From 894d80ca904430f95ece00358c07e6fc47e48129 Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Wed, 19 Apr 2023 11:30:03 +0200
Subject: [PATCH] Minor wording changes; Added MAP and ML equivalence note

---
 .../chapters/theoretical_background.tex       | 21 ++++++++++++-------
 1 file changed, 13 insertions(+), 8 deletions(-)

diff --git a/latex/thesis/chapters/theoretical_background.tex b/latex/thesis/chapters/theoretical_background.tex
index 80bf172..9ef72c0 100644
--- a/latex/thesis/chapters/theoretical_background.tex
+++ b/latex/thesis/chapters/theoretical_background.tex
@@ -24,13 +24,12 @@ For example:%
     c \in \mathbb{F}_2           &\to \tilde{c} \in \left[ 0, 1 \right] \subseteq \mathbb{R}
 .\end{align*}
 %
-Additionally, a shorthand notation will be used to denote a set of indices:%
+Additionally, a shorthand notation will be used, denoting a set of indices as%
 %
 \begin{align*}
     \left[ m:n \right]      &:= \left\{ m, m+1, \ldots, n-1, n \right\},
         \hspace{5mm} m < n, \hspace{2mm} m,n\in\mathbb{Z}
 .\end{align*}
-\todo{Not really slicing. How should it be denoted?}
 %
 In order to designate elemen-twise operations, in particular the \textit{Hadamard product}
 and the \textit{Hadamard power}, the operator $\circ$ will be used:%
@@ -58,7 +57,7 @@ This is known as modulation. The modulation scheme chosen here is \ac{BPSK}:%
     \boldsymbol{x} = 1 - 2\boldsymbol{c}
 .\end{align*}
 %
-The transmitted symbol is distorted by the channel and denoted by
+The transmitted symbol is distorted by the channel and denoted as
 $\boldsymbol{y} \in \mathbb{R}^n$.
 This distortion is described by the channel model, which in the context of
 this thesis is chosen to be \ac{AWGN}:%
@@ -170,17 +169,23 @@ criterion:%
         \right)
 .\end{align*}%
 %
-The \ac{MAP}- and \ac{ML}-criteria are closely connected through
-\textit{Bayes' theorem}:%
+The two criteria are closely connected through Bayes' theorem and are equivalent
+when the prior probability of transmitting a codeword is the same for all
+codewords:
 %
 \begin{align*}
     \argmax_{c\in\mathcal{C}} p_{\boldsymbol{C} \mid \boldsymbol{Y}}
         \left( \boldsymbol{c} \mid \boldsymbol{y} \right)
-    = TODO
+    &= \argmax_{c\in\mathcal{C}} \frac{f_{\boldsymbol{Y} \mid \boldsymbol{C}}
+        \left( \boldsymbol{y} \mid \boldsymbol{c} \right) p_{\boldsymbol{C}}
+        \left( \boldsymbol{c} \right)}{f_{\boldsymbol{Y}}\left( \boldsymbol{y} \right) } \\
+    &= \argmax_{c\in\mathcal{C}} f_{\boldsymbol{Y} \mid \boldsymbol{C}}
+        \left( \boldsymbol{y} \mid \boldsymbol{c} \right) p_{\boldsymbol{C}}
+        \left( \boldsymbol{c} \right) \\
+    &= \argmax_{c\in\mathcal{C}}f_{\boldsymbol{Y} \mid \boldsymbol{C}}
+        \left( \boldsymbol{y} \mid \boldsymbol{c} \right)
 .\end{align*}
 %
-This has the consequence that if the probability \ldots, the two criteria are
-equivalent.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%