` which convergence criterium
to use to draw Gaussian samples.
See also

`Encoding prior knowledge in the structure of the likelihood`,
Jakob Knollmüller, Torsten A. Ensslin,
`<https://arxiv.org/abs/1812.04403>`_
"""
...
...
@@ 318,50 +320,53 @@ class Hamiltonian(EnergyOperator):
class
AveragedEnergy
(
EnergyOperator
):
"""Class for KullbachLeibler (KL) Divergence or Gibbs free energies
Precisely a sample averaged Hamiltonian (or other energy) that represents
approximatively the relevant part of a KL to be used in Variational Bayes
inference if the samples are drawn from the approximating Gaussian.
Let :math:`Q(f) = G(fm,D)` Gaussian used to approximate
:math:`P(fd)`, the correct posterior with information Hamiltonian
:math:`H(d,f) = 
\\
log P(d,f) = 
\\
log P(fd) +
\\
text{const.}`
The KL divergence between those should then be optimized for m. It is
"""Computes KullbachLeibler (KL) divergence or Gibbs free energies.
:math:`KL(Q,P) =
\\
int Df Q(f)
\\
log Q(f)/P(f)
\\\\
=
\\
left<
\\
log Q(f)
\\
right>_Q(f) 
\\
left<
\\
log P(f)
\\
right>_Q(f)
\\\\
=
\\
text{const} +
\\
left< H(f)
\\
right>_G(fm,D)`
A sampleaveraged energy, e.g. an Hamiltonian, approximates the relevant
part of a KL to be used in Variational Bayes inference if the samples are
drawn from the approximating Gaussian:
in essence the information Hamiltonian averaged over a Gaussian
distribution centered on the mean m.
.. math ::
\\
text{KL}(m) =
\\
frac1{
\\
#\{v_i\}}
\\
sum_{v_i} H(m+v_i),
``AveragedEnergy(H)`` approximates
:math:`
\\
left< H(f)
\\
right>_{G(fm,D)}` if the residuals
:math:`fm` are drawn from covariance :math:`D`.
where :math:`v_i` are the residual samples and :math:`m` is the mean field
around which the samples are drawn.
Parameters

h: Hamiltonian
t
he
Hamiltonian/
energy to be averaged
T
he energy to be averaged
.
res_samples : iterable of Fields
set of residual sample points to be added to mean field
for approximate estimation of the KL
Notes

``KL = AveragedEnergy(H, samples)`` represents
:math:`
\\
text{KL}(m) =
\\
sum_i H(m+v_i) / N`,
where :math:`v_i` are the residual samples, :math:`N` is their number,
and :math:`m` is the mean field around which the samples are drawn.
Set of residual sample points to be added to mean field for approximate
estimation of the KL.
Note

Having symmetrized residual samples, with both v_i and v_i being present
ensures that the distribution mean is exactly represented. This reduces
sampling noise and helps the numerics of the KL minimization process in the
variational Bayes inference.
See also

Let :math:`Q(f) = G(fm,D)` be the Gaussian distribution
which is used to approximate the accurate posterior :math:`P(fd)` with
information Hamiltonian
:math:`H(d,f) = 
\\
log P(d,f) = 
\\
log P(fd) +
\\
text{const}`. In
Variational Bayes one needs to optimize the KL divergence between those
two distributions for m. It is:
:math:`KL(Q,P) =
\\
int Df Q(f)
\\
log Q(f)/P(f)
\\\\
=
\\
left<
\\
log Q(f)
\\
right>_Q(f) 
\\
left<
\\
log P(f)
\\
right>_Q(f)
\\\\
=
\\
text{const} +
\\
left< H(f)
\\
right>_G(fm,D)`
in essence the information Hamiltonian averaged over a Gaussian
distribution centered on the mean m.
:class:`AveragedEnergy(h)` approximates
:math:`
\\
left< H(f)
\\
right>_{G(fm,D)}` if the residuals
:math:`fm` are drawn from a Gaussian distribution with covariance
:math:`D`.
"""
def
__init__
(
self
,
h
,
res_samples
):
...
...
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