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diff --git a/src/year2/natural-language-processing/sections/_classification.tex b/src/year2/natural-language-processing/sections/_classification.tex
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--- a/src/year2/natural-language-processing/sections/_classification.tex
+++ b/src/year2/natural-language-processing/sections/_classification.tex
@@ -63,7 +63,7 @@
         Given a vocabulary $V$ and a document $d$, the bag-of-words embedding of $d$ is a vector in $\mathbb{N}^{\vert V \vert}$ where the $i$-th position contains the number of occurrences of the $i$-th token in $d$.
 
     \item[Multinomial naive Bayes classifier] \marginnote{Multinomial naive Bayes classifier}
-        Probabilistic (i.e., generative) classifier based on the assumption that features are independent given the class.
+        Generative probabilistic classifier based on the assumption that features are independent given the class.
 
         Given a document $d = \{ w_1, \dots, w_n \}$, a naive Bayes classifier returns the class $\hat{c}$ with maximum posterior probability:
         \[
@@ -128,4 +128,327 @@
             \prob{\texttt{-}} = \frac{3}{5} \qquad \prob{\texttt{predictable} | \texttt{-}} = \frac{1+1}{14+20} \quad \prob{\texttt{no} | \texttt{-}} = \frac{1+1}{14+20} \quad \prob{\texttt{fun} | \texttt{-}} = \frac{0+1}{14+20}
         \end{gathered}
     \]
-\end{example}
\ No newline at end of file
+\end{example}
+
+
+\subsection{Optimizations}
+
+Possible optimizations for naive Bayes applied to sentiment analysis are the following:
+\begin{descriptionlist}
+    \item[Binarization] \marginnote{Binarization}
+        Generally, the information regarding the occurrence of a word is more important than its frequency. Therefore, instead of applying bag-of-words by counting, it is possible to produce a one-hot encoded vector to indicate which words are in the document.
+
+    \item[Negation encoding] \marginnote{Negation encoding}
+        To encode negations, two approaches can be taken:
+        \begin{description}
+            \item[Negation annotation]
+                Add to negated words an annotation so that they are treated as a new word.
+                \begin{example}
+                    Prepend \texttt{NOT\char`_} to each word between a negation and the next punctuation:
+                    \[ \text{didn't like this movie.} \mapsto \text{didn't \texttt{NOT\char`_}like \texttt{NOT\char`_}this \texttt{NOT\char`_}movie.} \]
+                \end{example}
+
+            \item[Parse tree]
+                Build a tree to encode the sentiment and interactions of the words. By propagating the sentiments bottom-up it is possible to determine the overall sentiment of the sequence.
+
+                \begin{example}
+                    The parse tree for the sentence ``\texttt{This film doesn't care about cleverness, wit or any other kind of intelligent humor.}'' is the following:
+                    \begin{figure}[H]
+                        \centering
+                        \includegraphics[width=0.6\linewidth]{./img/_sentiment_parse_tree.pdf}
+                    \end{figure}
+                    Due to the negation (\texttt{doesn't}), the whole positive sequence is negated.
+                \end{example}
+        \end{description}
+
+    \item[Sentiment lexicon] \marginnote{Sentiment lexicon}
+        If training data is insufficient, external domain knowledge, such as sentiment lexicon, can be used.
+
+        \begin{example}
+            A possible way to use a lexicon is to count the number of positive and negative words according to that corpus.
+        \end{example}
+
+        \begin{remark}
+            Possible ways to create a lexicon are through:
+            \begin{itemize}
+                \item Expert annotators.
+                \item Crowdsourcing in a two-step procedure:
+                    \begin{enumerate}
+                        \item Ask questions related to synonyms (e.g., which word is closest in meaning to \textit{startle}?).
+                        \item Rate the association of words with emotions (e.g., how does \textit{startle} associate with \textit{joy}, \textit{fear}, \textit{anger}, \dots?).
+                    \end{enumerate}
+                \item Semi-supervised induction of labels from a small set of annotated data (i.e., seed labels). It works by looking for words that appear together with the ones with a known sentiment.
+                \item Supervised learning using annotated data.
+            \end{itemize}
+        \end{remark}
+\end{descriptionlist}
+
+
+\subsection{Properties}
+
+Naive Bayes has the following properties:
+\begin{itemize}
+    \item It is generally effective with short sequences and fewer data samples.
+    \item It is robust to irrelevant features (i.e., words that appear in both negative and positive sentences) as they cancel out each other.
+    \item It has good performance in domains with many equally important features (contrarily to decision trees).
+    \item The independence assumption might produce overestimated predictions.
+\end{itemize}
+
+\begin{remark}
+    Naive Bayes is a good baseline when experimenting with text classifications.
+\end{remark}
+
+
+
+\section{Logistic regression}
+
+\begin{description}
+    \item[Features engineering] \marginnote{Features engineering}
+        Determine features by hand from the data (e.g., number of positive and negative lexicon).
+
+
+    \item[Binary logistic regression] \marginnote{Binary logistic regression}
+        Discriminative probabilistic model that computes the joint distribution $\prob{c | d}$ of a class $c$ and a document $d$.
+
+        Given the input features $\vec{x} = [x_1, \dots, x_n]$, logistic regression computes the following:
+        \[ 
+            \sigma\left( \sum_{i=1}^{n} w_i x_i \right) + b = \sigma(\vec{w}\vec{x} + b)
+        \]
+        where $\sigma$ is the sigmoid function.
+
+        \begin{description}
+            \item[Loss]
+                The loss function should aim to maximize the probability of predicting the correct label $\hat{y}$ given the observation $\vec{x}$. This can be expressed as a Bernoulli distribution:
+                \[ 
+                    \prob{y | x} = \hat{y}^y (1-\hat{y})^{1-y} = \begin{cases}
+                        1 - \hat{y} & \text{if $y=0$} \\
+                        \hat{y} & \text{if $y=1$} \\
+                    \end{cases} 
+                \]
+                By applying a log-transformation and inverting the sign, this corresponds to the cross-entropy loss in the binary case:
+                \[ \mathcal{L}_{\text{BCE}}(\hat{y}, y) = -\log \prob{y | x} = -[y \log(\hat{y}) + (1-y)\log(1-\hat{y})] \]
+
+            \item[Optimization]
+                As cross-entropy is convex, SGD is well suited to find the parameters $\vec{\theta}$ of a logistic regressor $f$ over batches of $m$ examples by solving:
+                \[ \arg\min_{\vec{\theta}} \sum_{i=1}^{m} \mathcal{L}_\text{BCE}(\hat{y}^{(i)}, f(x^{(i)}; \vec{\theta})) + \alpha \mathcal{R}(\vec{\theta}) \]
+                where $\alpha$ is the regularization factor and $\mathcal{R}(\vec{\theta})$ is the regularization term. Typical regularization approaches are:
+                \begin{descriptionlist}
+                    \item[Ridge regression (L1)] $\mathcal{R}(\vec{\theta}) = \Vert \vec{\theta} \Vert^2_2 = \sum_{j=1}^{n} \vec{\theta}_j^2$.
+                    \item[Lasso regression (L2)] $\mathcal{R}(\vec{\theta}) = \Vert \vec{\theta} \Vert_1 = \sum_{j=1}^{n} \vert \vec{\theta}_j \vert$.
+                \end{descriptionlist}
+        \end{description}
+
+    \item[Multinomial logistic regression] \marginnote{Multinomial logistic regression}
+        Extension of logistic regression to the multi-class case. The joint probability becomes $\prob{y = c | x}$ and softmax is used in place of the sigmoid.
+
+        Cross-entropy is extended over the classes $C$:
+        \[ \mathcal{L}_\text{CE}(\hat{y}, y) = - \sum_{c \in C} \mathbbm{1}\{y = c\} \log \prob{y = c | x} \]
+\end{description}
+
+
+\subsection{Properties}
+
+Logistic regression has the following properties:
+\begin{itemize}
+    \item It is generally effective with large documents or datasets.
+    \item It is robust to correlated features.
+\end{itemize}
+
+\begin{remark}
+    Logistic regression is also a good baseline when experimenting with text classifications.
+
+    As they are lightweight to train, it is a good idea to test both naive Bayes and logistic regression to determine the best baseline for other experiments.
+\end{remark}
+
+
+
+\section{Metrics}
+
+
+\subsection{Binary classification}
+
+\begin{description}
+    \item[Contingency table] \marginnote{Contingency table}
+        $2 \times 2$ table matching predictions to ground truths. It contains true positives (\texttt{TP}), false positives (\texttt{FP}), false negatives (\texttt{FN}), and true negatives (\texttt{TN}).
+
+    \item[Recall] \marginnote{Recall} 
+        $\frac{\texttt{TP}}{\texttt{TP} + \texttt{FN}}$.
+
+    \item[Precision] \marginnote{Precision} 
+        $\frac{\texttt{TP}}{\texttt{TP} + \texttt{FP}}$.
+
+    \item[Accuracy] \marginnote{Accuracy} 
+        $\frac{\texttt{TP} + \texttt{TN}}{\texttt{TP} + \texttt{FP} + \texttt{FN} + \texttt{TN}}$.
+
+        \begin{remark}
+            Accuracy is a reasonable metric only when classes are balanced.
+        \end{remark}
+
+    \item[F1 score] \marginnote{F1 score} 
+        $\frac{2 \cdot \texttt{recall} \cdot \texttt{precision}}{\texttt{recall} + \texttt{precision}}$.
+\end{description}
+
+
+\subsection{Multi-class classification}
+
+\begin{description}
+    \item[Confusion matrix] \marginnote{Confusion matrix}
+        $c \times c$ table matching predictions to ground truths.
+
+    \item[Precision/Recall] \marginnote{Precision/Recall}
+        Precision and recall can be defined class-wise (i.e., consider a class as the positive label and the others as the negative).
+
+    \item[Micro-average precision/recall] \marginnote{Micro-average precision/recall}
+        Compute the contingency table of each class and collapse them into a single table. Compute precision or recall on the pooled contingency table.
+
+        \begin{remark}
+            This approach is sensitive to the most frequent class.
+        \end{remark}
+
+    \item[Macro-average precision/recall] \marginnote{Macro-average precision/recall}
+        Compute precision or recall class-wise and then average over the classes.
+
+        \begin{remark}
+            This approach is reasonable if the classes are equally important.
+        \end{remark}
+
+        \begin{remark}
+            Macro-average is more common in NLP.
+        \end{remark}
+\end{description}
+
+\begin{example}
+    \phantom{}
+
+    \begin{figure}[H]
+        \centering
+        \includegraphics[width=0.5\linewidth]{./img/_confusion_matrix_example.pdf}
+        \caption{Confusion matrix}
+    \end{figure}
+
+    \begin{figure}[H]
+        \centering
+        \includegraphics[width=0.85\linewidth]{./img/_micro_macro_average_example.pdf}
+        \caption{
+            \parbox[t]{0.6\linewidth}{
+                Class-wise contingency tables, pooled contingency table, and micro/macro-average precision
+            }
+        }
+    \end{figure}
+\end{example}
+
+
+\subsection{Cross-validation}
+
+\begin{description}
+    \item[$\mathbf{n}$-fold cross-validation] \marginnote{$n$-fold cross-validation}
+        Tune a classifier on different sections of the training data:
+        \begin{enumerate}
+            \item Randomly choose a training and validation set.
+            \item Train the classifier.
+            \item Evaluate the classifier on a held-out test set.
+            \item Repeat for $n$ times.
+        \end{enumerate}
+\end{description}
+
+\subsection{Statistical significance}
+
+\begin{description}
+    \item[$\mathbf{p}$-value] \marginnote{$p$-value}
+        Measure to determine whether a model $A$ is outperforming a model $B$ on a given test set by chance (i.e., test the null hypothesis $H_0$ or, in other words, test that there is no relation between $A$ and $B$).
+
+        Given:
+        \begin{itemize}
+            \item A test set $x$,
+            \item A random variable $X$ over the test sets (i.e., another test set),
+            \item Two models $A$ and $B$, such that $A$ is better than $B$ by $\delta(x)$ on the test set $x$,
+        \end{itemize}
+        the $p$-value is defined as:
+        \[ p\text{-value}(x) = \prob{\delta(X) > \delta(x) | H_0} \]
+        There are two cases:
+        \begin{itemize}
+            \item $p\text{-value}(x)$ is big: the null hypothesis holds (i.e., $\prob{\delta(X) > \delta(x)}$ is high under the assumption that $A$ and $B$ are not related), so $A$ outperforms $B$ by chance.
+            \item $p\text{-value}(x)$ is small (i.e., $< 0.05$ or $< 0.01$): the null hypothesis is rejected, so $A$ actually outperforms $B$.
+        \end{itemize}
+
+    \item[Bootstrapping test] \marginnote{Bootstrapping test}
+        Approach to compute $p$-values.
+
+        Given a test set $x$, multiple virtual test sets $\bar{x}^{(i)}$ are created by sampling with replacement (it is assumed that the new sets are representative). The performance difference $\delta(\cdot)$ is computed between two models and the $p$-value is determined as the frequency of:
+        \[ \delta(\bar{x}^{(i)}) > 2\delta(x) \]
+
+        \begin{remark}
+            $\delta(x)$ is doubled due to theoretical reasons.
+        \end{remark}
+
+        \begin{example}
+            Consider two models $A$ and $B$, and a test set $x$ with 10 samples. From $x$, multiple new sets (in this case of the same size) can be sampled. In the following table, each cell indicates which model correctly predicted the class:
+            \begin{center}
+                \footnotesize
+                \begin{tabular}{ccccccccccc|ccc}
+                    \toprule
+                                    & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & $A\%$ & $B\%$ & $\delta(\cdot)$ \\
+                    \midrule
+                    $x$             & $AB$ & $A$ & $AB$ & $B$ & $A$ & $B$ & $A$ & $AB$ & -- & $A$ & $0.7$ & $0.5$ & $0.2$ \\
+                    $\bar{x}^{(1)}$   & $A$ & $AB$ & $A$ & $B$ & $B$ & $A$ & $B$ & $AB$ & -- & $AB$ & $0.6$ & $0.6$ & $0.0$ \\
+                    $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \\
+                    \bottomrule
+                \end{tabular}
+            \end{center}
+            A possible way to sample $\bar{x}^{(1)}$ is (w.r.t. the indexes of the examples in $x$) $[2, 3, 3, 2, 4, 6, 2, 4, 1, 9, 1]$.
+        \end{example}
+\end{description}
+
+
+
+\section{Affective meaning}
+
+The affective meaning of a text corpus can vary depending on:
+\begin{descriptionlist}
+    \item[Personality traits] 
+        Stable behavior and personality of a person (e.g., nervous, anxious, reckless, \dots).
+
+    \item[Attitude] 
+        Enduring sentiment towards objects or people (e.g., liking, loving, hating, \dots).
+
+    \item[Interpersonal stance] 
+        Affective stance taken in a specific interaction (e.g., distant, cold, warm, \dots).
+
+    \item[Mood] 
+        Affective state of low intensity and long duration often without an apparent cause (e.g., cheerful, gloomy, irritable, \dots).
+
+    \item[Emotion] 
+        Brief response to an external or internal event of major significance (e.g., angry, sad, joyful, \dots).
+
+        \begin{remark}
+            Emotion is the most common subject in affective computing.
+        \end{remark}
+\end{descriptionlist}
+
+
+\subsection{Emotion}
+
+\begin{description}
+    \item[Theory of emotion]
+        There are two main theories of emotion:
+        \begin{descriptionlist}
+            \item[Basic emotions] \marginnote{Basic emotions}
+                Discrete and fixed range of atomic emotions.
+
+                \begin{remark}
+                    Emotions associated to a word might be in contrast. For instance, in the NRC Word-Emotion Association Lexicon the word \texttt{thirst} is associated to both \textit{anticipation} and \textit{surprise}.
+                \end{remark}
+
+            \item[Continuum emotions] \marginnote{Continuum emotions}
+                Describe emotions in a 2 or 3 dimensional space with the following features:
+                \begin{descriptionlist}
+                    \item[Valence] Pleasantness of a stimulus.
+                    \item[Arousal] Intensity of emotion.
+                    \item[Dominance] Degree of control on the stimulus.
+                \end{descriptionlist}
+
+                \begin{remark}
+                    Valence is often used as a measure of sentiment.
+                \end{remark}
+        \end{descriptionlist}
+\end{description}
\ No newline at end of file