From 3836519896e9de3c113ac2577cc90265ad6bad19 Mon Sep 17 00:00:00 2001 From: NotXia <35894453+NotXia@users.noreply.github.com> Date: Fri, 17 Nov 2023 17:09:53 +0100 Subject: [PATCH] Add FAIKR3 compact conditional distributions --- .../module3/img/_dynamic_bn_example.pdf | Bin 0 -> 58930 bytes .../module3/img/_linear_gaussian_example.pdf | Bin 0 -> 51969 bytes .../module3/img/_noisy_or_example.pdf | Bin 0 -> 18056 bytes .../module3/sections/_bayesian_net.tex | 146 ++++++++++++++++++ 4 files changed, 146 insertions(+) create mode 100644 src/fundamentals-of-ai-and-kr/module3/img/_dynamic_bn_example.pdf create mode 100644 src/fundamentals-of-ai-and-kr/module3/img/_linear_gaussian_example.pdf create mode 100644 src/fundamentals-of-ai-and-kr/module3/img/_noisy_or_example.pdf diff --git a/src/fundamentals-of-ai-and-kr/module3/img/_dynamic_bn_example.pdf b/src/fundamentals-of-ai-and-kr/module3/img/_dynamic_bn_example.pdf new file mode 100644 index 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HcmV?d00001 diff --git a/src/fundamentals-of-ai-and-kr/module3/sections/_bayesian_net.tex b/src/fundamentals-of-ai-and-kr/module3/sections/_bayesian_net.tex index 1c162f2..a9c6ed6 100644 --- a/src/fundamentals-of-ai-and-kr/module3/sections/_bayesian_net.tex +++ b/src/fundamentals-of-ai-and-kr/module3/sections/_bayesian_net.tex @@ -408,4 +408,150 @@ that respects the causality allows to obtain more compact networks. \[ \prob{\texttt{WetGrass} \mid \texttt{Sprinkler} = \texttt{true}} \] \end{minipage} \end{example} +\end{description} + + + +\section{Compact conditional distributions} + +Use canonical distributions (standard patterns) to reduce +the number of variables in a conditional probability table. + + +\subsection{Noisy-OR} +\marginnote{Noisy-OR} +Noisy-OR distributions model a network of non-interacting causes with a common effect. +A node $X$ has $k$ parents $U_1, \dots, U_k$ and eventually a leak node $U_L$ to capture unmodeled concepts. + +\begin{figure}[h] + \centering + \includegraphics[width=0.3\textwidth]{img/_noisy_or_example.pdf} + \caption{Example of noisy-OR network} +\end{figure} + +Each node $U_i$ has a failure (inhibition) probability $q_i$: +\[ q_i = \prob{\lnot x \mid u_i, \lnot u_j \text{ for } j \neq i} \] +The CRT can be built by computing the probabilities as: +\[ \prob{\lnot x \mid \texttt{Parents($X$)}} = \prod_{j:\, U_j = \texttt{true}} q_j \] +In other words: +\[ \prob{\lnot x \mid u_1, \dots, u_n} = + \prob{\lnot x \mid u_1} \cdot \prob{\lnot x \mid u_2} \cdot \text{\dots} \cdot \prob{\lnot x \mid u_n} \] + +Because only the failure probabilities are required, the number of parameters is linear in the number of parents. + +\begin{example} + We have as causes \texttt{Cold}, \texttt{Flu} and \texttt{Malaria} and as effect \texttt{Fever}. + For simplicity there are no leak nodes. + The failure probabilities are: + \[ + \begin{split} + q_\texttt{cold} &= \prob{\lnot \texttt{fever} \mid \texttt{cold}, \lnot\texttt{flu}, \lnot\texttt{malaria}} = 0.6 \\ + q_\texttt{flu} &= \prob{\lnot \texttt{fever} \mid \lnot\texttt{cold}, \texttt{flu}, \lnot\texttt{malaria}} = 0.2 \\ + q_\texttt{malaria} &= \prob{\lnot \texttt{fever} \mid \lnot\texttt{cold}, \lnot\texttt{flu}, \texttt{malaria}} = 0.1 + \end{split} + \] + + Known the failure probabilities, the entire CRT can be computed: + \begin{center} + \begin{tabular}{c|c|c|rc|c} + \hline + \texttt{Cold} & \texttt{Flu} & \texttt{Malaria} & \multicolumn{2}{c|}{$\prob{\lnot\texttt{fever}}$} & $1-\prob{\lnot\texttt{fever}}$ \\ + \hline + F & F & F & & 0.0 & 1.0 \\ + F & F & T & $q_\texttt{malaria} =$ & 0.1 & 0.9 \\ + F & T & F & $q_\texttt{flu} =$ & 0.2 & 0.8 \\ + F & T & T & $q_\texttt{flu} \cdot q_\texttt{malaria} =$ & 0.02 & 0.98 \\ + T & F & F & $q_\texttt{cold} =$ & 0.6 & 0.4 \\ + T & F & T & $q_\texttt{cold} \cdot q_\texttt{malaria} =$ & 0.06 & 0.94 \\ + T & T & F & $q_\texttt{cold} \cdot q_\texttt{flu} =$ & 0.12 & 0.88 \\ + T & T & T & $q_\texttt{cold} \cdot q_\texttt{flu} \cdot q_\texttt{malaria} =$ & 0.012 & 0.988 \\ + \hline + \end{tabular} + \end{center} +\end{example} + + +\subsection{Hybrid Bayesian networks} +\marginnote{Hybrid Bayesian networks} + +Network with discrete and continuous random variables. +Continuous variables must be converted into a finite representation. +Possible approaches are: +\begin{description} + \item[Discretization] \marginnote{Discretization} + Values are divided into a fixed set of intervals. + This approach may introduce large errors and large CPTs. + + \item[Finitely parametrized canonical families] \marginnote{Finitely parametrized canonical families} + There are two cases to handle using this approach: + \begin{descriptionlist} + \item[Continuous child] + Given the continuous variables $X$ and $C$ and a discrete (boolean, for simplicity) variable $D$, + we want to compute the distribution $\textbf{P}(X \mid C, D)$. + + The discrete parent is handled by enumeration, by computing the probability over the domain of $D$. + + For the continuous parent, an arbitrarily chosen distribution over the values of $X$ is used. + A common choice is the \textbf{linear Gaussian} \marginnote{Linear Gaussian} + whose mean is a linear combination of the values of the parents and the variance is fixed. + + A network with all continuous linear Gaussian distributions has the property + of having a multivariate Gaussian distribution as joint distribution. + Moreover, if a continuous variable has some discrete parents, it defines a conditional Gaussian distribution + where, fixed the values of the discrete variables, the distribution over the continuous variable is a multivariate Gaussian. + + \begin{example} + Let \texttt{Subsidy} and \texttt{Buys} be discrete variables and + \texttt{Harvest} and \texttt{Cost} be continuous variables. + \begin{center} + \includegraphics[width=0.3\textwidth]{img/_linear_gaussian_example.pdf} + \end{center} + + To compute $\textbf{P}(\texttt{Cost} \mid \texttt{Harvest}, \texttt{Subsidy})$, + we split the probabilities over the values of the discrete variable \texttt{Subsidy} + and use a linear Gaussian for \texttt{Harvest}. + We therefore have that: + \[ + \begin{split} + \prob{\texttt{C} = \texttt{c} \mid \texttt{Harvest} = \texttt{h}, \texttt{Subsidy} = \texttt{true}} + &= \mathcal{N}(a_t h + b_t, \sigma_t)(c) \\ + \prob{\texttt{C} = \texttt{c} \mid \texttt{Harvest} = \texttt{h}, \texttt{Subsidy} = \texttt{false}} + &= \mathcal{N}(a_f h + b_f, \sigma_f)(c) + \end{split} + \] + where $a_t$, $b_t$, $\sigma_t$, $a_f$, $b_f$ and $\sigma_t$ are parameters. + \end{example} + + \item[Discrete child with continuous parents] + Given the continuous variable $C$ and a discrete variable $X$, + the probability of $X$ given $C$ in obtained by using a threshold function. + For instance, probit or sigmoid distributions can be used. + \end{descriptionlist} +\end{description} + + +\subsection{Other methods} + +\begin{description} + \item[Dynamic Bayesian network] \marginnote{Dynamic Bayesian network} + Useful to model the evolution through time. + A template variable $X_i$ is instantiated as $X_i^{(t)}$ at each time step. + \begin{figure}[h] + \centering + \includegraphics[width=0.3\textwidth]{img/_dynamic_bn_example.pdf} + \caption{Example of dynamic Bayesian network} + \end{figure} + + \item[Density estimation] \marginnote{Density estimation} + Parameters of the conditional distribution are learnt: + \begin{description} + \item[Bayesian learning] calculate the probability of each hypothesis. + \item[Approximations] using the maximum-a-posteriori and maximum-likelihood hypothesis. + \item[Expectation-maximization algorithm{\normalfont.}] + \end{description} + + \item[Undirected graphical models] \marginnote{Undirected graphical models} + Markov networks are an alternative to probabilistic graphical models (as Bayesian networks). + Markov networks are undirected graphs with factors (instead of probabilities) and + are able to naturally capture independence relations. \end{description} \ No newline at end of file