corrected typos

main.tex

@@ -67,7 +67,7 @@ building blocks.
   the data, the statistics of the data including the error bars and a
   description of the measurement device itself.
 \item The \emph{prior} $\mathcal P(s)$ describes the knowledge the scientist has
-  \emph{before} executing the experiment. In order to define prior one needs to
+  \emph{before} executing the experiment. In order to define a prior one needs to
   make one's knowledge about the physical process which was observed by the
   measurement device explicit.
 \item Finally, one needs an algorithmic and computational framework which is
@@ -83,8 +83,8 @@ The IFT group led by Torsten En{\ss}lin has knowledge and GPL-licensed software
 to address the third part.
 
 The interesting part is the second one, i.e. thinking about the priors. Here one
-needs to have an understanding both of the underlying physics and probability
-theory and Bayesian statistics. This is where an interesting interaction between
+needs to have an understanding both of the underlying physics and Bayesian probability
+theory. This is where an interesting interaction between
 observers, i.e. you, and information field theorists, i.e. Torsten and his
 group, will happen.
 
@@ -125,7 +125,7 @@ If the likelihood is Gaussian
 \begin{align*}
 \mathcal H(d|s) \propto (d-R(s))^\dagger N^{-1}(d-R(s))
 \end{align*}
-or Poissonian
+with noise covariance $N$ and response $R$,\\ or Poissonian
 \begin{align*}
 \mathcal H(d|s) \propto - \log (R(s))^\dagger d+\sum_i R(s)_i,
 \end{align*}
@@ -151,11 +151,11 @@ the course of the algorithm. In \texttt{s}, NIFTy will store the reconstruction
 of the physical field which the user wants to infer. \texttt{s} would be a
 one-dimensional array for a time-series, two-dimensional for a multi-frequency
 time-series or for a single-frequency image of the sky, three-dimensional for a
-multi-frequency image of sky, etc. To cut a long story short: be aware of the
+multi-frequency image of the sky, etc. To cut a long story short: be aware of the
 shape of \texttt{s} and \texttt{d}!
 
 \subsection*{The response}
-When we talk about \enquote{the response} we mean a function \texttt{R(s)} which
+When we talk about \enquote{the response}, we mean a function \texttt{R(s)} which
 takes an array of shape \texttt{s.shape} and returns an array of shape
 \texttt{d.shape}. This function shall be made such that it simulates the
 measurement device. Assume one knows the signal \texttt{s}, what would be the
@@ -175,7 +175,7 @@ response_out = R(np.ones(shp))
 if response_out.shape == d.shape:
     print('Yay!')
 else:
-    raise ValueError('Output of response doesn't have the correct shape.')
+    raise ValueError('Output of response does not have the correct shape.')
 \end{lstlisting}
 
 \subsection*{Derivative of response}
@@ -189,7 +189,7 @@ following shape: \footnote{There are various ways to think about derivatives and
   gradients of multi-dimensional functions. A different view on gradients would
   be that at a given point $s=\text{position}$ the gradient is a matrix with
   \texttt{s.size} columns and \texttt{d.size} rows. Obviously, it is not
-  feasible to store such a matrix on a computer due to its size. There we think
+  feasible to store such a matrix on a computer due to its size. Therefore we think
   of this matrix in terms of a linear map which maps an array of shape
   \texttt{s.shape} to an array of shape \texttt{d.shape}. This linear map shall
   be implemented on the computer in terms of a function. Think of this map as a
@@ -200,7 +200,7 @@ following shape: \footnote{There are various ways to think about derivatives and
 What needs to be implemented is a function \texttt{R\_prime(position, s0)} which
 takes the arguments \texttt{position} (which is an array of shape \texttt{s.shape}
 and determines the position at which we want to calculate the derivative) and
-the array \texttt{s0} which shall be the derivative taken of.
+the array \texttt{s0} of which the derivative shall be taken.
 \texttt{R\_prime} is nonlinear in \texttt{position} in general and linear in
 \texttt{s0}. The output of \texttt{R\_prime} is of shape \texttt{d.shape}.
 
@@ -304,8 +304,8 @@ Why is this already sufficient?
 \item The Hamiltonian $\mathcal H$ is given by: $\mathcal H (d|s) = - \log
   (R(s))^\dagger d+\sum_i R(s)_i$. 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
+  \item Since $R$ is a composition of a convolution and a sampling, both of which
+    are linear operations, $R$ 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}