Francisco Coelho / zugzwang

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updated amartins tasks

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code/asp/pdist.lp
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  1 +# Tarefa 2: Ler Redes Bayesianas, Escrever Programas Lógicos
  2 +
  3 +> **Estado da Tarefa.** Importação de Redes Bayesianas - OK; Construção de Programa Lógico a Partir de uma RB - Em Curso.
  4 +
  5 +## Importar uma Rede Bayesiana
  6 +
  7 +Passos:
  8 +
  9 +- [x] Implementar
  10 +- [ ] Testar e Documentar
  11 +- [x] Usar
  12 +
  13 +Função `summary_dag(filename)` no módulo `bninput`. **Deve ser testada e documentada.**
  14 +
  15 +## Construir um Programa Lógico dada uma Rede Bayesiana
  16 +
  17 +Passos:
  18 +
  19 +- [/] Implementar
  20 +- [ ] Testar e Documentar
  21 +- [ ] Usar
  22 +
  23 +### 2023-07-20
  24 +
  25 +O ficheiro `tarefa2.py` está **quase** adequado para esta tarefa. Em particular, tem código para converter a descrição de uma bn em _algo que se assemelha a um programa lógico_. No entanto:
  26 +
  27 +**Criar funções.** À semelhança do que fez no `bninput`, deve **colocar o código "essencial" em funções**. Isto é, o essencial de
  28 +
  29 +```python
  30 +if __name__ == "__main__":
  31 + summary = summary_dag("asia2.bif")
  32 + model = summary["bnmodel"]
  33 + probabilities = get_yes_probabilities(model)
  34 + for node, yes_prob in probabilities.items():
  35 + parents = model.get_parents(node)
  36 + s = ""
  37 + if len(parents) == 0:
  38 +...
  39 +```
  40 +
  41 +deve ir para uma função. A minha sugestão é que o argumento dessa função seja um `model` que poderá resultar de, por exemplo, `summary_dag(...)`.
  42 +
  43 +**Adaptar a notação dos programas lógicos.**
  44 +
  45 +A sintaxe para os programas lógicos é a seguinte:
  46 +```prolog
  47 +f. /* Facto Determinista */
  48 +h :- b1, ..., bN. /* Regra Determinista */
  49 +p::f. /* Facto Probabilístico */
  50 +p::h :- b1, ..., bN./* Regra Probabilística */
  51 +```
  52 +
  53 +em que `p` é uma probabilidade (um `float` entre 0 e 1); `f` é um "facto" (por exemplo, `asia`) e `h :- b1, ..., bN` é uma "regra" em que `h` é a "cabeça" (_"head"_) e o "corpo" (_"body"_) tem "literais" (factos ou negações de factos) `b1, ..., bN`. O símbolo "`,`" denota a _conjunção_ ($\wedge$), "`-`" a negação ($\neg$) e "`:-`" (em vez de "`<-`", e lê-se "_if_" ou "se") denota $\leftarrow$.
  54 +
  55 +Além disso, em relação ao que o seu programa produz, cada regra e cada facto termina em "`.`". Portanto, **falta acertar a sintaxe com a dos programas lógicos.**
  56 +
  57 +**Sintaxe, parte 2**
  58 +
  59 +Há, ainda, um aspeto adicional: Os programas que processam os programas lógicos não suportam (mais ou menos, em geral, por enquanto) factos e regras probabilísticas. Isso significa que a sintaxe
  60 +```prolog
  61 +p::f. /* Facto Probabilístico */
  62 +p::h :- b1, ..., bN./* Regra Probabilística */
  63 +```
  64 +está "errada" para esses programas. O que podemos fazer, por enquanto, é escrever
  65 +```prolog
  66 +%* p::f. *%
  67 +f ; -f.
  68 +%* p::h. *%
  69 +h ; -h :- b1, ..., bN.
  70 +```
  71 +
  72 +Por exemplo,
  73 +```prolog
  74 +%* 0.01::asia. *%
  75 +asia ; -asia.
  76 +```
  77 +em vez de
  78 +```prolog
  79 +0.01::asia.
  80 +```
  81 +
  82 +Nestes exemplos a sintaxe dos programas lógicos está acrescentada com "`;`" para denotar a disjunção ($\vee$) e "`%* ... *%`" para blocos de comentários. Isto é,
  83 +```prolog
  84 +%* 0.01::asia. *%
  85 +asia ; -asia.
  86 +```
  87 +diz que temos um **facto disjuntivo**, `asia ; -asia` que indica que ou "acontece" `asia` ou "acontece" não `asia`. O comentário `%* 0.01::asia. *%` serve para "transportar" a informação sobre as probabilidades. Esta informação será tratada posteriormente, talvez na tarefa 4 ou na 5.
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students/amartins/tarefas/tarefa2.pdf 0 → 100644
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text/paper_01/pre-paper.pdf
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text/paper_01/pre-paper.tex
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... ... @@ -5,6 +5,7 @@ bibstyle=numeric,
5 5 citestyle=numeric
6 6 ]{biblatex} %Imports biblatex package
7 7 \addbibresource{zugzwang.bib} %Import the bibliography file
  8 +
8 9 \usepackage[x11colors]{xcolor}
9 10  
10 11 \usepackage{tikz}
... ... @@ -95,7 +96,7 @@ citecolor=blue,
95 96 \acrodef{KL}[KL]{Kullback-Leibler}
96 97  
97 98 \title{An Algebraic Approach to Stochastic ASP
98   - %Zugzwang\\\emph{Logic and Artificial Intelligence}\\{\bruno Why this title?}
  99 + %Zugzwang\\\emph{Logic and Artificial Intelligence}\\
99 100 }
100 101  
101 102 \author{
... ... @@ -132,7 +133,7 @@ citecolor=blue,
132 133 \Acf{ASP} is a logic programming paradigm based on the \ac{SM} semantics of \acp{NP} that can be implemented using the latest advances in SAT solving technology. Unlike ProLog, \ac{ASP} is a truly declarative language that supports language constructs such as disjunction in the head of a clause, choice rules, and hard and weak constraints.
133 134  
134 135 \todo{references}
135   -The \ac{DS} is a key approach to extend logical representations with probabilistic reasoning. \Acp{PF} are the most basic \ac{DS} stochastic primitives and take the form of logical facts, $a$, labelled with probabilities, $p$, such as $\probfact{p}{a}$; Each \ac{PF} represents a boolean random variable that is true with probability $p$ and false with probability $\co{p} = 1 - p$. A (consistent) combination of the \acp{PF} defines a \acf{TC} $t = \set{\probfact{p}{a}, \ldots}$ such that \franc{changed \acl{TC} $c$ to $t$ everywhere.}
  136 +The \ac{DS} is a key approach to extend logical representations with probabilistic reasoning. \Acp{PF} are the most basic \ac{DS} stochastic primitives and take the form of logical facts, $a$, labelled with probabilities, $p$, such as $\probfact{p}{a}$; each \ac{PF} represents a boolean random variable that is true with probability $p$ and false with probability $\co{p} = 1 - p$. A (consistent) combination of the \acp{PF} defines a \acf{TC} $t = \set{\probfact{p}{a}, \ldots}$ such that %\franc{changed \acl{TC} $c$ to $t$ everywhere.}
136 137  
137 138 \begin{equation}
138 139 \pr{T = t} = \prod_{a\in t} p \prod_{a \not\in t} \co{p}.
... ... @@ -146,9 +147,9 @@ Our goal is to extend this probability, from \acp{TC}, to cover the \emph{specif
146 147 \item Also, given a dataset and a divergence measure, the specification can be scored (by the divergence w.r.t.\ the \emph{empiric} distribution of the dataset), and weighted or sorted amongst other specifications. These are key ingredients in algorithms searching, for example, optimal specifications of a dataset.
147 148 \end{enumerate}
148 149  
149   -Our idea to extend probabilities starts with the stance that a specification describes an \emph{observable system} and that observed events must be related with the \acp{SM} of that specification. From here, probabilities must be extended from \aclp{TC} to \acp{SM} and then from \acp{SM} to any event.
  150 +Our idea to extend probabilities from \acp{TC} starts with the stance that a specification describes an \emph{observable system} and that observed events must be related with the \acp{SM} of that specification. From here, probabilities must be extended from \aclp{TC} to \acp{SM} and then from \acp{SM} to any event.
150 151  
151   -Extending probability from \acp{TC} to \acp{SM} faces a critical problem, illustrated by the example in \cref{sec:example.1}, concerning situations where multiple \acp{SM}, $ab$ and $ac$, result from a single \ac{TC}, $a$, but there is not enough information (in the specification) to assign a single probability to each \ac{SM}. We propose to address this issue by using algebraic variables to describe that lack of information and then estimate the value of those variables from empirical data.
  152 +Extending probabilities from \acp{TC} to \acp{SM} faces a critical problem, illustrated by the example in \cref{sec:example.1}, concerning situations where multiple \acp{SM}, $ab$ and $ac$, result from a single \ac{TC}, $a$, but there is not enough information (in the specification) to assign a single probability to each \ac{SM}. We propose to address this issue by using algebraic variables to describe that lack of information and then estimate the value of those variables from empirical data.
152 153  
153 154 In a related work, \cite{verreet2022inference}, epistemic uncertainty (or model uncertainty) is considered as a lack of knowledge about the underlying model, that may be mitigated via further observations. This seems to presuppose a Bayesian approach to imperfect knowledge in the sense that having further observations allows to improve/correct the model. Indeed, the approach in that work uses Beta distributions in order to be able to learn the full distribution. This approach seems to be specially fitted to being able to tell when some probability lies beneath some given value. \todo{Our approach seems to be similar in spirit. If so, we should mention this in the introduction.}
154 155 \todo{Also remark that our apporach remains algebraic in the way that we address the problems concerning the extension of probabilities.}
... ... @@ -161,7 +162,10 @@ In a related work, \cite{verreet2022inference}, epistemic uncertainty (or model
161 162  
162 163 \section{A simple but fruitful example}\label{sec:example.1}
163 164  
164   -\todo{Write an introduction to the section}
  165 +%\todo{Write an introduction to the section}
  166 +
  167 +{\bruno In this section we consider a somewhat simple example that showcases the problem of extending probabilities from \aclp{TC} to \aclp{SM}. As mentioned before, the main issue arises from the lack of information in the specification to assign a single probability to each \aclp{SM}. This becomes a crucial problem in situations where multiple \aclp{SM} result from a single \aclp{TC}. We will come back to the example given in this section in \cref{S:SBF_developed}, after we present our proposal for extending the probabilities from \aclp{TC} to \aclp{SM} in \cref{sec:extending.probalilities}.}
  168 +
165 169  
166 170 \begin{example}\label{running.example}
167 171 Consider the following specification
... ... @@ -268,12 +272,12 @@ The \aclp{SM} $ab, ac$ from \cref{running.example} result from the clause $b \v
268 272 \label{fig:running.example}
269 273 \end{figure}
270 274  
271   -\todo{Somewhere, we need to shift the language from extending \emph{probabilities} to extending \emph{measures}}
272   -
273   -\note{$\emptyevent$ notation introduced in \cref{fig:running.example}.}
  275 +%\note{$\emptyevent$ notation introduced in \cref{fig:running.example}.}
274 276  
275 277 The diagram in \cref{fig:running.example} illustrates the problem of extending probabilities from \acp{TC} nodes to \acp{SM} and then to general events in a \emph{node-wise} process. This quickly leads to \remark{coherence problems}{for example?} concerning probability, with no clear systematic approach --- Instead, weight extension can be based in the relation an event has with the \aclp{SM}.
276 278  
  279 +{\bruno We will consider first the problem of extending measures, since the problem of extending probabilities easily follows by means of a suitable normalization (see \eqref{E:Normalization} and \eqref{E:measure_to_prob}).}
  280 +
277 281 \subsection{An Equivalence Relation}\label{subsec:equivalence.relation}
278 282  
279 283 \begin{figure}[t]
... ... @@ -383,11 +387,10 @@ The diagram in \cref{fig:running.example} illustrates the problem of extending p
383 387 \label{fig:running.example.classes}
384 388 \end{figure}
385 389  
386   -Given an ASP specification,
387   -\remark{{\bruno Introduce also the sets mentioned below}}{how?}
388   -we consider the \emph{atoms} $a \in \fml{A}$ and \emph{literals}, $z \in \fml{L}$, \emph{events} $e \in \fml{E} \iff e \subseteq \fml{L}$ and \emph{worlds} $w \in \fml{W}$ (consistent events), \emph{\aclp{TC} } $t \in \fml{T} \iff t = a \vee \neg a$ and \emph{\aclp{SM} } $s \in \fml{S}\subset\fml{W}$.
  390 +Given an ASP specification, we consider {\bruno a set of \emph{atoms} $ \fml{A}$, a set of \emph{literals}, $\fml{L}$, and a set of \emph{events} $\fml{E}$ such that $e \in \fml{E} \iff e \subseteq \fml{L}$. We also consider a set of \emph{worlds} $\fml{W}$ (consistent events), a set of \emph{\aclp{TC} } such that for every $a \in \fml{A}$ we have $t \in \fml{T} \iff t = a \vee \neg a$, and a set of \emph{\aclp{SM} } such that $ \fml{S}\subset\fml{W}$.}
  391 +%the \emph{atoms} $a \in \fml{A}$ and \emph{literals}, $z \in \fml{L}$, \emph{events} $e \in \fml{E} \iff e \subseteq \fml{L}$ and \emph{worlds} $w \in \fml{W}$ (consistent events), \emph{\aclp{TC} } $t \in \fml{T} \iff t = a \vee \neg a$ and \emph{\aclp{SM} } $s \in \fml{S}\subset\fml{W}$.
389 392  
390   -Our path starts with a perspective of \aclp{SM} as playing a role similar to \emph{prime} factors. The \aclp{SM} of a specification are the irreducible events entailed from that specification and any event must be \replace{interpreted}{considered} under its relation with the \aclp{SM}.
  393 +Our path starts with a perspective of \aclp{SM} as playing a role similar to \emph{prime} factors. The \aclp{SM} of a specification are the irreducible events entailed from that specification and any event must be considered under its relation with the \aclp{SM}.
391 394  
392 395 %\remark{\todo{Introduce a structure with worlds, events, and \aclp{SM} }}{seems irrelevant}
393 396 This focus on the \acp{SM} leads to the following definition:
... ... @@ -427,7 +430,7 @@ Observe that the minimality of \aclp{SM} implies that, in \cref{def:stable.core
427 430 \end{cases}\label{eq:event.class}
428 431 \end{equation}
429 432  
430   -The subsets of the \aclp{SM}, together with $\inconsistent$, form a set of representatives. Consider again Example~\ref{running.example}. As previously mentioned, the \aclp{SM} are $\fml{S} = \co{a}, ab, ac$ so the quotient set of this relation is:
  433 +The subsets of the \aclp{SM}, together with $\inconsistent$, form a set of representatives. Consider again \cref{running.example}. As previously mentioned, the \aclp{SM} are $\fml{S} = \co{a}, ab, ac$ so the quotient set of this relation is:
431 434 \begin{equation}
432 435 \class{\fml{E}} = \set{
433 436 \inconsistent,
... ... @@ -521,7 +524,7 @@ where $\indepclass$ denotes both the class of \emph{independent} events $e$ such
521 524 \item Normalization of the weights.
522 525 \end{enumerate}
523 526  
524   -The ``extension'' phase, traced by equations (\ref{eq:prob.total.choice}) and (\ref{eq:weight.tchoice} --- \ref{eq:weight.events}), starts with the weight (probability) of \aclp{TC}, $\pw{t} = \pr{T = t}$, expands it to \aclp{SM}, $\pw{s}$, and then, within the equivalence relation from \cref{eq:equiv.rel}, to (general) events, $\pw{e}$, including (consistent) worlds.
  527 +The ``extension'' phase, traced by \cref{eq:prob.total.choice} and eqs. \eqref{eq:weight.tchoice} to \eqref{eq:weight.events}, starts with the weight (probability) of \aclp{TC}, $\pw{t} = \pr{T = t}$, expands it to \aclp{SM}, $\pw{s}$, and then, within the equivalence relation from \cref{eq:equiv.rel}, to (general) events, $\pw{e}$, including (consistent) worlds.
525 528  
526 529 \begin{description}
527 530 %
... ... @@ -555,7 +558,7 @@ The ``extension&#39;&#39; phase, traced by equations (\ref{eq:prob.total.choice}) and (\
555 558 \pw{\indepclass, t} := 0.
556 559 \label{eq:weight.class.independent}
557 560 \end{equation}
558   - \item[Other Classes.] The extension must be constant within a class, its value should result from the elements in the \acl{SC}, and respect the assumption \ref{assumption:smodels.independence} (\aclp{SM} independence):
  561 + \item[Other Classes.] The extension must be constant within a class, its value should result from the elements in the \acl{SC}, and respect assumption \ref{assumption:smodels.independence} (\aclp{SM} independence):
559 562 \begin{equation}
560 563 \pw{\class{e}, t} := \sum_{k=1}^{n}\pw{s_k, t},~\text{if}~\stablecore{e} = \set{s_1, \ldots, s_n}.
561 564 \label{eq:weight.class.other}
... ... @@ -605,9 +608,9 @@ Equation \eqref{eq:weight.class.other} results from conditional independence of
605 608  
606 609 \section{Developed Examples}
607 610  
608   -\subsection{The SBF Example}
  611 +\subsection{The SBF Example}\label{S:SBF_developed}
609 612  
610   -We continue with the specification from Equation \eqref{eq:example.1}.
  613 +We continue with the specification from \cref{eq:example.1}.
611 614  
612 615 \begin{description}
613 616 %
... ... @@ -682,14 +685,14 @@ We continue with the specification from Equation \eqref{eq:example.1}.
682 685 \end{array}
683 686 \end{equation*}
684 687 \item[Normalization.] To get a weight that sums up to one, we compute the \emph{normalization factor}. Since $\pw{\cdot}$ is constant on classes,\todo{prove that we get a probability.}
685   - \begin{equation*}
  688 + \begin{equation}\label{E:Normalization}
686 689 Z := \sum_{e\in\fml{E}} \pw{e}
687 690 = \sum_{\class{e} \in\class{\fml{E}}} \frac{\pw{\class{e}}}{\#\class{e}},
688   - \end{equation*}
  691 + \end{equation}
689 692 that divides the weight function into a normalized weight
690   - \begin{equation*}
  693 + \begin{equation}\label{E:measure_to_prob}
691 694 \pr{e} := \frac{\pw{e}}{Z}.
692   - \end{equation*}
  695 + \end{equation}
693 696 such that
694 697 $$
695 698 \sum_{e \in \fml{E}} \pr{e} = 1.
... ... @@ -782,83 +785,75 @@ We continue with the specification from Equation \eqref{eq:example.1}.
782 785 %
783 786 \subsection{An example involving Bayesian networks}
784 787  
785   -\franc{Comentários:}
786   -\begin{itemize}
787   - \item Há uma macro, $\backslash\text{pr}\{A\}$, para denotar a função de probabilidade, $\pr{A}$ em vez de $P(A)$. Já agora, para a condicional também há um comando, $\backslash\text{given}$: $\pr{A \given B}$.
788   - \item E, claro, para factos+probabilidades: $\probfact{p}{a}$.
789   - \item A designação dos `pesos' não está consistente: $pj\_a$ e $a\_be$. Fiz uma macro (\emph{hehe}) para sistematizar isto: \condsymb{a}{bnc}.
790   - \item Nos programas, alinhei pelos factos. Isto é, $\probfact{0.3}{a}$ e $a \leftarrow b$ alinham pelo (fim do) $a$.
791   -\end{itemize}
792   -
793 788  
794   -As it turns out, our framework is suitable to deal with more sophisticated cases, \replace{for example}{in particular} cases involving Bayesian networks. In order to illustrate this, in this section we see how the classical example of the Burglary, Earthquake, Alarm \cite{Judea88} works in our setting. This example is a commonly used example in Bayesian networks because it illustrates reasoning under uncertainty. The gist of example is given in \cref{Figure_Alarm}. It involves a simple network of events and conditional probabilities.
  789 +As it turns out, our framework is suitable to deal with more sophisticated cases, in particular cases involving Bayesian networks. In order to illustrate this, in this section we see how the classical example of the Burglary, Earthquake, Alarm \cite{Judea88} works in our setting. This example is a commonly used example in Bayesian networks because it illustrates reasoning under uncertainty. The gist of the example is given in \cref{Figure_Alarm}. It involves a simple network of events and conditional probabilities.
795 790  
796   -The events are: Burglary ($B$), Earthquake ($E$), Alarm ($A$), Mary calls ($M$) and John calls ($J$). The initial events $B$ and $E$ are assumed to be independent events that occur with probabilities $P(B)$ and $P(E)$, respectively. There is an alarm system that can be triggered by either of the initial events $B$ and $E$. The probability of the alarm going off is a conditional probability given that $B$ and $E$ have occurred. One denotes these probabilities, as per usual, by $P(A|B)$, and $P(A|E)$. There are two neighbours, Mary and John who have agreed to call if they hear the alarm. The probability that they do actually call is also a conditional probability denoted by $P(M|A)$ and $P(J|A)$, respectively.
  791 +The events are: Burglary ($B$), Earthquake ($E$), Alarm ($A$), Mary calls ($M$) and John calls ($J$). The initial events $B$ and $E$ are assumed to be independent events that occur with probabilities $\pr{B}$ and $\pr{E}$, respectively. There is an alarm system that can be triggered by either of the initial events $B$ and $E$. The probability of the alarm going off is a conditional probability given that $B$ and $E$ have occurred. One denotes these probabilities, as per usual, by $\pr{A \given B}$, and $\pr{A \given E}$. There are two neighbours, Mary and John who have agreed to call if they hear the alarm. The probability that they do actually call is also a conditional probability denoted by $\pr{M \given A}$ and $\pr{J \given A}$, respectively.
797 792  
798 793  
799 794  
800 795 \begin{figure}
801 796 \begin{center}
802 797 \begin{tikzpicture}[node distance=2.5cm]
803   -
  798 +
804 799 % Nodes
805 800 \node[smodel, circle] (A) {A};
806 801 \node[tchoice, above right of=A] (B) {B};
807 802 \node[tchoice, above left of=A] (E) {E};
808 803 \node[tchoice, below left of=A] (M) {M};
809 804 \node[tchoice, below right of=A] (J) {J};
810   -
  805 +
811 806 % Edges
812   - \draw[->] (B) to[bend left] (A) node[right,xshift=1.1cm,yshift=0.8cm] {\footnotesize{$P(B)=0.001$}} ;
813   - \draw[->] (E) to[bend right] (A) node[left, xshift=-1.4cm,yshift=0.8cm] {\footnotesize{$P(E)=0.002$}} ;
814   - \draw[->] (A) to[bend right] (M) node[left,xshift=0.2cm,yshift=0.7cm] {\footnotesize{$P(M|A)$}};
815   - \draw[->] (A) to[bend left] (J) node[right,xshift=-0.2cm,yshift=0.7cm] {\footnotesize{$P(J|A)$}} ;
  807 + \draw[->] (B) to[bend left] (A) node[right,xshift=1.1cm,yshift=0.8cm] {\footnotesize{$\pr{B}=0.001$}} ;
  808 + \draw[->] (E) to[bend right] (A) node[left, xshift=-1.4cm,yshift=0.8cm] {\footnotesize{$\pr{E}=0.002$}} ;
  809 + \draw[->] (A) to[bend right] (M) node[left,xshift=0.2cm,yshift=0.7cm] {\footnotesize{$\pr{M \given A}$}};
  810 + \draw[->] (A) to[bend left] (J) node[right,xshift=-0.2cm,yshift=0.7cm] {\footnotesize{$\pr{J \given A}$}} ;
816 811 \end{tikzpicture}
817 812 \end{center}
818   -
  813 +
819 814 \begin{multicols}{3}
820   -
  815 +
821 816 \footnotesize{
822   - \begin{equation*}
823   - \begin{split}
824   - &P(M|A)\\
825   - & \begin{array}{c|cc}
826   - & m & \neg m \\
827   - \hline
828   - a & 0.9 & 0.1 \\
829   - \neg a & 0.05 & 0.95
830   - \end{array}
831   - \end{split}
832   - \end{equation*}
  817 + \begin{equation*}
  818 + \begin{split}
  819 + &\pr{M \given A}\\
  820 + & \begin{array}{c|cc}
  821 + & m & \neg m \\
  822 + \hline
  823 + a & 0.9 & 0.1\\
  824 + \neg a& 0.05 & 0.95
  825 + \end{array}
  826 + \end{split}
  827 + \end{equation*}
833 828 }
834   -
  829 +
835 830 \footnotesize{
836   - \begin{equation*}
837   - \begin{split}
838   - &P(J|A)\\
839   - & \begin{array}{c|cc}
840   - & j & \neg j \\
841   - \hline
842   - a & 0.7 & 0.3 \\
843   - \neg a & 0.01 & 0.99
844   - \end{array}
845   - \end{split}
846   - \end{equation*}
  831 + \begin{equation*}
  832 + \begin{split}
  833 + &\pr{J \given A}\\
  834 + & \begin{array}{c|cc}
  835 + & j & \neg j \\
  836 + \hline
  837 + a & 0.7 & 0.3\\
  838 + \neg a& 0.01 & 0.99
  839 + \end{array}
  840 + \end{split}
  841 + \end{equation*}
847 842 }
848 843 \footnotesize{
849   - \begin{equation*}
850   - \begin{split}
851   - P(A|B \wedge E)\\
852   - \begin{array}{c|c|cc}
853   - & & a & \neg a \\
854   - \hline
855   - b & e & 0.95 & 0.05 \\
856   - b & \neg e & 0.94 & 0.06 \\
857   - \neg b & e & 0.29 & 0.71 \\
858   - \neg b & \neg e & 0.001 & 0.999
859   - \end{array}
860   - \end{split}
861   - \end{equation*}
  844 + \begin{equation*}
  845 + \begin{split}
  846 + \pr{A \given B \wedge E}\\
  847 + \begin{array}{c|c|cc}
  848 + & & a & \neg a \\
  849 + \hline
  850 + b & e & 0.95 & 0.05\\
  851 + b & \neg e & 0.94 & 0.06\\
  852 + \neg b & e & 0.29 & 0.71\\
  853 + \neg b & \neg e & 0.001 & 0.999
  854 + \end{array}
  855 + \end{split}
  856 + \end{equation*}
862 857 }
863 858 \end{multicols}
864 859 \caption{The Earthquake, Burglary, Alarm model}
... ... @@ -866,64 +861,63 @@ The events are: Burglary ($B$), Earthquake ($E$), Alarm ($A$), Mary calls ($M$)
866 861 \end{figure}
867 862  
868 863  
869   -Considering the probabilities given in \cref{Figure_Alarm} we obtain the following spe\-ci\-fi\-ca\-tion
  864 +Considering the probabilities given in \cref{Figure_Alarm} we obtain the following spe\-ci\-fi\-ca\-tion:
870 865  
871 866 \begin{equation*}
872 867 \begin{aligned}
873   - \probfact{0.001}{b} & ,\cr
874   - \probfact{0.002}{e} & ,\cr
875   - \end{aligned}
  868 + \probfact{0.001}{b}&,\cr
  869 + \probfact{0.002}{e}&,\cr
  870 + \end{aligned}
876 871 \label{eq:not_so_simple_example}
877 872 \end{equation*}
878 873  
879   -For the table giving the probability $P(M|A)$ we obtain the specification:
  874 +For the table giving the probability $\pr{M \given A}$ we obtain the specification:
880 875  
881 876  
882 877 \begin{equation*}
883 878 \begin{aligned}
884   - \probfact{0.9}{pm\_a} & ,\cr
885   - \probfact{0.05}{pm\_na} & ,\cr
886   - m & \leftarrow a, pm\_a,\cr
887   - \neg m & \leftarrow a, \neg pm\_a.
888   - \end{aligned}
  879 + \probfact{0.9}{\condsymb{m}{a}}&,\cr
  880 + \probfact{0.05}{\condsymb{m}{na}}&,\cr
  881 + m & \leftarrow a, \condsymb{m}{a},\cr
  882 + \neg m & \leftarrow a, \neg \condsymb{m}{a}.
  883 + \end{aligned}
889 884 \end{equation*}
890 885  
891 886 This latter specification can be simplified by writing $\probfact{0.9}{m \leftarrow a}$ and $\probfact{0.05}{m \leftarrow \neg a}$.
892 887  
893   -Similarly, for the probability $P(J|A)$ we obtain
  888 +Similarly, for the probability $\pr{J \given A}$ we obtain
894 889  
895 890 \begin{equation*}
896 891 \begin{aligned}
897   - \probfact{0.7}{pj\_a} & ,\cr
898   - \probfact{0.01}{pj\_na} & ,\cr
899   - j & \leftarrow a, pj\_a,\cr
900   - \neg j & \leftarrow a, \neg pj\_a.\cr
901   - \end{aligned}
  892 + \probfact{0.7}{\condsymb{j}{a}}&,\cr
  893 + \probfact{0.01}{\condsymb{j}{na}}&,\cr
  894 + j & \leftarrow a, \condsymb{j}{a},\cr
  895 + \neg j & \leftarrow a, \neg \condsymb{j}{a}.\cr
  896 + \end{aligned}
902 897 \end{equation*}
903 898  
904 899 Again, this can be simplified by writing $\probfact{0.7}{j \leftarrow a}$ and $\probfact{0.01}{j \leftarrow \neg a}$.
905 900  
906   -Finally, for the probability $P(A|B \wedge E)$ we obtain
  901 +Finally, for the probability $\pr{A \given B \wedge E}$ we obtain
907 902  
908 903 \begin{equation*}
909 904 \begin{aligned}
910   - \probfact{0.95}{a\_be} & ,\cr
911   - \probfact{0.94}{a\_bne} & ,\cr
912   - \probfact{0.29}{a\_nbe} & ,\cr
913   - \probfact{0.001}{a\_nbne} & ,\cr
914   - a & \leftarrow b, e, a\_be,\cr
915   - \neg a & \leftarrow b,e, \neg a\_be, \cr
916   - a & \leftarrow b,e, a\_bne,\cr
917   - \neg a & \leftarrow b,e, \neg a\_bne, \cr
918   - a & \leftarrow b,e, a\_nbe,\cr
919   - \neg a & \leftarrow b,e, \neg a\_nbe, \cr
920   - a & \leftarrow b,e, a\_nbne,\cr
921   - \neg a & \leftarrow b,e, \neg a\_nbne. \cr
922   - \end{aligned}
  905 + \probfact{0.95}{\condsymb{a}{be}}&,\cr
  906 + \probfact{0.94}{\condsymb{a}{bne}}&,\cr
  907 + \probfact{0.29}{\condsymb{a}{nbe}}&,\cr
  908 + \probfact{0.001}{\condsymb{a}{nbne}}&,\cr
  909 + a & \leftarrow b, e, \condsymb{a}{be},\cr
  910 + \neg a & \leftarrow b,e, \neg \condsymb{a}{be}, \cr
  911 + a & \leftarrow b,e, \condsymb{a}{bne},\cr
  912 + \neg a & \leftarrow b,e, \neg \condsymb{a}{bne}, \cr
  913 + a & \leftarrow b,e, \condsymb{a}{nbe},\cr
  914 + \neg a & \leftarrow b,e, \neg \condsymb{a}{nbe}, \cr
  915 + a & \leftarrow b,e, \condsymb{a}{nbne},\cr
  916 + \neg a & \leftarrow b,e, \neg \condsymb{a}{nbne}. \cr
  917 + \end{aligned}
923 918 \end{equation*}
924 919  
925   -One can then proceed as in the previous subsection and analyse this example. The details of such analysis are not given here since they are analogous, albeit admittedly more cumbersome.
926   -
  920 +One can then proceed as in the previous subsection and analyse this example. The details of such analysis are not given here since they are analogous, albeit admittedly more cumbersome.
927 921  
928 922 \section{Discussion}
929 923  
... ...