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Philipp Arras
whatisalikelihood
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0023f8f8
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0023f8f8
authored
Aug 24, 2018
by
Philipp Arras
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@@ 108,18 +108,22 @@ in a fashion which is compatible with our inference machinery called NIFTy.
\end{itemize}
\section*
{
Mathematical description of what NIFTy needs
}
For first order minimization:
The Bayesian inference will be a minimization problem in the end. NIFTy comes
with a variety of minimizers in order to solve this problem. These minimizers
differ in their order. If you want to use only first order minimizers, you need
to provide the following two functions:
\begin{itemize}
\item
$
\mathcal
H
(
ds
)
=

\log
\mathcal
P
(
ds
)
+
\text
{
constant in
}
s
$
\item
$
\mathcal
H'
=
\frac
{
\partial
}{
\partial
s
}
\mathcal
H
(
ds
)
$
\item
The loglikelihood
$
\mathcal
H
(
ds
)
:
=

\log
\mathcal
P
(
ds
)
$
as a function of
$
s
$
.
\item
Its derivative
$
\mathcal
H'
=
\frac
{
\partial
}{
\partial
s
}
\mathcal
H
(
ds
)
$
.
\end{itemize}
For second order minimization additionally:
If you want to use second order minimizers (which is definitely worth it),
another object is needed:
\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
do
es it automatically.
need to be calculated and implemented because NIFTy
comput
es it automatically.
\section*
{
Becoming more specific
}
If the likelihood is Gaussian (
$
\mathcal
H
(
ds
)
\propto
(
d

R
(
s
))
^
\dagger
N
^{

1
}
...
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