diff --git a/text/paper_01/pre-paper.pdf b/text/paper_01/pre-paper.pdf index 9767a67..7863c0f 100644 Binary files a/text/paper_01/pre-paper.pdf and b/text/paper_01/pre-paper.pdf differ diff --git a/text/paper_01/pre-paper.tex b/text/paper_01/pre-paper.tex index 11e9b74..d43c474 100644 --- a/text/paper_01/pre-paper.tex +++ b/text/paper_01/pre-paper.tex @@ -121,7 +121,7 @@ citecolor=blue, \begin{abstract} \todo{rewrite} - A major limitation of logical representations in real world applications is the implicit assumption that the \acl{BK} is perfect. This assumption is problematic if data is noisy, which is often the case. Here we aim to explore how \acl{ASP} specifications with probabilistic facts can lead to \remark{characterizations of probability functions}{Why is this important? Is this what `others in sota' are trying do to?} on the specification's domain. + A major limitation of logical representations in real world applications is the implicit assumption that the \acl{BK} is perfect. This assumption is problematic if data is noisy, which is often the case. Here we aim to explore how \acl{ASP} specifications with probabilistic facts can lead to \remark{characterizations of probability functions}{Why is this important? Is this what `others in sota' are trying do to?} on the specification's domain. \end{abstract} \section{Introduction and Motivation} @@ -143,10 +143,10 @@ Our goal is to extend this probability, from \acp{TC}, to cover the \emph{specif \begin{enumerate} \item Support probabilistic reasoning/tasks on the specification domain. - \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. + \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. \end{enumerate} -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. +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. 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. @@ -165,48 +165,48 @@ In a related work, \cite{verreet2022inference}, epistemic uncertainty (or model \begin{example}\label{running.example} Consider the following specification - + \begin{equation} \begin{aligned} - \probfact{0.3}{a}&,\cr - b \vee c& \leftarrow a. + \probfact{0.3}{a} & ,\cr + b \vee c & \leftarrow a. \end{aligned} \label{eq:example.1} \end{equation} - + This specification has three \aclp{SM}, $\co{a}, ab$ and $ac$ (see \cref{fig:running.example}). While it is straightforward to set $P(\co{a})=0.7$, there is no further information to assign values to $P(ab)$ and $P(ac)$. Assuming that the \acfp{SM} are (probabilistically) independent, we can use a parameter $\theta$ such that $$ - \begin{aligned} - P(ab) &= 0.3 \theta,\cr - P(ac) &= 0.3 (1 - \theta). - \end{aligned} + \begin{aligned} + P(ab) & = 0.3 \theta,\cr + P(ac) & = 0.3 (1 - \theta). + \end{aligned} $$ -\end{example} +\end{example} While uncertainty is inherent to the specification it can be mitigated with the help of a dataset: the parameter $\theta$ can be estimated from a empirical distribution \todo{or we can have a distribution of $\theta$}. \todo{point to examples of this in following sections.} In summary, if an \ac{ASP} specification is intended to describe some observable system then: \begin{enumerate} - + \item Observations can be used to estimate the value of the parameters (such as $\theta$ above and others entailed from further clauses). - + \item \todo{What about the case where we already know a distribution of $\theta$?} - - \item With a probability set for the \aclp{SM}, we want to extend it to all the events of the specification domain. - + + \item With a probability set for the \aclp{SM}, we want to extend it to all the events of the specification domain. + \item This extended probability can then be related to the \emph{empirical distribution}, using a probability divergence, such as \ac{KL}; and the divergence value used as a \emph{performance} measure of the specification with respect to the observations. - + \item If that specification is only but one of many possible candidates then that performance measure can be used, \emph{e.g.} as fitness, by algorithms searching (optimal) specifications of a dataset of observations. - + \end{enumerate} \begin{quote} - \todo{Expand this:} If observations are not consistent with the models of the specification, then the specification is wrong and must be changed. + \todo{Expand this:} If observations are not consistent with the models of the specification, then the specification is wrong and must be changed. \end{quote} -Currently, we are addressing the problem of extending a probability function (possibly using parameters such as $\theta$), defined on the \acp{SM} of a specification, to all the events of that specification. Of course, this extension must satisfy the Kolmogorov axioms of probability so that probabilistic reasoning is consistent with the \ac{ASP} specification. +Currently, we are addressing the problem of extending a probability function (possibly using parameters such as $\theta$), defined on the \acp{SM} of a specification, to all the events of that specification. Of course, this extension must satisfy the Kolmogorov axioms of probability so that probabilistic reasoning is consistent with the \ac{ASP} specification. The conditional independence of stable worlds asserts the \remark{least informed strategy}{references?} that we discussed in the introduction and make explicit here: @@ -237,7 +237,7 @@ The \aclp{SM} $ab, ac$ from \cref{running.example} result from the clause $b \v % ---- \draw[doubt] (a) to[bend left] (ab); \draw[doubt] (a) to[bend right] (ac); - + \draw[doubt] (ab) to[bend left] (abc); \draw[doubt] (ab) to[bend right] (abC); @@ -245,14 +245,14 @@ The \aclp{SM} $ab, ac$ from \cref{running.example} result from the clause $b \v \draw[doubt] (ac) to[bend left] (aBc); \draw[doubt, dash dot] (Ac) to (Abc); - + \draw[doubt] (A) to (Ac); \draw[doubt] (A) to (Abc); - + \draw[doubt] (ab) to[bend right] (E); \draw[doubt] (ac) to[bend right] (E); \draw[doubt] (A) to[bend left] (E); - + \draw[doubt] (ab) to (b); \draw[doubt] (ac) to (c); % \draw[doubt] (ab) to[bend left] (a); @@ -263,7 +263,7 @@ The \aclp{SM} $ab, ac$ from \cref{running.example} result from the clause $b \v \draw[doubt, dash dot] (c) to[bend right] (Ac); \end{tikzpicture} \end{center} - + \caption{Events related to the \aclp{SM} of \cref{running.example}. The circle nodes are \aclp{TC} and shaded nodes are \aclp{SM}. The \emph{empty event}, with no literals, is denoted by $\emptyevent$. Notice that the event $bc$ is not related with any \acl{SM}.} \label{fig:running.example} \end{figure} @@ -293,99 +293,99 @@ The diagram in \cref{fig:running.example} illustrates the problem of extending p \node[event, above = of A] (Ac) {$\co{a}c$}; \node[event, above right = of Ac] (Abc) {$\co{a}bc$}; % ---- - \path[draw, rounded corners, pattern=north west lines, opacity=0.2] - (ab.west) -- - (ab.north west) -- - % - (abC.south west) -- - (abC.north west) -- - (abC.north) -- - % - (abc.north east) -- - (abc.east) -- - (abc.south east) -- - % - (ab.north east) -- - (ab.east) -- - (ab.south east) -- - % - (a.north east) -- - % - (E.north east) -- - (E.east) -- - (E.south east) -- - (E.south) -- - (E.south west) -- - % - (b.south west) -- - % - (ab.west) + \path[draw, rounded corners, pattern=north west lines, opacity=0.2] + (ab.west) -- + (ab.north west) -- + % + (abC.south west) -- + (abC.north west) -- + (abC.north) -- + % + (abc.north east) -- + (abc.east) -- + (abc.south east) -- + % + (ab.north east) -- + (ab.east) -- + (ab.south east) -- + % + (a.north east) -- + % + (E.north east) -- + (E.east) -- + (E.south east) -- + (E.south) -- + (E.south west) -- + % + (b.south west) -- + % + (ab.west) ; % ---- - \path[draw, rounded corners, pattern=north east lines, opacity=0.2] - (ac.south west) -- - (ac.west) -- - (ac.north west) -- - % - (abc.south west) -- - (abc.west) -- - (abc.north west) -- - % - (aBc.north east) -- - (aBc.east) -- - (aBc.south east) -- - % - (ac.north east) -- - % - (c.east) -- - % - (E.east) -- - (E.south east) -- - (E.south) -- - (E.south west) -- - % - (a.south west) -- - (a.west) -- - (a.north west) -- - (a.north) -- - % - (ac.south west) + \path[draw, rounded corners, pattern=north east lines, opacity=0.2] + (ac.south west) -- + (ac.west) -- + (ac.north west) -- + % + (abc.south west) -- + (abc.west) -- + (abc.north west) -- + % + (aBc.north east) -- + (aBc.east) -- + (aBc.south east) -- + % + (ac.north east) -- + % + (c.east) -- + % + (E.east) -- + (E.south east) -- + (E.south) -- + (E.south west) -- + % + (a.south west) -- + (a.west) -- + (a.north west) -- + (a.north) -- + % + (ac.south west) ; % ---- \path[draw, rounded corners, pattern=horizontal lines, opacity=0.2] - % (A.north west) -- - % - (Ac.north west) -- - % - (Abc.north west) -- - (Abc.north) -- - (Abc.north east) -- - (Abc.south east) -- - % - % (Ac.north east) -- - % (Ac.east) -- - % - % (A.east) -- - (A.south east) -- - % - (E.south east) -- - (E.south) -- - (E.south west) -- - (E.west) -- - (E.north west) -- - % - (Ac.north west) + % (A.north west) -- + % + (Ac.north west) -- + % + (Abc.north west) -- + (Abc.north) -- + (Abc.north east) -- + (Abc.south east) -- + % + % (Ac.north east) -- + % (Ac.east) -- + % + % (A.east) -- + (A.south east) -- + % + (E.south east) -- + (E.south) -- + (E.south west) -- + (E.west) -- + (E.north west) -- + % + (Ac.north west) ; \end{tikzpicture} \end{center} - + \caption{Classes (of consistent events) related to the \aclp{SM} of \cref{running.example} are defined through intersections and inclusions. \todo{write the caption}} \label{fig:running.example.classes} \end{figure} -Given an ASP specification, +Given an ASP specification, \remark{{\bruno Introduce also the sets mentioned below}}{how?} - 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} } $s \in \fml{S}\subset\fml{W}$. +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} } $s \in \fml{S}\subset\fml{W}$. 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}. @@ -400,11 +400,11 @@ This focus on the \acp{SM} leads to the following definition: \todo{expand this text to explain how the \aclp{SM} form the basis of the equivalence relation}. %This \replace{stance}{} leads to definition \ref{def:rel.events}: \begin{definition}\label{def:stable.core} - The \emph{\ac{SC}} of the event $e\in \fml{E}$ is + The \emph{\ac{SC}} of the event $e\in \fml{E}$ is \begin{equation} \stablecore{e} := \set{s \in \fml{S} \given s \subseteq e \vee e \subseteq s} \label{eq:stable.core} \end{equation} - + \end{definition} We now define an equivalence relation, $\sim$, so that two events are related if either both are inconsistent or both are consistent with the same \acl{SC}. @@ -420,102 +420,102 @@ Observe that the minimality of \aclp{SM} implies that, in \cref{def:stable.core \begin{equation} \class{e} = \begin{cases} - \inconsistent := \fml{E} \setminus \fml{W} - &\text{if~} e \in \fml{E} \setminus \fml{W}, \\ + \inconsistent := \fml{E} \setminus \fml{W} + & \text{if~} e \in \fml{E} \setminus \fml{W}, \\ \set{u \in \fml{W} \given \stablecore{u} = \stablecore{e}} - &\text{if~} e \in \fml{W}, + & \text{if~} e \in \fml{W}, \end{cases}\label{eq:event.class} \end{equation} 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: \begin{equation} \class{\fml{E}} = \set{ - \inconsistent, - \indepclass, - \class{\co{a}}, - \class{ab}, - \class{ac}, - \class{\co{a}, ab}, - \class{\co{a}, ac}, - \class{ab, ac}, - \class{\co{a}, ab, ac} + \inconsistent, + \indepclass, + \class{\co{a}}, + \class{ab}, + \class{ac}, + \class{\co{a}, ab}, + \class{\co{a}, ac}, + \class{ab, ac}, + \class{\co{a}, ab, ac} } \end{equation} where $\indepclass$ denotes both the class of \emph{independent} events $e$ such that $\stablecore{e} = \emptyset$ and its core (which is the emptyset). We have: \begin{equation*} \begin{array}{l|lr} \text{\textbf{Core}}, \stablecore{e} - & \text{\textbf{Class}}, \class{e} - & \text{\textbf{Size}}, \# \class{e}\\ - \hline + & \text{\textbf{Class}}, \class{e} + & \text{\textbf{Size}}, \# \class{e} \\ + \hline % \inconsistent - & a\co{a}, \ldots - & 37 + & a\co{a}, \ldots + & 37 \\ % - \indepclass - & \co{b}, \co{c}, bc, \co{b}a, \co{b}c, \co{b}\co{c}, \co{c}a, \co{c}b, \co{b}\co{c}a - & 9 + \indepclass + & \co{b}, \co{c}, bc, \co{b}a, \co{b}c, \co{b}\co{c}, \co{c}a, \co{c}b, \co{b}\co{c}a + & 9 \\ % - \co{a} - & \co{a}, \co{a}b, \co{a}c, \co{a}\co{b}, \co{a}\co{c}, \co{a}bc, \co{a}b\co{c}, \co{a}\co{b}c, \co{a}\co{b}\co{c} - & 9 + \co{a} + & \co{a}, \co{a}b, \co{a}c, \co{a}\co{b}, \co{a}\co{c}, \co{a}bc, \co{a}b\co{c}, \co{a}\co{b}c, \co{a}\co{b}\co{c} + & 9 \\ % ab - & b, ab, ab\co{c} - & 3 + & b, ab, ab\co{c} + & 3 \\ % ac - & c, ac, a\co{b}c - & 3 + & c, ac, a\co{b}c + & 3 \\ % \co{a}, ab - & \emptyset - & 0 + & \emptyset + & 0 \\ % \co{a}, ac - & \emptyset - & 0 + & \emptyset + & 0 % \\ % ab, ac - & a, abc - & 2 + & a, abc + & 2 \\ % \co{a}, ab, ac - & \emptyevent - & 1 + & \emptyevent + & 1 \\ % \hline \Omega - & \text{all events} - & 64 + & \text{all events} + & 64 \end{array} \end{equation*} \begin{itemize} \item Since all events within an equivalence class are in relation with a specific set of \aclp{SM}, \emph{weights, including probability, should be constant within classes}: - \[ - \forall u\in \class{e} \left(\mu\at{u} = \mu\at{e} \right). - \] + \[ + \forall u\in \class{e} \left(\mu\at{u} = \mu\at{e} \right). + \] \item So, instead of dealing with $64 = 2^6$ events, we consider the $9 = 2^3 + 1$ classes, well defined in terms of combinations of the \aclp{SM}. In general, we have \emph{much more} \aclp{SM} than literals. Nevertheless, the equivalence classes allow us to propagate probabilities from \aclp{TC} to events, as explained in the next subsection. - % \item The extended probability \emph{events} are the \emph{classes}. + % \item The extended probability \emph{events} are the \emph{classes}. \end{itemize} \subsection{From Total Choices to Events}\label{subsec:from.tchoices.to.events} -\todo{Check adaptation} Our path to set a probability measure on $\fml{E}$ has two phases: +\todo{Check adaptation} Our path to set a probability measure on $\fml{E}$ has two phases: \begin{enumerate} \item Extending the probabilities, \emph{as weights}, from the \aclp{TC} to events. \item Normalization of the weights. @@ -525,68 +525,68 @@ The ``extension'' phase, traced by equations (\ref{eq:prob.total.choice}) and (\ \begin{description} % - \item[Total Choices.] Using \eqref{eq:prob.total.choice}, this case is given by - \begin{equation} - \pw{t} := \pr{T = t}= \prod_{a\in t} p \prod_{a \not\in t} \co{p} - \label{eq:weight.tchoice} - \end{equation} - % - \item[Stable Models.] Each \acl{TC} $t$, together with the rules and the other facts of a specification, defines a set of \aclp{SM} associated with that choice, that \remark{we denote by $\tcgen{t}$}{put this in the introduction, where core concepts are presented}. - - Given a \acl{SM} $s \in \fml{S}$, a \acl{TC} $t$, and variables/values $\theta_{s,t} \in \intcc{0, 1}$, - \begin{equation} - \pw{s, t} := \begin{cases} - \theta_{s,t} & \text{if~} s \in \tcgen{t}\cr - 0&\text{otherwise} - \end{cases} - \label{eq:weight.stablemodel} - \end{equation} - such that $\sum_{s\in \tcgen{t}} \theta_{s,t} = 1$. - % - \item[Classes.] \label{item:class.cases} Each class is either the inconsistent class, $\inconsistent$, or is represented by some set of \aclp{SM}. - \begin{description} - \item[Inconsistent Class.] The inconsistent class contains events that are logically inconsistent, thus should never be observed: + \item[Total Choices.] Using \eqref{eq:prob.total.choice}, this case is given by \begin{equation} - \pw{\inconsistent, t} := 0. - \label{eq:weight.class.inconsistent} + \pw{t} := \pr{T = t}= \prod_{a\in t} p \prod_{a \not\in t} \co{p} + \label{eq:weight.tchoice} \end{equation} - \item[Independent Class.] A world that neither contains nor is contained in a \acl{SM} describes a case that, according to the specification, shouldn't exist. So the respective weight is set to zero: + % + \item[Stable Models.] Each \acl{TC} $t$, together with the rules and the other facts of a specification, defines a set of \aclp{SM} associated with that choice, that \remark{we denote by $\tcgen{t}$}{put this in the introduction, where core concepts are presented}. + + Given a \acl{SM} $s \in \fml{S}$, a \acl{TC} $t$, and variables/values $\theta_{s,t} \in \intcc{0, 1}$, \begin{equation} - \pw{\indepclass, t} := 0. - \label{eq:weight.class.independent} + \pw{s, t} := \begin{cases} + \theta_{s,t} & \text{if~} s \in \tcgen{t}\cr + 0 & \text{otherwise} + \end{cases} + \label{eq:weight.stablemodel} \end{equation} - \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): + such that $\sum_{s\in \tcgen{t}} \theta_{s,t} = 1$. + % + \item[Classes.] \label{item:class.cases} Each class is either the inconsistent class, $\inconsistent$, or is represented by some set of \aclp{SM}. + \begin{description} + \item[Inconsistent Class.] The inconsistent class contains events that are logically inconsistent, thus should never be observed: + \begin{equation} + \pw{\inconsistent, t} := 0. + \label{eq:weight.class.inconsistent} + \end{equation} + \item[Independent Class.] A world that neither contains nor is contained in a \acl{SM} describes a case that, according to the specification, shouldn't exist. So the respective weight is set to zero: + \begin{equation} + \pw{\indepclass, t} := 0. + \label{eq:weight.class.independent} + \end{equation} + \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): + \begin{equation} + \pw{\class{e}, t} := \sum_{k=1}^{n}\pw{s_k, t},~\text{if}~\stablecore{e} = \set{s_1, \ldots, s_n}. + \label{eq:weight.class.other} + \end{equation} + and + \begin{equation} + \pw{\class{e}} := \sum_{t \in \fml{T}} \pw{\class{e}, t}\pw{t}. + \label{eq:weight.class.unconditional} + \end{equation} + + \end{description} + % + \item[Events.] \label{item:event.cases} Each (general) event $e$ is in the class defined by its \acl{SC}, $\stablecore{e}$. So, we set: \begin{equation} - \pw{\class{e}, t} := \sum_{k=1}^{n}\pw{s_k, t},~\text{if}~\stablecore{e} = \set{s_1, \ldots, s_n}. - \label{eq:weight.class.other} + \pw{e, t} := \frac{\pw{\class{e}, t}}{\# \class{e}} . + \label{eq:weight.events} \end{equation} - and + and \begin{equation} - \pw{\class{e}} := \sum_{t \in \fml{T}} \pw{\class{e}, t}\pw{t}. - \label{eq:weight.class.unconditional} + \pw{e} := \sum_{t\in\fml{T}} \pw{e, t} \pw{t}. + \label{eq:weight.events.unconditional} \end{equation} - - \end{description} - % - \item[Events.] \label{item:event.cases} Each (general) event $e$ is in the class defined by its \acl{SC}, $\stablecore{e}$. So, we set: - \begin{equation} - \pw{e, t} := \frac{\pw{\class{e}, t}}{\# \class{e}} . - \label{eq:weight.events} - \end{equation} - and - \begin{equation} - \pw{e} := \sum_{t\in\fml{T}} \pw{e, t} \pw{t}. - \label{eq:weight.events.unconditional} - \end{equation} - % \remark{instead of that equation}{if we set $\pw{s,t} := \theta_{s,t}$ in equation \eqref{eq:weight.stablemodel} here we do: - % $$ - % \pw{e} := \sum_{t\in\fml{T}} \pw{e, t}\pw{t}. - % $$ - % By the way, this is the \emph{marginalization + bayes theorem} in statistics: - % $$ - % P(A) = \sum_b P(A | B=b)P(B=b) - % $$ - % } + % \remark{instead of that equation}{if we set $\pw{s,t} := \theta_{s,t}$ in equation \eqref{eq:weight.stablemodel} here we do: + % $$ + % \pw{e} := \sum_{t\in\fml{T}} \pw{e, t}\pw{t}. + % $$ + % By the way, this is the \emph{marginalization + bayes theorem} in statistics: + % $$ + % P(A) = \sum_b P(A | B=b)P(B=b) + % $$ + % } \end{description} % PARAMETERS FOR UNCERTAINTY @@ -600,180 +600,180 @@ The ``extension'' phase, traced by equations (\ref{eq:prob.total.choice}) and (\ The $\theta_{s,t}$ parameters in equation \eqref{eq:weight.stablemodel} express the \emph{specification's} lack of knowledge about the weight assignment, when a single \acl{TC} entails more than one \acl{SM}. In that case, how to distribute the respective weights? Our proposal to address this problem consists in assigning an unknown weight, $\theta_{s,t}$, conditional on the \acl{TC}, $t$, to each \acl{SM} $s$. This approach allows the expression of an unknown quantity and future estimation, given observed data. % SUPERSET -Equation \eqref{eq:weight.class.other} results from conditional independence of \aclp{SM}. +Equation \eqref{eq:weight.class.other} results from conditional independence of \aclp{SM}. \section{Developed Examples} \subsection{The SBF Example} -We continue with the specification from Equation \eqref{eq:example.1}. +We continue with the specification from Equation \eqref{eq:example.1}. \begin{description} % \item[\Aclp{TC}.] The \aclp{TC}, and respective \aclp{SM}, are - \begin{center} - \begin{tabular}{ll|r} - \textbf{\Acl{TC}} & \textbf{\Aclp{SM}} & \textbf{$\pw{t}$}\\ - \hline - $a$ & $ab, ac$ & $0.3$\\ - $\co{a} = \neg a$ & $\co{a}$ & $\co{0.3} = 0.7$ - \end{tabular} - \end{center} - % + \begin{center} + \begin{tabular}{ll|r} + \textbf{\Acl{TC}} & \textbf{\Aclp{SM}} & \textbf{$\pw{t}$} \\ + \hline + $a$ & $ab, ac$ & $0.3$ \\ + $\co{a} = \neg a$ & $\co{a}$ & $\co{0.3} = 0.7$ + \end{tabular} + \end{center} + % \item[\Aclp{SM}.] The $\theta_{s,t}$ parameters in this example are - $$ - \theta_{ab,\co{a}} = \theta_{ac,\co{a}} = \theta_{\co{a}, a} = 0 - % - \text{~and~} - % - \theta_{\co{a}, \co{a}} = 1, \theta_{ab, a} = \theta, \theta_{ac, a} = \co{\theta} - $$ - with $\theta \in \intcc{0, 1}$. - \item[Classes.] Following the definitions in \cref{eq:stable.core,eq:equiv.rel,eq:event.class} and in \cref{eq:weight.class.inconsistent,eq:weight.class.independent,eq:weight.class.other} we get the following quotient set (ignoring $\inconsistent$ and $\indepclass$), and weights: - \begin{equation*} - \begin{array}{l|ll|r} - \stablecore{e} - & \pw{s_k, t= \co{a}} - & \pw{s_k, t= a} - & \pw{\class{e}}=\sum_{t}\pw{\class{e},t}\pw{t} - \\ - \hline - \co{a} - & 1 - & - & 0.7 - \\ - % - ab - & - & \theta - & 0.3\theta - \\ - % - ac - & - & \co{\theta} - & 0.3\co{\theta} - \\ - % - \co{a}, ab - & 1, 0 - & 0, \theta - & 0.7 + 0.3\theta - \\ - % - \co{a}, ac - & 1, 0 - & 0, \co{\theta} - & 0.7 + 0.3\co{\theta} - \\ + $$ + \theta_{ab,\co{a}} = \theta_{ac,\co{a}} = \theta_{\co{a}, a} = 0 % - ab, ac - & - & \theta, \co{\theta} - & 0.3 - \\ + \text{~and~} % - \co{a}, ab, ac - & 1, 0, 0 - & 0, \theta, \co{\theta} - & 1 - \end{array} - \end{equation*} + \theta_{\co{a}, \co{a}} = 1, \theta_{ab, a} = \theta, \theta_{ac, a} = \co{\theta} + $$ + with $\theta \in \intcc{0, 1}$. + \item[Classes.] Following the definitions in \cref{eq:stable.core,eq:equiv.rel,eq:event.class} and in \cref{eq:weight.class.inconsistent,eq:weight.class.independent,eq:weight.class.other} we get the following quotient set (ignoring $\inconsistent$ and $\indepclass$), and weights: + \begin{equation*} + \begin{array}{l|ll|r} + \stablecore{e} + & \pw{s_k, t= \co{a}} + & \pw{s_k, t= a} + & \pw{\class{e}}=\sum_{t}\pw{\class{e},t}\pw{t} + \\ + \hline + \co{a} + & 1 + & + & 0.7 + \\ + % + ab + & + & \theta + & 0.3\theta + \\ + % + ac + & + & \co{\theta} + & 0.3\co{\theta} + \\ + % + \co{a}, ab + & 1, 0 + & 0, \theta + & 0.7 + 0.3\theta + \\ + % + \co{a}, ac + & 1, 0 + & 0, \co{\theta} + & 0.7 + 0.3\co{\theta} + \\ + % + ab, ac + & + & \theta, \co{\theta} + & 0.3 + \\ + % + \co{a}, ab, ac + & 1, 0, 0 + & 0, \theta, \co{\theta} + & 1 + \end{array} + \end{equation*} \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.} - \begin{equation*} - Z := \sum_{e\in\fml{E}} \pw{e} - = \sum_{\class{e} \in\class{\fml{E}}} \frac{\pw{\class{e}}}{\#\class{e}}, - \end{equation*} - that divides the weight function into a normalized weight - \begin{equation*} - \pr{e} := \frac{\pw{e}}{Z}. - \end{equation*} - such that - $$ - \sum_{e \in \fml{E}} \pr{e} = 1. - $$ - For the SBF example, - \begin{equation*} - \begin{array}{lr|r|rr} - \stablecore{e} - & \# \class{e} - & \pw{\class{e}} - & \pw{e} - & \pr{e} - \\ - \hline - % - \inconsistent - & 37 - & 0 - & 0 - & 0 - \\[4pt] - % - \indepclass - & 9 - & 0 - & 0 - & 0 - \\[4pt] - % - \co{a} - & 9 - & \frac{7}{10} - & \frac{7}{90} - & \frac{7}{792} - \\[4pt] - % - ab - & 3 - & \frac{3\theta}{10} - & \frac{\theta}{10} - & \frac{\theta}{88} - \\[4pt] - % - ac - & 3 - & \frac{3\co{\theta}}{10} - & \frac{\co{\theta}}{10} - & \frac{\co{\theta}}{88} - \\[4pt] - % - \co{a}, ab - & 0 - & \frac{7 + 3\theta}{10} - & 0 - & 0 - \\[4pt] - % - \co{a}, ac - & 0 - & \frac{7 + 3\co{\theta}}{10} - & 0 - & 0 - % - \\[4pt] - % - ab, ac - & 2 - & \frac{3}{10} - & \frac{3}{20} - & \frac{3}{176} - \\[4pt] - % - \co{a}, ab, ac - & 1 - & 1 - & 1 - & \frac{5}{176} - \\[4pt] - % - \hline - & - & Z = \frac{44}{5} - \end{array} - \end{equation*} + \begin{equation*} + Z := \sum_{e\in\fml{E}} \pw{e} + = \sum_{\class{e} \in\class{\fml{E}}} \frac{\pw{\class{e}}}{\#\class{e}}, + \end{equation*} + that divides the weight function into a normalized weight + \begin{equation*} + \pr{e} := \frac{\pw{e}}{Z}. + \end{equation*} + such that + $$ + \sum_{e \in \fml{E}} \pr{e} = 1. + $$ + For the SBF example, + \begin{equation*} + \begin{array}{lr|r|rr} + \stablecore{e} + & \# \class{e} + & \pw{\class{e}} + & \pw{e} + & \pr{e} + \\ + \hline + % + \inconsistent + & 37 + & 0 + & 0 + & 0 + \\[4pt] + % + \indepclass + & 9 + & 0 + & 0 + & 0 + \\[4pt] + % + \co{a} + & 9 + & \frac{7}{10} + & \frac{7}{90} + & \frac{7}{792} + \\[4pt] + % + ab + & 3 + & \frac{3\theta}{10} + & \frac{\theta}{10} + & \frac{\theta}{88} + \\[4pt] + % + ac + & 3 + & \frac{3\co{\theta}}{10} + & \frac{\co{\theta}}{10} + & \frac{\co{\theta}}{88} + \\[4pt] + % + \co{a}, ab + & 0 + & \frac{7 + 3\theta}{10} + & 0 + & 0 + \\[4pt] + % + \co{a}, ac + & 0 + & \frac{7 + 3\co{\theta}}{10} + & 0 + & 0 + % + \\[4pt] + % + ab, ac + & 2 + & \frac{3}{10} + & \frac{3}{20} + & \frac{3}{176} + \\[4pt] + % + \co{a}, ab, ac + & 1 + & 1 + & 1 + & \frac{5}{176} + \\[4pt] + % + \hline + & + & Z = \frac{44}{5} + \end{array} + \end{equation*} \end{description} \todo{Continue this example with a set of observations to estimate $\theta$ and try to show some more. For example, that the resulting distribution is not very good when $t = \co{a}$. Also gather a sample following the specification.} @@ -782,15 +782,16 @@ We continue with the specification from Equation \eqref{eq:example.1}. % \subsection{An example involving Bayesian networks} -\franc{Cometários:} +\franc{Comentários:} \begin{itemize} \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, $\backslash\text{given}$: $\pr{A \given B}$. \item E, claro, para factos+probabilidades: $\probfact{p}{a}$. - \item A designação dos `pesos' não está consistente: $pj\_a$ e $a\_be$. Fiz uma macro (hehe) para sistematizar isto: \condsymb{a}{bnc} + \item A designação dos `pesos' não está consistente: $pj\_a$ e $a\_be$. Fiz uma macro (\emph{hehe}) para sistematizar isto: \condsymb{a}{bnc}. + \item Nos programas, alinhei pelos factos. Isto é, $\probfact{0.3}{a}$ e $a \leftarrow b$ alinham pelo (fim do) $a$. \end{itemize} -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. +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. 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. @@ -799,14 +800,14 @@ The events are: Burglary ($B$), Earthquake ($E$), Alarm ($A$), Mary calls ($M$) \begin{figure} \begin{center} \begin{tikzpicture}[node distance=2.5cm] - + % Nodes \node[smodel, circle] (A) {A}; \node[tchoice, above right of=A] (B) {B}; \node[tchoice, above left of=A] (E) {E}; \node[tchoice, below left of=A] (M) {M}; \node[tchoice, below right of=A] (J) {J}; - + % Edges \draw[->] (B) to[bend left] (A) node[right,xshift=1.1cm,yshift=0.8cm] {\footnotesize{$P(B)=0.001$}} ; \draw[->] (E) to[bend right] (A) node[left, xshift=-1.4cm,yshift=0.8cm] {\footnotesize{$P(E)=0.002$}} ; @@ -814,50 +815,50 @@ The events are: Burglary ($B$), Earthquake ($E$), Alarm ($A$), Mary calls ($M$) \draw[->] (A) to[bend left] (J) node[right,xshift=-0.2cm,yshift=0.7cm] {\footnotesize{$P(J|A)$}} ; \end{tikzpicture} \end{center} - + \begin{multicols}{3} - + \footnotesize{ - \begin{equation*} - \begin{split} - &P(M|A)\\ - & \begin{array}{c|cc} - & m & \neg m \\ - \hline - a & 0.9 & 0.1\\ - \neg a& 0.05 & 0.95 - \end{array} - \end{split} - \end{equation*} + \begin{equation*} + \begin{split} + &P(M|A)\\ + & \begin{array}{c|cc} + & m & \neg m \\ + \hline + a & 0.9 & 0.1 \\ + \neg a & 0.05 & 0.95 + \end{array} + \end{split} + \end{equation*} } - + \footnotesize{ - \begin{equation*} - \begin{split} - &P(J|A)\\ - & \begin{array}{c|cc} - & j & \neg j \\ - \hline - a & 0.7 & 0.3\\ - \neg a& 0.01 & 0.99 - \end{array} - \end{split} - \end{equation*} + \begin{equation*} + \begin{split} + &P(J|A)\\ + & \begin{array}{c|cc} + & j & \neg j \\ + \hline + a & 0.7 & 0.3 \\ + \neg a & 0.01 & 0.99 + \end{array} + \end{split} + \end{equation*} } \footnotesize{ - \begin{equation*} - \begin{split} - P(A|B \wedge E)\\ - \begin{array}{c|c|cc} - & & a & \neg a \\ - \hline - b & e & 0.95 & 0.05\\ - b & \neg e & 0.94 & 0.06\\ - \neg b & e & 0.29 & 0.71\\ - \neg b & \neg e & 0.001 & 0.999 - \end{array} - \end{split} - \end{equation*} + \begin{equation*} + \begin{split} + P(A|B \wedge E)\\ + \begin{array}{c|c|cc} + & & a & \neg a \\ + \hline + b & e & 0.95 & 0.05 \\ + b & \neg e & 0.94 & 0.06 \\ + \neg b & e & 0.29 & 0.71 \\ + \neg b & \neg e & 0.001 & 0.999 + \end{array} + \end{split} + \end{equation*} } \end{multicols} \caption{The Earthquake, Burglary, Alarm model} @@ -869,9 +870,9 @@ Considering the probabilities given in \cref{Figure_Alarm} we obtain the followi \begin{equation*} \begin{aligned} - \probfact{0.001}{b}&,\cr - \probfact{0.002}{e}&,\cr - \end{aligned} + \probfact{0.001}{b} & ,\cr + \probfact{0.002}{e} & ,\cr + \end{aligned} \label{eq:not_so_simple_example} \end{equation*} @@ -880,11 +881,11 @@ For the table giving the probability $P(M|A)$ we obtain the specification: \begin{equation*} \begin{aligned} - \probfact{0.9}{pm\_a}&,\cr - \probfact{0.05}{pm\_na}&,\cr - m & \leftarrow a, pm\_a,\cr - \neg m & \leftarrow a, \neg pm\_a. - \end{aligned} + \probfact{0.9}{pm\_a} & ,\cr + \probfact{0.05}{pm\_na} & ,\cr + m & \leftarrow a, pm\_a,\cr + \neg m & \leftarrow a, \neg pm\_a. + \end{aligned} \end{equation*} This latter specification can be simplified by writing $\probfact{0.9}{m \leftarrow a}$ and $\probfact{0.05}{m \leftarrow \neg a}$. @@ -893,11 +894,11 @@ Similarly, for the probability $P(J|A)$ we obtain \begin{equation*} \begin{aligned} - &\probfact{0.7}{pj\_a},\cr - &\probfact{0.01}{pj\_na},\cr - j & \leftarrow a, pj\_a,\cr - \neg j & \leftarrow a, \neg pj\_a.\cr - \end{aligned} + \probfact{0.7}{pj\_a} & ,\cr + \probfact{0.01}{pj\_na} & ,\cr + j & \leftarrow a, pj\_a,\cr + \neg j & \leftarrow a, \neg pj\_a.\cr + \end{aligned} \end{equation*} Again, this can be simplified by writing $\probfact{0.7}{j \leftarrow a}$ and $\probfact{0.01}{j \leftarrow \neg a}$. @@ -906,22 +907,22 @@ Finally, for the probability $P(A|B \wedge E)$ we obtain \begin{equation*} \begin{aligned} - &\probfact{0.95}{a\_be},\cr - &\probfact{0.94}{a\_bne},\cr - &\probfact{0.29}{a\_nbe},\cr - &\probfact{0.001}{a\_nbne},\cr - a & \leftarrow b, e, a\_be,\cr - \neg a & \leftarrow b,e, \neg a\_be, \cr - a & \leftarrow b,e, a\_bne,\cr - \neg a & \leftarrow b,e, \neg a\_bne, \cr - a & \leftarrow b,e, a\_nbe,\cr - \neg a & \leftarrow b,e, \neg a\_nbe, \cr - a & \leftarrow b,e, a\_nbne,\cr - \neg a & \leftarrow b,e, \neg a\_nbne. \cr - \end{aligned} + \probfact{0.95}{a\_be} & ,\cr + \probfact{0.94}{a\_bne} & ,\cr + \probfact{0.29}{a\_nbe} & ,\cr + \probfact{0.001}{a\_nbne} & ,\cr + a & \leftarrow b, e, a\_be,\cr + \neg a & \leftarrow b,e, \neg a\_be, \cr + a & \leftarrow b,e, a\_bne,\cr + \neg a & \leftarrow b,e, \neg a\_bne, \cr + a & \leftarrow b,e, a\_nbe,\cr + \neg a & \leftarrow b,e, \neg a\_nbe, \cr + a & \leftarrow b,e, a\_nbne,\cr + \neg a & \leftarrow b,e, \neg a\_nbne. \cr + \end{aligned} \end{equation*} -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. +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. \section{Discussion} @@ -934,119 +935,119 @@ One can then proceed as in the previous subsection and analyse this example. The % % My first guess was % \begin{equation*} - % \pr{W = w \given C = c} = \sum_{s \supset w}\pr{S = s \given C = c}. - % \end{equation*} - % - % $\pr{W = w \given C = c}$ already separates $\pr{W}$ into \textbf{disjoint} events! - % - % Also, I am assuming that \aclp{SM} are independent. - % - % This would entail $p(w) = p(s_1) + p(s_2) - p(s_1)p(s_2)$ \emph{if I'm bound to set inclusion}. But I'm not. I'm defining a relation - % - % Also, if I set $p(w) = p(s_1) + p(s_2)$ and respect the laws of probability, this entails $p(s_1)p(s_2) = 0$. - % - % So, maybe what I want is (1) to define the cover $\hat{w} = \cup_{s \supset w} s$ - % - % \begin{equation*} - % \pr{W = w \given C = c} = \sum_{s \supset w}\pr{S = s \given C = c} - \pr{W = \hat{w} \given C = c}. - % \end{equation*} - % - % But this doesn't works, because we'd get $\pr{W = a \given C = a} < 1$. - % % - % - % % - % \bigskip - % \hrule - % - % INDEPENDENCE - % - %, per equation (\ref{eq:weight.class.independent}). - % - % ================================================================ - % - \begin{itemize} - \item Changed from $\prod$ to $\sum$ to represent ``either'' instead of ``both'' since the later is not consistent with the ``only one stable model at a time'' assumption. - \item \todo{The `up and down' choice in the equivalence relation and the possibility of describing any probability distribution.} - \item \todo{Remark that no benchmark was done with other SOTA efforts.} - \item \todo{The possibility to `import' bayesian theory and tools to this study.} - \end{itemize} - - - \subsection{Dependence} - \label{subsec:dependence} - - Our basic assertion about dependence relations between atoms of the underlying system is that they can be \emph{explicitly expressed in the specification}. And, in that case, they should be. - - For example, a dependence relation between $b$ and $c$ can be expressed by $b \leftarrow c \wedge d$, where $d$ is an atomic choice that explicitly expresses the dependence between $b$ and $c$. One would get, for example, a specification such as - $$ - \probfact{0.3}{a}, b \vee c \leftarrow a, \probfact{0.2}{d}, b \leftarrow c \wedge d. - $$ - with \aclp{SM} - $ - \co{ad}, \co{a}d, a\co{d}b, a\co{d}c, adb - $. - - - The interesting case is the subtree of the \acl{TC} $ad$. Notice that no \acl{SM} $s$ contains $adc$ because $(i)$ $adb$ is a \acl{SM} and $(ii)$ if $adc \subset s$ then $b \in s$ so $adb \subset s$. - - Following equations \eqref{eq:world.fold.stablemodel} and \eqref{eq:world.fold.independent} {\bruno What are these equations?} this entails - \begin{equation*} - \begin{cases} - \pr{W = adc \given C = ad} = 0,\cr - \pr{W = adb \given C = ad} = 1 - \end{cases} - \end{equation*} - which concentrates all probability mass from the \acl{TC} $ad$ in the $adb$ branch, including the node $W = adbc$. This leads to the following cases: - $$ - \begin{array}{l|c} - x & \pr{W = x \given C = ad}\\ - \hline - ad & 1 \\ - adb & 1\\ - adc & 0\\ - adbc & 1 - \end{array} - $$ - so, for $C = ad$, - $$ - \begin{aligned} - \pr{W = b} &= \frac{2}{4} \cr - \pr{W = c} &= \frac{1}{4} \cr - \pr{W = bc} &= \frac{1}{4} \cr - &\not= \pr{W = b}\pr{W = c} - \end{aligned} - $$ - \emph{i.e.} the events $W = b$ and $W = c$ are dependent and that dependence results directly from the segment $\probfact{0.2}{d}, b \leftarrow c \wedge d$ in the specification. - - {\bruno Why does this not contradict Assumption 1?} - - % - - % - \hrule - \begin{quotation}\note{Todo} - - Prove the four world cases (done), support the product (done) and sum (tbd) options, with the independence assumptions. - \end{quotation} - - \subsection{Future Work} - - \todo{develop this section.} - - \begin{itemize} - \item The measure of the inconsistent events doesn't need to be set to $0$ and, maybe, in some cases, it shouldn't. - \item The physical system might have \emph{latent} variables, possibly also represented in the specification. These variables are never observed, so observations should be concentrated \emph{somewhere else}. - \item Comment on the possibility of extending equation \eqref{eq:weight.events.unconditional} with parameters expressing further uncertainties, enabling a tuning of the model's \aclp{TC}, given observations. - \begin{equation*} - \pw{e} := \sum_{c\in\fml{T}} \pw{e, c}\theta_c. - \end{equation*} - \end{itemize} - - - \section*{Acknowledgements} - - This work is supported by NOVA\textbf{LINCS} (UIDB/04516/2020) with the financial support of FCT.IP. - - \printbibliography - - \end{document} \ No newline at end of file +% \pr{W = w \given C = c} = \sum_{s \supset w}\pr{S = s \given C = c}. +% \end{equation*} +% +% $\pr{W = w \given C = c}$ already separates $\pr{W}$ into \textbf{disjoint} events! +% +% Also, I am assuming that \aclp{SM} are independent. +% +% This would entail $p(w) = p(s_1) + p(s_2) - p(s_1)p(s_2)$ \emph{if I'm bound to set inclusion}. But I'm not. I'm defining a relation +% +% Also, if I set $p(w) = p(s_1) + p(s_2)$ and respect the laws of probability, this entails $p(s_1)p(s_2) = 0$. +% +% So, maybe what I want is (1) to define the cover $\hat{w} = \cup_{s \supset w} s$ +% +% \begin{equation*} +% \pr{W = w \given C = c} = \sum_{s \supset w}\pr{S = s \given C = c} - \pr{W = \hat{w} \given C = c}. +% \end{equation*} +% +% But this doesn't works, because we'd get $\pr{W = a \given C = a} < 1$. +% % +% +% % +% \bigskip +% \hrule +% +% INDEPENDENCE +% +%, per equation (\ref{eq:weight.class.independent}). +% +% ================================================================ +% +\begin{itemize} + \item Changed from $\prod$ to $\sum$ to represent ``either'' instead of ``both'' since the later is not consistent with the ``only one stable model at a time'' assumption. + \item \todo{The `up and down' choice in the equivalence relation and the possibility of describing any probability distribution.} + \item \todo{Remark that no benchmark was done with other SOTA efforts.} + \item \todo{The possibility to `import' bayesian theory and tools to this study.} +\end{itemize} + + +\subsection{Dependence} +\label{subsec:dependence} + +Our basic assertion about dependence relations between atoms of the underlying system is that they can be \emph{explicitly expressed in the specification}. And, in that case, they should be. + +For example, a dependence relation between $b$ and $c$ can be expressed by $b \leftarrow c \wedge d$, where $d$ is an atomic choice that explicitly expresses the dependence between $b$ and $c$. One would get, for example, a specification such as +$$ + \probfact{0.3}{a}, b \vee c \leftarrow a, \probfact{0.2}{d}, b \leftarrow c \wedge d. +$$ +with \aclp{SM} +$ + \co{ad}, \co{a}d, a\co{d}b, a\co{d}c, adb +$. + + +The interesting case is the subtree of the \acl{TC} $ad$. Notice that no \acl{SM} $s$ contains $adc$ because $(i)$ $adb$ is a \acl{SM} and $(ii)$ if $adc \subset s$ then $b \in s$ so $adb \subset s$. + +Following equations \eqref{eq:world.fold.stablemodel} and \eqref{eq:world.fold.independent} {\bruno What are these equations?} this entails +\begin{equation*} + \begin{cases} + \pr{W = adc \given C = ad} = 0,\cr + \pr{W = adb \given C = ad} = 1 + \end{cases} +\end{equation*} +which concentrates all probability mass from the \acl{TC} $ad$ in the $adb$ branch, including the node $W = adbc$. This leads to the following cases: +$$ + \begin{array}{l|c} + x & \pr{W = x \given C = ad} \\ + \hline + ad & 1 \\ + adb & 1 \\ + adc & 0 \\ + adbc & 1 + \end{array} +$$ +so, for $C = ad$, +$$ + \begin{aligned} + \pr{W = b} & = \frac{2}{4} \cr + \pr{W = c} & = \frac{1}{4} \cr + \pr{W = bc} & = \frac{1}{4} \cr + & \not= \pr{W = b}\pr{W = c} + \end{aligned} +$$ +\emph{i.e.} the events $W = b$ and $W = c$ are dependent and that dependence results directly from the segment $\probfact{0.2}{d}, b \leftarrow c \wedge d$ in the specification. + + {\bruno Why does this not contradict Assumption 1?} + +% + +% +\hrule +\begin{quotation}\note{Todo} + + Prove the four world cases (done), support the product (done) and sum (tbd) options, with the independence assumptions. +\end{quotation} + +\subsection{Future Work} + +\todo{develop this section.} + +\begin{itemize} + \item The measure of the inconsistent events doesn't need to be set to $0$ and, maybe, in some cases, it shouldn't. + \item The physical system might have \emph{latent} variables, possibly also represented in the specification. These variables are never observed, so observations should be concentrated \emph{somewhere else}. + \item Comment on the possibility of extending equation \eqref{eq:weight.events.unconditional} with parameters expressing further uncertainties, enabling a tuning of the model's \aclp{TC}, given observations. + \begin{equation*} + \pw{e} := \sum_{c\in\fml{T}} \pw{e, c}\theta_c. + \end{equation*} +\end{itemize} + + +\section*{Acknowledgements} + +This work is supported by NOVA\textbf{LINCS} (UIDB/04516/2020) with the financial support of FCT.IP. + +\printbibliography + +\end{document} \ No newline at end of file -- libgit2 0.21.2