Add FAIKR3 joint distribution inference

src/fundamentals-of-ai-and-kr/module3/sections/_intro.tex

@@ -45,89 +45,4 @@
         Defined as:
         \[ \text{Decision theory} = \text{Utility theory} + \text{Probability theory} \]
         where the utility theory depends on one's preferences.
-\end{description}
-
-
-\subsection{Probability}
-
-\begin{description}
-    \item[Sample space] \marginnote{Sample space}
-        Set $\Omega$ of all possible worlds.
-        \begin{descriptionlist}
-            \item[Event] \marginnote{Event}
-                Subset $A \subseteq \Omega$.
-            \item[Sample point/Possible world/Atomic event] \marginnote{Sample point}
-                Element $\omega \in \Omega$.
-        \end{descriptionlist}
-
-    \item[Probability space] \marginnote{Probability space}
-        A probability space/model is a function $\prob{\cdot}: \Omega \rightarrow [0, 1]$ assigned to a sample space such that:
-        \begin{itemize}
-            \item $0 \leq \prob{\omega} \leq 1$
-            \item $\sum_{\omega \in \Omega} \prob{\omega} = 1$
-            \item $\prob{A} = \sum_{\omega \in A} \prob{\omega}$
-        \end{itemize}
-
-    \item[Random variable] \marginnote{Random variable}
-        A function from an event to some range (e.g. reals, booleans, \dots).
-
-    \item[Probability distribution] \marginnote{Probability distribution}
-        For any random variable $X$:
-        \[ \prob{X = x_i} = \sum_{\omega \text{ st } X(\omega)=x_i} \prob{\omega} \]
-
-    \item[Proposition] \marginnote{Proposition}
-        Event where a random variable has a certain value.
-        \[ a = \{ \omega \,\vert\, A(\omega) = \texttt{true} \} \]
-        \[ \lnot  a = \{ \omega \,\vert\, A(\omega) = \texttt{false} \} \]
-        \[ (\texttt{Weather} = \texttt{rain}) = \{ \omega \,\vert\, B(\omega) = \texttt{rain} \} \]
-
-    \item[Prior probability] \marginnote{Prior probability}
-        Prior/unconditional probability of a proposition based on known evidence.
-        
-    \item[Probability distribution (all)] \marginnote{Probability distribution (all)}
-        Gives all the probabilities of a random variable.
-        \[ \textbf{P}(A) = \langle \prob{A=a_1}, \dots, \prob{A=a_n} \rangle \]
-    
-    \item[Joint probability distribution] \marginnote{Joint probability distribution}
-        The joint probability distribution of a set of random variables gives 
-        the probability of all the different combinations of their atomic events.
-
-        Note: Every question on a domain can, in theory, be answered using the joint distribution.
-        In practice, it is hard to apply.
-
-        \begin{example}
-            $\textbf{P}(\texttt{Weather}, \texttt{Cavity}) = $
-            \begin{center}
-                \small
-                \begin{tabular}{c | cccc}
-                                            & \texttt{Weather=sunny} & \texttt{Weather=rain} & \texttt{Weather=cloudy} & \texttt{Weather=snow} \\
-                    \hline
-                    \texttt{Cavity=true}    & 0.144 & 0.02 & 0.016 & 0.02 \\
-                    \texttt{Cavity=false}   & 0.576 & 0.08 & 0.064 & 0.08
-                \end{tabular}
-            \end{center}
-        \end{example}
-
-    \item[Probability density function] \marginnote{Probability density function}
-        The probability density function (PDF) of a random variable $X$ is a function $p: \mathbb{R} \rightarrow \mathbb{R}$
-        such that:
-        \[ \int_{\mathcal{T}_X} p(x) \,dx = 1 \]
-        \begin{descriptionlist}
-            \item[Uniform distribution] \marginnote{Uniform distribution}
-                \[ 
-                    p(x) = \text{Unif}[a, b](x) = 
-                    \begin{cases}
-                        \frac{1}{b-a} & a \leq x \leq b \\
-                        0 & \text{otherwise}
-                    \end{cases} 
-                \]
-            \item[Gaussian (normal) distribution] \marginnote{Gaussian (normal) distribution}
-                \[ \mathcal{N}(\mu, \sigma^2) = \frac{1}{\sigma\sqrt{2\pi}}e^{\frac{-(x-\mu)^2}{2\sigma^2}} \]
-
-                $\mathcal{N}(0, 1)$ is the standard gaussian.
-        \end{descriptionlist}
-
-    \item[Conditional probability] \marginnote{Conditional probability}
-        Probability of a prior knowledge with new evidence:
-        \[ \prob{a \vert b} = \frac{\prob{a \land b}}{\prob{b}} \]
 \end{description}