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Philipp Arras
whatisalikelihood
Commits
8cb33343
Commit
8cb33343
authored
Aug 24, 2018
by
Philipp Arras
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main.tex
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8cb33343
...
...
@@ 3,7 +3,7 @@
\usepackage
{
csquotes
}
\usepackage
{
hyperref
}
\usepackage
{
graphicx
}
\usepackage
{
graphicx
, amsmath
}
\graphicspath
{{
img/
}}
\usepackage
[backend=biber, sorting=none]
{
biblatex
}
...
...
@@ 17,9 +17,54 @@
\begin{document}
\maketitle
\begin{abstract}
Abstract
\end{abstract}
\section*
{
The need for a guide
}
\section*
{
Mathematical description of what NIFTy needs
}
For first order minimization:
\begin{itemize}
\item
$
\mathcal
H
(
ds
)
=

\log
\mathcal
P
(
ds
)
+
\text
{
constant in
}
s
$
\item
$
\mathcal
H'
=
\frac
{
\partial
}{
\partial
s
}
\mathcal
H
(
ds
)
$
\end{itemize}
For second order minimization additionally:
\begin{itemize}
\item
$
\langle
\mathcal
H'
\mathcal
H'
^
\dagger
\rangle
_{
\mathcal
P
(
ds
)
}$
.
\end{itemize}
Note, that for Gaussian, Poissonian and Bernoulli likelihoods this term doesn't
need to be calculated and implemented because NIFTy does it automatically.
\section*
{
Becoming more specific
}
If the likelihood is Gaussian (
$
\mathcal
H
(
ds
)
\propto
(
d

R
(
s
))
^
\dagger
N
^{

1
}
(
d

R
(
s
))
$
) or Poissonian (
$
\mathcal
H
(
ds
)
\propto

\log
(
R
(
s
))
^
\dagger
d
+
\sum
_
i R
(
s
)
_
i
$
), NIFTy needs:
\begin{itemize}
\item
$
R
$
.
\item
$
R'
^
\dagger
:
=
(
\frac
{
\partial
}{
\partial
s
}
R
(
s
))
^
\dagger
$
.
\item
$
R'
=
\frac
{
\partial
}{
\partial
s
}
R
(
s
)
$
. (only for 2nd order minimization)
\end{itemize}
\section*
{
Example
$
\gamma
$
ray imaging
}
The information a
$
\gamma
$
ray astronomer would provide to the algorithm (in the
simplest case):
\begin{itemize}
\item
Data has Poissonian statistics.
\item
Two functions:
\verb

R(s)

and
\verb

R_adjoint(d)

where
$
R
$
applies a convolution with
a Gaussian kernel of given size to
\verb

s

and picks out the positions to which
there exists a data point in
\verb

d

.
\verb

R_adjoint(d)

implements
$
R
^
\dagger
$
.
\end{itemize}
Why is this already sufficient?
\begin{itemize}
\item
The Hamiltonian
$
\mathcal
H
$
is given by:
$
\mathcal
H
(
ds
)
=

\log
(
R
(
s
))
^
\dagger
d
+
\sum
_
i R
(
s
)
_
i
$
. Implementing
$
R
$
and stating that the data is
Poissonian determines this form.
\item
Since
$
R
$
is a composition of a convolution and a sampling both of which
is a linear operation,
$
R
$
itself is a linear operator.
\footnote
{
I.e.
$
R
(
\alpha
s
_
1
+
s
_
2
)
=
\alpha
R
(
s
_
1
)
+
R
(
s
_
2
)
$
.
}
Thus,
$
R'
=
R
$
and
$
R'
^
\dagger
=
R
^
\dagger
$
. All in all, we need an implementation for
$
R
$
and
$
R
^
\dagger
$
.
\end{itemize}
% \section*{Bibliography test}
% RESOLVE was first presented in \cite{Resolve2016}.
...
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