Commit 0023f8f8 authored by Philipp Arras's avatar Philipp Arras
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Rework first paragraph

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......@@ -108,18 +108,22 @@ in a fashion which is compatible with our inference machinery called NIFTy.
\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:
\item $\mathcal H (d|s) = - \log \mathcal P (d|s) + \text{constant in }s$
\item $\mathcal H' = \frac{\partial}{\partial s} \mathcal H (d|s)$
\item The log-likelihood $\mathcal H (d|s) := - \log \mathcal P (d|s)$ as a function of $s$.
\item Its derivative $\mathcal H' = \frac{\partial}{\partial s} \mathcal H (d|s)$.
For second order minimization additionally:
If you want to use second order minimizers (which is definitely worth it),
another object is needed:
\item $\langle \mathcal H' \mathcal H'^\dagger \rangle_{\mathcal P (d|s)}$.
Note, that for Gaussian, Poissonian and Bernoulli likelihoods this term doesn't
need to be calculated and implemented because NIFTy does it automatically.
need to be calculated and implemented because NIFTy computes it automatically.
\section*{Becoming more specific}
If the likelihood is Gaussian ($\mathcal H(d|s) \propto (d-R(s))^\dagger N^{-1}
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