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comentários e macros em nno exemplo bayesiano
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text/paper_01/pre-paper.tex
| ... | ... | @@ -121,7 +121,7 @@ citecolor=blue, |
| 121 | 121 | |
| 122 | 122 | \begin{abstract} |
| 123 | 123 | \todo{rewrite} |
| 124 | - 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. | |
| 124 | + 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. | |
| 125 | 125 | \end{abstract} |
| 126 | 126 | |
| 127 | 127 | \section{Introduction and Motivation} |
| ... | ... | @@ -143,10 +143,10 @@ Our goal is to extend this probability, from \acp{TC}, to cover the \emph{specif |
| 143 | 143 | |
| 144 | 144 | \begin{enumerate} |
| 145 | 145 | \item Support probabilistic reasoning/tasks on the specification domain. |
| 146 | - \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. | |
| 146 | + \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 | 147 | \end{enumerate} |
| 148 | 148 | |
| 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. | |
| 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 | 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 | 152 | |
| ... | ... | @@ -165,48 +165,48 @@ In a related work, \cite{verreet2022inference}, epistemic uncertainty (or model |
| 165 | 165 | |
| 166 | 166 | \begin{example}\label{running.example} |
| 167 | 167 | Consider the following specification |
| 168 | - | |
| 168 | + | |
| 169 | 169 | \begin{equation} |
| 170 | 170 | \begin{aligned} |
| 171 | - \probfact{0.3}{a}&,\cr | |
| 172 | - b \vee c& \leftarrow a. | |
| 171 | + \probfact{0.3}{a} & ,\cr | |
| 172 | + b \vee c & \leftarrow a. | |
| 173 | 173 | \end{aligned} |
| 174 | 174 | \label{eq:example.1} |
| 175 | 175 | \end{equation} |
| 176 | - | |
| 176 | + | |
| 177 | 177 | 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 |
| 178 | 178 | |
| 179 | 179 | $$ |
| 180 | - \begin{aligned} | |
| 181 | - P(ab) &= 0.3 \theta,\cr | |
| 182 | - P(ac) &= 0.3 (1 - \theta). | |
| 183 | - \end{aligned} | |
| 180 | + \begin{aligned} | |
| 181 | + P(ab) & = 0.3 \theta,\cr | |
| 182 | + P(ac) & = 0.3 (1 - \theta). | |
| 183 | + \end{aligned} | |
| 184 | 184 | $$ |
| 185 | -\end{example} | |
| 185 | +\end{example} | |
| 186 | 186 | |
| 187 | 187 | 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.} |
| 188 | 188 | |
| 189 | 189 | In summary, if an \ac{ASP} specification is intended to describe some observable system then: |
| 190 | 190 | |
| 191 | 191 | \begin{enumerate} |
| 192 | - | |
| 192 | + | |
| 193 | 193 | \item Observations can be used to estimate the value of the parameters (such as $\theta$ above and others entailed from further clauses). |
| 194 | - | |
| 194 | + | |
| 195 | 195 | \item \todo{What about the case where we already know a distribution of $\theta$?} |
| 196 | - | |
| 197 | - \item With a probability set for the \aclp{SM}, we want to extend it to all the events of the specification domain. | |
| 198 | - | |
| 196 | + | |
| 197 | + \item With a probability set for the \aclp{SM}, we want to extend it to all the events of the specification domain. | |
| 198 | + | |
| 199 | 199 | \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. |
| 200 | - | |
| 200 | + | |
| 201 | 201 | \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. |
| 202 | - | |
| 202 | + | |
| 203 | 203 | \end{enumerate} |
| 204 | 204 | |
| 205 | 205 | \begin{quote} |
| 206 | - \todo{Expand this:} If observations are not consistent with the models of the specification, then the specification is wrong and must be changed. | |
| 206 | + \todo{Expand this:} If observations are not consistent with the models of the specification, then the specification is wrong and must be changed. | |
| 207 | 207 | \end{quote} |
| 208 | 208 | |
| 209 | -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. | |
| 209 | +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. | |
| 210 | 210 | |
| 211 | 211 | The conditional independence of stable worlds asserts the \remark{least informed strategy}{references?} that we discussed in the introduction and make explicit here: |
| 212 | 212 | |
| ... | ... | @@ -237,7 +237,7 @@ The \aclp{SM} $ab, ac$ from \cref{running.example} result from the clause $b \v |
| 237 | 237 | % ---- |
| 238 | 238 | \draw[doubt] (a) to[bend left] (ab); |
| 239 | 239 | \draw[doubt] (a) to[bend right] (ac); |
| 240 | - | |
| 240 | + | |
| 241 | 241 | \draw[doubt] (ab) to[bend left] (abc); |
| 242 | 242 | \draw[doubt] (ab) to[bend right] (abC); |
| 243 | 243 | |
| ... | ... | @@ -245,14 +245,14 @@ The \aclp{SM} $ab, ac$ from \cref{running.example} result from the clause $b \v |
| 245 | 245 | \draw[doubt] (ac) to[bend left] (aBc); |
| 246 | 246 | |
| 247 | 247 | \draw[doubt, dash dot] (Ac) to (Abc); |
| 248 | - | |
| 248 | + | |
| 249 | 249 | \draw[doubt] (A) to (Ac); |
| 250 | 250 | \draw[doubt] (A) to (Abc); |
| 251 | - | |
| 251 | + | |
| 252 | 252 | \draw[doubt] (ab) to[bend right] (E); |
| 253 | 253 | \draw[doubt] (ac) to[bend right] (E); |
| 254 | 254 | \draw[doubt] (A) to[bend left] (E); |
| 255 | - | |
| 255 | + | |
| 256 | 256 | \draw[doubt] (ab) to (b); |
| 257 | 257 | \draw[doubt] (ac) to (c); |
| 258 | 258 | % \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 |
| 263 | 263 | \draw[doubt, dash dot] (c) to[bend right] (Ac); |
| 264 | 264 | \end{tikzpicture} |
| 265 | 265 | \end{center} |
| 266 | - | |
| 266 | + | |
| 267 | 267 | \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}.} |
| 268 | 268 | \label{fig:running.example} |
| 269 | 269 | \end{figure} |
| ... | ... | @@ -293,99 +293,99 @@ The diagram in \cref{fig:running.example} illustrates the problem of extending p |
| 293 | 293 | \node[event, above = of A] (Ac) {$\co{a}c$}; |
| 294 | 294 | \node[event, above right = of Ac] (Abc) {$\co{a}bc$}; |
| 295 | 295 | % ---- |
| 296 | - \path[draw, rounded corners, pattern=north west lines, opacity=0.2] | |
| 297 | - (ab.west) -- | |
| 298 | - (ab.north west) -- | |
| 299 | - % | |
| 300 | - (abC.south west) -- | |
| 301 | - (abC.north west) -- | |
| 302 | - (abC.north) -- | |
| 303 | - % | |
| 304 | - (abc.north east) -- | |
| 305 | - (abc.east) -- | |
| 306 | - (abc.south east) -- | |
| 307 | - % | |
| 308 | - (ab.north east) -- | |
| 309 | - (ab.east) -- | |
| 310 | - (ab.south east) -- | |
| 311 | - % | |
| 312 | - (a.north east) -- | |
| 313 | - % | |
| 314 | - (E.north east) -- | |
| 315 | - (E.east) -- | |
| 316 | - (E.south east) -- | |
| 317 | - (E.south) -- | |
| 318 | - (E.south west) -- | |
| 319 | - % | |
| 320 | - (b.south west) -- | |
| 321 | - % | |
| 322 | - (ab.west) | |
| 296 | + \path[draw, rounded corners, pattern=north west lines, opacity=0.2] | |
| 297 | + (ab.west) -- | |
| 298 | + (ab.north west) -- | |
| 299 | + % | |
| 300 | + (abC.south west) -- | |
| 301 | + (abC.north west) -- | |
| 302 | + (abC.north) -- | |
| 303 | + % | |
| 304 | + (abc.north east) -- | |
| 305 | + (abc.east) -- | |
| 306 | + (abc.south east) -- | |
| 307 | + % | |
| 308 | + (ab.north east) -- | |
| 309 | + (ab.east) -- | |
| 310 | + (ab.south east) -- | |
| 311 | + % | |
| 312 | + (a.north east) -- | |
| 313 | + % | |
| 314 | + (E.north east) -- | |
| 315 | + (E.east) -- | |
| 316 | + (E.south east) -- | |
| 317 | + (E.south) -- | |
| 318 | + (E.south west) -- | |
| 319 | + % | |
| 320 | + (b.south west) -- | |
| 321 | + % | |
| 322 | + (ab.west) | |
| 323 | 323 | ; |
| 324 | 324 | % ---- |
| 325 | - \path[draw, rounded corners, pattern=north east lines, opacity=0.2] | |
| 326 | - (ac.south west) -- | |
| 327 | - (ac.west) -- | |
| 328 | - (ac.north west) -- | |
| 329 | - % | |
| 330 | - (abc.south west) -- | |
| 331 | - (abc.west) -- | |
| 332 | - (abc.north west) -- | |
| 333 | - % | |
| 334 | - (aBc.north east) -- | |
| 335 | - (aBc.east) -- | |
| 336 | - (aBc.south east) -- | |
| 337 | - % | |
| 338 | - (ac.north east) -- | |
| 339 | - % | |
| 340 | - (c.east) -- | |
| 341 | - % | |
| 342 | - (E.east) -- | |
| 343 | - (E.south east) -- | |
| 344 | - (E.south) -- | |
| 345 | - (E.south west) -- | |
| 346 | - % | |
| 347 | - (a.south west) -- | |
| 348 | - (a.west) -- | |
| 349 | - (a.north west) -- | |
| 350 | - (a.north) -- | |
| 351 | - % | |
| 352 | - (ac.south west) | |
| 325 | + \path[draw, rounded corners, pattern=north east lines, opacity=0.2] | |
| 326 | + (ac.south west) -- | |
| 327 | + (ac.west) -- | |
| 328 | + (ac.north west) -- | |
| 329 | + % | |
| 330 | + (abc.south west) -- | |
| 331 | + (abc.west) -- | |
| 332 | + (abc.north west) -- | |
| 333 | + % | |
| 334 | + (aBc.north east) -- | |
| 335 | + (aBc.east) -- | |
| 336 | + (aBc.south east) -- | |
| 337 | + % | |
| 338 | + (ac.north east) -- | |
| 339 | + % | |
| 340 | + (c.east) -- | |
| 341 | + % | |
| 342 | + (E.east) -- | |
| 343 | + (E.south east) -- | |
| 344 | + (E.south) -- | |
| 345 | + (E.south west) -- | |
| 346 | + % | |
| 347 | + (a.south west) -- | |
| 348 | + (a.west) -- | |
| 349 | + (a.north west) -- | |
| 350 | + (a.north) -- | |
| 351 | + % | |
| 352 | + (ac.south west) | |
| 353 | 353 | ; |
| 354 | 354 | % ---- |
| 355 | 355 | \path[draw, rounded corners, pattern=horizontal lines, opacity=0.2] |
| 356 | - % (A.north west) -- | |
| 357 | - % | |
| 358 | - (Ac.north west) -- | |
| 359 | - % | |
| 360 | - (Abc.north west) -- | |
| 361 | - (Abc.north) -- | |
| 362 | - (Abc.north east) -- | |
| 363 | - (Abc.south east) -- | |
| 364 | - % | |
| 365 | - % (Ac.north east) -- | |
| 366 | - % (Ac.east) -- | |
| 367 | - % | |
| 368 | - % (A.east) -- | |
| 369 | - (A.south east) -- | |
| 370 | - % | |
| 371 | - (E.south east) -- | |
| 372 | - (E.south) -- | |
| 373 | - (E.south west) -- | |
| 374 | - (E.west) -- | |
| 375 | - (E.north west) -- | |
| 376 | - % | |
| 377 | - (Ac.north west) | |
| 356 | + % (A.north west) -- | |
| 357 | + % | |
| 358 | + (Ac.north west) -- | |
| 359 | + % | |
| 360 | + (Abc.north west) -- | |
| 361 | + (Abc.north) -- | |
| 362 | + (Abc.north east) -- | |
| 363 | + (Abc.south east) -- | |
| 364 | + % | |
| 365 | + % (Ac.north east) -- | |
| 366 | + % (Ac.east) -- | |
| 367 | + % | |
| 368 | + % (A.east) -- | |
| 369 | + (A.south east) -- | |
| 370 | + % | |
| 371 | + (E.south east) -- | |
| 372 | + (E.south) -- | |
| 373 | + (E.south west) -- | |
| 374 | + (E.west) -- | |
| 375 | + (E.north west) -- | |
| 376 | + % | |
| 377 | + (Ac.north west) | |
| 378 | 378 | ; |
| 379 | 379 | \end{tikzpicture} |
| 380 | 380 | \end{center} |
| 381 | - | |
| 381 | + | |
| 382 | 382 | \caption{Classes (of consistent events) related to the \aclp{SM} of \cref{running.example} are defined through intersections and inclusions. \todo{write the caption}} |
| 383 | 383 | \label{fig:running.example.classes} |
| 384 | 384 | \end{figure} |
| 385 | 385 | |
| 386 | -Given an ASP specification, | |
| 386 | +Given an ASP specification, | |
| 387 | 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} } $s \in \fml{S}\subset\fml{W}$. | |
| 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} } $s \in \fml{S}\subset\fml{W}$. | |
| 389 | 389 | |
| 390 | 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}. |
| 391 | 391 | |
| ... | ... | @@ -400,11 +400,11 @@ This focus on the \acp{SM} leads to the following definition: |
| 400 | 400 | \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}: |
| 401 | 401 | |
| 402 | 402 | \begin{definition}\label{def:stable.core} |
| 403 | - The \emph{\ac{SC}} of the event $e\in \fml{E}$ is | |
| 403 | + The \emph{\ac{SC}} of the event $e\in \fml{E}$ is | |
| 404 | 404 | \begin{equation} |
| 405 | 405 | \stablecore{e} := \set{s \in \fml{S} \given s \subseteq e \vee e \subseteq s} \label{eq:stable.core} |
| 406 | 406 | \end{equation} |
| 407 | - | |
| 407 | + | |
| 408 | 408 | \end{definition} |
| 409 | 409 | |
| 410 | 410 | 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 |
| 420 | 420 | \begin{equation} |
| 421 | 421 | \class{e} = |
| 422 | 422 | \begin{cases} |
| 423 | - \inconsistent := \fml{E} \setminus \fml{W} | |
| 424 | - &\text{if~} e \in \fml{E} \setminus \fml{W}, \\ | |
| 423 | + \inconsistent := \fml{E} \setminus \fml{W} | |
| 424 | + & \text{if~} e \in \fml{E} \setminus \fml{W}, \\ | |
| 425 | 425 | \set{u \in \fml{W} \given \stablecore{u} = \stablecore{e}} |
| 426 | - &\text{if~} e \in \fml{W}, | |
| 426 | + & \text{if~} e \in \fml{W}, | |
| 427 | 427 | \end{cases}\label{eq:event.class} |
| 428 | 428 | \end{equation} |
| 429 | 429 | |
| 430 | 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: |
| 431 | 431 | \begin{equation} |
| 432 | 432 | \class{\fml{E}} = \set{ |
| 433 | - \inconsistent, | |
| 434 | - \indepclass, | |
| 435 | - \class{\co{a}}, | |
| 436 | - \class{ab}, | |
| 437 | - \class{ac}, | |
| 438 | - \class{\co{a}, ab}, | |
| 439 | - \class{\co{a}, ac}, | |
| 440 | - \class{ab, ac}, | |
| 441 | - \class{\co{a}, ab, ac} | |
| 433 | + \inconsistent, | |
| 434 | + \indepclass, | |
| 435 | + \class{\co{a}}, | |
| 436 | + \class{ab}, | |
| 437 | + \class{ac}, | |
| 438 | + \class{\co{a}, ab}, | |
| 439 | + \class{\co{a}, ac}, | |
| 440 | + \class{ab, ac}, | |
| 441 | + \class{\co{a}, ab, ac} | |
| 442 | 442 | } |
| 443 | 443 | \end{equation} |
| 444 | 444 | 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: |
| 445 | 445 | \begin{equation*} |
| 446 | 446 | \begin{array}{l|lr} |
| 447 | 447 | \text{\textbf{Core}}, \stablecore{e} |
| 448 | - & \text{\textbf{Class}}, \class{e} | |
| 449 | - & \text{\textbf{Size}}, \# \class{e}\\ | |
| 450 | - \hline | |
| 448 | + & \text{\textbf{Class}}, \class{e} | |
| 449 | + & \text{\textbf{Size}}, \# \class{e} \\ | |
| 450 | + \hline | |
| 451 | 451 | % |
| 452 | 452 | \inconsistent |
| 453 | - & a\co{a}, \ldots | |
| 454 | - & 37 | |
| 453 | + & a\co{a}, \ldots | |
| 454 | + & 37 | |
| 455 | 455 | \\ |
| 456 | 456 | % |
| 457 | - \indepclass | |
| 458 | - & \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 | |
| 459 | - & 9 | |
| 457 | + \indepclass | |
| 458 | + & \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 | |
| 459 | + & 9 | |
| 460 | 460 | \\ |
| 461 | 461 | % |
| 462 | - \co{a} | |
| 463 | - & \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} | |
| 464 | - & 9 | |
| 462 | + \co{a} | |
| 463 | + & \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} | |
| 464 | + & 9 | |
| 465 | 465 | \\ |
| 466 | 466 | % |
| 467 | 467 | ab |
| 468 | - & b, ab, ab\co{c} | |
| 469 | - & 3 | |
| 468 | + & b, ab, ab\co{c} | |
| 469 | + & 3 | |
| 470 | 470 | \\ |
| 471 | 471 | % |
| 472 | 472 | ac |
| 473 | - & c, ac, a\co{b}c | |
| 474 | - & 3 | |
| 473 | + & c, ac, a\co{b}c | |
| 474 | + & 3 | |
| 475 | 475 | \\ |
| 476 | 476 | % |
| 477 | 477 | \co{a}, ab |
| 478 | - & \emptyset | |
| 479 | - & 0 | |
| 478 | + & \emptyset | |
| 479 | + & 0 | |
| 480 | 480 | \\ |
| 481 | 481 | % |
| 482 | 482 | \co{a}, ac |
| 483 | - & \emptyset | |
| 484 | - & 0 | |
| 483 | + & \emptyset | |
| 484 | + & 0 | |
| 485 | 485 | % |
| 486 | 486 | \\ |
| 487 | 487 | % |
| 488 | 488 | ab, ac |
| 489 | - & a, abc | |
| 490 | - & 2 | |
| 489 | + & a, abc | |
| 490 | + & 2 | |
| 491 | 491 | \\ |
| 492 | 492 | % |
| 493 | 493 | \co{a}, ab, ac |
| 494 | - & \emptyevent | |
| 495 | - & 1 | |
| 494 | + & \emptyevent | |
| 495 | + & 1 | |
| 496 | 496 | \\ |
| 497 | 497 | % |
| 498 | 498 | \hline |
| 499 | 499 | \Omega |
| 500 | - & \text{all events} | |
| 501 | - & 64 | |
| 500 | + & \text{all events} | |
| 501 | + & 64 | |
| 502 | 502 | \end{array} |
| 503 | 503 | \end{equation*} |
| 504 | 504 | |
| 505 | 505 | \begin{itemize} |
| 506 | 506 | \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}: |
| 507 | - \[ | |
| 508 | - \forall u\in \class{e} \left(\mu\at{u} = \mu\at{e} \right). | |
| 509 | - \] | |
| 507 | + \[ | |
| 508 | + \forall u\in \class{e} \left(\mu\at{u} = \mu\at{e} \right). | |
| 509 | + \] | |
| 510 | 510 | \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. |
| 511 | - % \item The extended probability \emph{events} are the \emph{classes}. | |
| 511 | + % \item The extended probability \emph{events} are the \emph{classes}. | |
| 512 | 512 | \end{itemize} |
| 513 | 513 | |
| 514 | 514 | |
| 515 | 515 | |
| 516 | 516 | \subsection{From Total Choices to Events}\label{subsec:from.tchoices.to.events} |
| 517 | 517 | |
| 518 | -\todo{Check adaptation} Our path to set a probability measure on $\fml{E}$ has two phases: | |
| 518 | +\todo{Check adaptation} Our path to set a probability measure on $\fml{E}$ has two phases: | |
| 519 | 519 | \begin{enumerate} |
| 520 | 520 | \item Extending the probabilities, \emph{as weights}, from the \aclp{TC} to events. |
| 521 | 521 | \item Normalization of the weights. |
| ... | ... | @@ -525,68 +525,68 @@ The ``extension'' phase, traced by equations (\ref{eq:prob.total.choice}) and (\ |
| 525 | 525 | |
| 526 | 526 | \begin{description} |
| 527 | 527 | % |
| 528 | - \item[Total Choices.] Using \eqref{eq:prob.total.choice}, this case is given by | |
| 529 | - \begin{equation} | |
| 530 | - \pw{t} := \pr{T = t}= \prod_{a\in t} p \prod_{a \not\in t} \co{p} | |
| 531 | - \label{eq:weight.tchoice} | |
| 532 | - \end{equation} | |
| 533 | - % | |
| 534 | - \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}. | |
| 535 | - | |
| 536 | - Given a \acl{SM} $s \in \fml{S}$, a \acl{TC} $t$, and variables/values $\theta_{s,t} \in \intcc{0, 1}$, | |
| 537 | - \begin{equation} | |
| 538 | - \pw{s, t} := \begin{cases} | |
| 539 | - \theta_{s,t} & \text{if~} s \in \tcgen{t}\cr | |
| 540 | - 0&\text{otherwise} | |
| 541 | - \end{cases} | |
| 542 | - \label{eq:weight.stablemodel} | |
| 543 | - \end{equation} | |
| 544 | - such that $\sum_{s\in \tcgen{t}} \theta_{s,t} = 1$. | |
| 545 | - % | |
| 546 | - \item[Classes.] \label{item:class.cases} Each class is either the inconsistent class, $\inconsistent$, or is represented by some set of \aclp{SM}. | |
| 547 | - \begin{description} | |
| 548 | - \item[Inconsistent Class.] The inconsistent class contains events that are logically inconsistent, thus should never be observed: | |
| 528 | + \item[Total Choices.] Using \eqref{eq:prob.total.choice}, this case is given by | |
| 549 | 529 | \begin{equation} |
| 550 | - \pw{\inconsistent, t} := 0. | |
| 551 | - \label{eq:weight.class.inconsistent} | |
| 530 | + \pw{t} := \pr{T = t}= \prod_{a\in t} p \prod_{a \not\in t} \co{p} | |
| 531 | + \label{eq:weight.tchoice} | |
| 552 | 532 | \end{equation} |
| 553 | - \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: | |
| 533 | + % | |
| 534 | + \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}. | |
| 535 | + | |
| 536 | + Given a \acl{SM} $s \in \fml{S}$, a \acl{TC} $t$, and variables/values $\theta_{s,t} \in \intcc{0, 1}$, | |
| 554 | 537 | \begin{equation} |
| 555 | - \pw{\indepclass, t} := 0. | |
| 556 | - \label{eq:weight.class.independent} | |
| 538 | + \pw{s, t} := \begin{cases} | |
| 539 | + \theta_{s,t} & \text{if~} s \in \tcgen{t}\cr | |
| 540 | + 0 & \text{otherwise} | |
| 541 | + \end{cases} | |
| 542 | + \label{eq:weight.stablemodel} | |
| 557 | 543 | \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): | |
| 544 | + such that $\sum_{s\in \tcgen{t}} \theta_{s,t} = 1$. | |
| 545 | + % | |
| 546 | + \item[Classes.] \label{item:class.cases} Each class is either the inconsistent class, $\inconsistent$, or is represented by some set of \aclp{SM}. | |
| 547 | + \begin{description} | |
| 548 | + \item[Inconsistent Class.] The inconsistent class contains events that are logically inconsistent, thus should never be observed: | |
| 549 | + \begin{equation} | |
| 550 | + \pw{\inconsistent, t} := 0. | |
| 551 | + \label{eq:weight.class.inconsistent} | |
| 552 | + \end{equation} | |
| 553 | + \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: | |
| 554 | + \begin{equation} | |
| 555 | + \pw{\indepclass, t} := 0. | |
| 556 | + \label{eq:weight.class.independent} | |
| 557 | + \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): | |
| 559 | + \begin{equation} | |
| 560 | + \pw{\class{e}, t} := \sum_{k=1}^{n}\pw{s_k, t},~\text{if}~\stablecore{e} = \set{s_1, \ldots, s_n}. | |
| 561 | + \label{eq:weight.class.other} | |
| 562 | + \end{equation} | |
| 563 | + and | |
| 564 | + \begin{equation} | |
| 565 | + \pw{\class{e}} := \sum_{t \in \fml{T}} \pw{\class{e}, t}\pw{t}. | |
| 566 | + \label{eq:weight.class.unconditional} | |
| 567 | + \end{equation} | |
| 568 | + | |
| 569 | + \end{description} | |
| 570 | + % | |
| 571 | + \item[Events.] \label{item:event.cases} Each (general) event $e$ is in the class defined by its \acl{SC}, $\stablecore{e}$. So, we set: | |
| 559 | 572 | \begin{equation} |
| 560 | - \pw{\class{e}, t} := \sum_{k=1}^{n}\pw{s_k, t},~\text{if}~\stablecore{e} = \set{s_1, \ldots, s_n}. | |
| 561 | - \label{eq:weight.class.other} | |
| 573 | + \pw{e, t} := \frac{\pw{\class{e}, t}}{\# \class{e}} . | |
| 574 | + \label{eq:weight.events} | |
| 562 | 575 | \end{equation} |
| 563 | - and | |
| 576 | + and | |
| 564 | 577 | \begin{equation} |
| 565 | - \pw{\class{e}} := \sum_{t \in \fml{T}} \pw{\class{e}, t}\pw{t}. | |
| 566 | - \label{eq:weight.class.unconditional} | |
| 578 | + \pw{e} := \sum_{t\in\fml{T}} \pw{e, t} \pw{t}. | |
| 579 | + \label{eq:weight.events.unconditional} | |
| 567 | 580 | \end{equation} |
| 568 | - | |
| 569 | - \end{description} | |
| 570 | - % | |
| 571 | - \item[Events.] \label{item:event.cases} Each (general) event $e$ is in the class defined by its \acl{SC}, $\stablecore{e}$. So, we set: | |
| 572 | - \begin{equation} | |
| 573 | - \pw{e, t} := \frac{\pw{\class{e}, t}}{\# \class{e}} . | |
| 574 | - \label{eq:weight.events} | |
| 575 | - \end{equation} | |
| 576 | - and | |
| 577 | - \begin{equation} | |
| 578 | - \pw{e} := \sum_{t\in\fml{T}} \pw{e, t} \pw{t}. | |
| 579 | - \label{eq:weight.events.unconditional} | |
| 580 | - \end{equation} | |
| 581 | - % \remark{instead of that equation}{if we set $\pw{s,t} := \theta_{s,t}$ in equation \eqref{eq:weight.stablemodel} here we do: | |
| 582 | - % $$ | |
| 583 | - % \pw{e} := \sum_{t\in\fml{T}} \pw{e, t}\pw{t}. | |
| 584 | - % $$ | |
| 585 | - % By the way, this is the \emph{marginalization + bayes theorem} in statistics: | |
| 586 | - % $$ | |
| 587 | - % P(A) = \sum_b P(A | B=b)P(B=b) | |
| 588 | - % $$ | |
| 589 | - % } | |
| 581 | + % \remark{instead of that equation}{if we set $\pw{s,t} := \theta_{s,t}$ in equation \eqref{eq:weight.stablemodel} here we do: | |
| 582 | + % $$ | |
| 583 | + % \pw{e} := \sum_{t\in\fml{T}} \pw{e, t}\pw{t}. | |
| 584 | + % $$ | |
| 585 | + % By the way, this is the \emph{marginalization + bayes theorem} in statistics: | |
| 586 | + % $$ | |
| 587 | + % P(A) = \sum_b P(A | B=b)P(B=b) | |
| 588 | + % $$ | |
| 589 | + % } | |
| 590 | 590 | \end{description} |
| 591 | 591 | |
| 592 | 592 | % PARAMETERS FOR UNCERTAINTY |
| ... | ... | @@ -600,180 +600,180 @@ The ``extension'' phase, traced by equations (\ref{eq:prob.total.choice}) and (\ |
| 600 | 600 | 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. |
| 601 | 601 | |
| 602 | 602 | % SUPERSET |
| 603 | -Equation \eqref{eq:weight.class.other} results from conditional independence of \aclp{SM}. | |
| 603 | +Equation \eqref{eq:weight.class.other} results from conditional independence of \aclp{SM}. | |
| 604 | 604 | |
| 605 | 605 | |
| 606 | 606 | \section{Developed Examples} |
| 607 | 607 | |
| 608 | 608 | \subsection{The SBF Example} |
| 609 | 609 | |
| 610 | -We continue with the specification from Equation \eqref{eq:example.1}. | |
| 610 | +We continue with the specification from Equation \eqref{eq:example.1}. | |
| 611 | 611 | |
| 612 | 612 | \begin{description} |
| 613 | 613 | % |
| 614 | 614 | \item[\Aclp{TC}.] The \aclp{TC}, and respective \aclp{SM}, are |
| 615 | - \begin{center} | |
| 616 | - \begin{tabular}{ll|r} | |
| 617 | - \textbf{\Acl{TC}} & \textbf{\Aclp{SM}} & \textbf{$\pw{t}$}\\ | |
| 618 | - \hline | |
| 619 | - $a$ & $ab, ac$ & $0.3$\\ | |
| 620 | - $\co{a} = \neg a$ & $\co{a}$ & $\co{0.3} = 0.7$ | |
| 621 | - \end{tabular} | |
| 622 | - \end{center} | |
| 623 | - % | |
| 615 | + \begin{center} | |
| 616 | + \begin{tabular}{ll|r} | |
| 617 | + \textbf{\Acl{TC}} & \textbf{\Aclp{SM}} & \textbf{$\pw{t}$} \\ | |
| 618 | + \hline | |
| 619 | + $a$ & $ab, ac$ & $0.3$ \\ | |
| 620 | + $\co{a} = \neg a$ & $\co{a}$ & $\co{0.3} = 0.7$ | |
| 621 | + \end{tabular} | |
| 622 | + \end{center} | |
| 623 | + % | |
| 624 | 624 | \item[\Aclp{SM}.] The $\theta_{s,t}$ parameters in this example are |
| 625 | - $$ | |
| 626 | - \theta_{ab,\co{a}} = \theta_{ac,\co{a}} = \theta_{\co{a}, a} = 0 | |
| 627 | - % | |
| 628 | - \text{~and~} | |
| 629 | - % | |
| 630 | - \theta_{\co{a}, \co{a}} = 1, \theta_{ab, a} = \theta, \theta_{ac, a} = \co{\theta} | |
| 631 | - $$ | |
| 632 | - with $\theta \in \intcc{0, 1}$. | |
| 633 | - \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: | |
| 634 | - \begin{equation*} | |
| 635 | - \begin{array}{l|ll|r} | |
| 636 | - \stablecore{e} | |
| 637 | - & \pw{s_k, t= \co{a}} | |
| 638 | - & \pw{s_k, t= a} | |
| 639 | - & \pw{\class{e}}=\sum_{t}\pw{\class{e},t}\pw{t} | |
| 640 | - \\ | |
| 641 | - \hline | |
| 642 | - \co{a} | |
| 643 | - & 1 | |
| 644 | - & | |
| 645 | - & 0.7 | |
| 646 | - \\ | |
| 647 | - % | |
| 648 | - ab | |
| 649 | - & | |
| 650 | - & \theta | |
| 651 | - & 0.3\theta | |
| 652 | - \\ | |
| 653 | - % | |
| 654 | - ac | |
| 655 | - & | |
| 656 | - & \co{\theta} | |
| 657 | - & 0.3\co{\theta} | |
| 658 | - \\ | |
| 659 | - % | |
| 660 | - \co{a}, ab | |
| 661 | - & 1, 0 | |
| 662 | - & 0, \theta | |
| 663 | - & 0.7 + 0.3\theta | |
| 664 | - \\ | |
| 665 | - % | |
| 666 | - \co{a}, ac | |
| 667 | - & 1, 0 | |
| 668 | - & 0, \co{\theta} | |
| 669 | - & 0.7 + 0.3\co{\theta} | |
| 670 | - \\ | |
| 625 | + $$ | |
| 626 | + \theta_{ab,\co{a}} = \theta_{ac,\co{a}} = \theta_{\co{a}, a} = 0 | |
| 671 | 627 | % |
| 672 | - ab, ac | |
| 673 | - & | |
| 674 | - & \theta, \co{\theta} | |
| 675 | - & 0.3 | |
| 676 | - \\ | |
| 628 | + \text{~and~} | |
| 677 | 629 | % |
| 678 | - \co{a}, ab, ac | |
| 679 | - & 1, 0, 0 | |
| 680 | - & 0, \theta, \co{\theta} | |
| 681 | - & 1 | |
| 682 | - \end{array} | |
| 683 | - \end{equation*} | |
| 630 | + \theta_{\co{a}, \co{a}} = 1, \theta_{ab, a} = \theta, \theta_{ac, a} = \co{\theta} | |
| 631 | + $$ | |
| 632 | + with $\theta \in \intcc{0, 1}$. | |
| 633 | + \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: | |
| 634 | + \begin{equation*} | |
| 635 | + \begin{array}{l|ll|r} | |
| 636 | + \stablecore{e} | |
| 637 | + & \pw{s_k, t= \co{a}} | |
| 638 | + & \pw{s_k, t= a} | |
| 639 | + & \pw{\class{e}}=\sum_{t}\pw{\class{e},t}\pw{t} | |
| 640 | + \\ | |
| 641 | + \hline | |
| 642 | + \co{a} | |
| 643 | + & 1 | |
| 644 | + & | |
| 645 | + & 0.7 | |
| 646 | + \\ | |
| 647 | + % | |
| 648 | + ab | |
| 649 | + & | |
| 650 | + & \theta | |
| 651 | + & 0.3\theta | |
| 652 | + \\ | |
| 653 | + % | |
| 654 | + ac | |
| 655 | + & | |
| 656 | + & \co{\theta} | |
| 657 | + & 0.3\co{\theta} | |
| 658 | + \\ | |
| 659 | + % | |
| 660 | + \co{a}, ab | |
| 661 | + & 1, 0 | |
| 662 | + & 0, \theta | |
| 663 | + & 0.7 + 0.3\theta | |
| 664 | + \\ | |
| 665 | + % | |
| 666 | + \co{a}, ac | |
| 667 | + & 1, 0 | |
| 668 | + & 0, \co{\theta} | |
| 669 | + & 0.7 + 0.3\co{\theta} | |
| 670 | + \\ | |
| 671 | + % | |
| 672 | + ab, ac | |
| 673 | + & | |
| 674 | + & \theta, \co{\theta} | |
| 675 | + & 0.3 | |
| 676 | + \\ | |
| 677 | + % | |
| 678 | + \co{a}, ab, ac | |
| 679 | + & 1, 0, 0 | |
| 680 | + & 0, \theta, \co{\theta} | |
| 681 | + & 1 | |
| 682 | + \end{array} | |
| 683 | + \end{equation*} | |
| 684 | 684 | \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*} | |
| 686 | - Z := \sum_{e\in\fml{E}} \pw{e} | |
| 687 | - = \sum_{\class{e} \in\class{\fml{E}}} \frac{\pw{\class{e}}}{\#\class{e}}, | |
| 688 | - \end{equation*} | |
| 689 | - that divides the weight function into a normalized weight | |
| 690 | - \begin{equation*} | |
| 691 | - \pr{e} := \frac{\pw{e}}{Z}. | |
| 692 | - \end{equation*} | |
| 693 | - such that | |
| 694 | - $$ | |
| 695 | - \sum_{e \in \fml{E}} \pr{e} = 1. | |
| 696 | - $$ | |
| 697 | - For the SBF example, | |
| 698 | - \begin{equation*} | |
| 699 | - \begin{array}{lr|r|rr} | |
| 700 | - \stablecore{e} | |
| 701 | - & \# \class{e} | |
| 702 | - & \pw{\class{e}} | |
| 703 | - & \pw{e} | |
| 704 | - & \pr{e} | |
| 705 | - \\ | |
| 706 | - \hline | |
| 707 | - % | |
| 708 | - \inconsistent | |
| 709 | - & 37 | |
| 710 | - & 0 | |
| 711 | - & 0 | |
| 712 | - & 0 | |
| 713 | - \\[4pt] | |
| 714 | - % | |
| 715 | - \indepclass | |
| 716 | - & 9 | |
| 717 | - & 0 | |
| 718 | - & 0 | |
| 719 | - & 0 | |
| 720 | - \\[4pt] | |
| 721 | - % | |
| 722 | - \co{a} | |
| 723 | - & 9 | |
| 724 | - & \frac{7}{10} | |
| 725 | - & \frac{7}{90} | |
| 726 | - & \frac{7}{792} | |
| 727 | - \\[4pt] | |
| 728 | - % | |
| 729 | - ab | |
| 730 | - & 3 | |
| 731 | - & \frac{3\theta}{10} | |
| 732 | - & \frac{\theta}{10} | |
| 733 | - & \frac{\theta}{88} | |
| 734 | - \\[4pt] | |
| 735 | - % | |
| 736 | - ac | |
| 737 | - & 3 | |
| 738 | - & \frac{3\co{\theta}}{10} | |
| 739 | - & \frac{\co{\theta}}{10} | |
| 740 | - & \frac{\co{\theta}}{88} | |
| 741 | - \\[4pt] | |
| 742 | - % | |
| 743 | - \co{a}, ab | |
| 744 | - & 0 | |
| 745 | - & \frac{7 + 3\theta}{10} | |
| 746 | - & 0 | |
| 747 | - & 0 | |
| 748 | - \\[4pt] | |
| 749 | - % | |
| 750 | - \co{a}, ac | |
| 751 | - & 0 | |
| 752 | - & \frac{7 + 3\co{\theta}}{10} | |
| 753 | - & 0 | |
| 754 | - & 0 | |
| 755 | - % | |
| 756 | - \\[4pt] | |
| 757 | - % | |
| 758 | - ab, ac | |
| 759 | - & 2 | |
| 760 | - & \frac{3}{10} | |
| 761 | - & \frac{3}{20} | |
| 762 | - & \frac{3}{176} | |
| 763 | - \\[4pt] | |
| 764 | - % | |
| 765 | - \co{a}, ab, ac | |
| 766 | - & 1 | |
| 767 | - & 1 | |
| 768 | - & 1 | |
| 769 | - & \frac{5}{176} | |
| 770 | - \\[4pt] | |
| 771 | - % | |
| 772 | - \hline | |
| 773 | - & | |
| 774 | - & Z = \frac{44}{5} | |
| 775 | - \end{array} | |
| 776 | - \end{equation*} | |
| 685 | + \begin{equation*} | |
| 686 | + Z := \sum_{e\in\fml{E}} \pw{e} | |
| 687 | + = \sum_{\class{e} \in\class{\fml{E}}} \frac{\pw{\class{e}}}{\#\class{e}}, | |
| 688 | + \end{equation*} | |
| 689 | + that divides the weight function into a normalized weight | |
| 690 | + \begin{equation*} | |
| 691 | + \pr{e} := \frac{\pw{e}}{Z}. | |
| 692 | + \end{equation*} | |
| 693 | + such that | |
| 694 | + $$ | |
| 695 | + \sum_{e \in \fml{E}} \pr{e} = 1. | |
| 696 | + $$ | |
| 697 | + For the SBF example, | |
| 698 | + \begin{equation*} | |
| 699 | + \begin{array}{lr|r|rr} | |
| 700 | + \stablecore{e} | |
| 701 | + & \# \class{e} | |
| 702 | + & \pw{\class{e}} | |
| 703 | + & \pw{e} | |
| 704 | + & \pr{e} | |
| 705 | + \\ | |
| 706 | + \hline | |
| 707 | + % | |
| 708 | + \inconsistent | |
| 709 | + & 37 | |
| 710 | + & 0 | |
| 711 | + & 0 | |
| 712 | + & 0 | |
| 713 | + \\[4pt] | |
| 714 | + % | |
| 715 | + \indepclass | |
| 716 | + & 9 | |
| 717 | + & 0 | |
| 718 | + & 0 | |
| 719 | + & 0 | |
| 720 | + \\[4pt] | |
| 721 | + % | |
| 722 | + \co{a} | |
| 723 | + & 9 | |
| 724 | + & \frac{7}{10} | |
| 725 | + & \frac{7}{90} | |
| 726 | + & \frac{7}{792} | |
| 727 | + \\[4pt] | |
| 728 | + % | |
| 729 | + ab | |
| 730 | + & 3 | |
| 731 | + & \frac{3\theta}{10} | |
| 732 | + & \frac{\theta}{10} | |
| 733 | + & \frac{\theta}{88} | |
| 734 | + \\[4pt] | |
| 735 | + % | |
| 736 | + ac | |
| 737 | + & 3 | |
| 738 | + & \frac{3\co{\theta}}{10} | |
| 739 | + & \frac{\co{\theta}}{10} | |
| 740 | + & \frac{\co{\theta}}{88} | |
| 741 | + \\[4pt] | |
| 742 | + % | |
| 743 | + \co{a}, ab | |
| 744 | + & 0 | |
| 745 | + & \frac{7 + 3\theta}{10} | |
| 746 | + & 0 | |
| 747 | + & 0 | |
| 748 | + \\[4pt] | |
| 749 | + % | |
| 750 | + \co{a}, ac | |
| 751 | + & 0 | |
| 752 | + & \frac{7 + 3\co{\theta}}{10} | |
| 753 | + & 0 | |
| 754 | + & 0 | |
| 755 | + % | |
| 756 | + \\[4pt] | |
| 757 | + % | |
| 758 | + ab, ac | |
| 759 | + & 2 | |
| 760 | + & \frac{3}{10} | |
| 761 | + & \frac{3}{20} | |
| 762 | + & \frac{3}{176} | |
| 763 | + \\[4pt] | |
| 764 | + % | |
| 765 | + \co{a}, ab, ac | |
| 766 | + & 1 | |
| 767 | + & 1 | |
| 768 | + & 1 | |
| 769 | + & \frac{5}{176} | |
| 770 | + \\[4pt] | |
| 771 | + % | |
| 772 | + \hline | |
| 773 | + & | |
| 774 | + & Z = \frac{44}{5} | |
| 775 | + \end{array} | |
| 776 | + \end{equation*} | |
| 777 | 777 | \end{description} |
| 778 | 778 | |
| 779 | 779 | \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}. |
| 782 | 782 | % |
| 783 | 783 | \subsection{An example involving Bayesian networks} |
| 784 | 784 | |
| 785 | -\franc{Cometários:} | |
| 785 | +\franc{Comentários:} | |
| 786 | 786 | \begin{itemize} |
| 787 | 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 | 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 (hehe) para sistematizar isto: \condsymb{a}{bnc} | |
| 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) $a$. | |
| 790 | 791 | \end{itemize} |
| 791 | 792 | |
| 792 | 793 | |
| 793 | -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. | |
| 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. | |
| 794 | 795 | |
| 795 | 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. |
| 796 | 797 | |
| ... | ... | @@ -799,14 +800,14 @@ The events are: Burglary ($B$), Earthquake ($E$), Alarm ($A$), Mary calls ($M$) |
| 799 | 800 | \begin{figure} |
| 800 | 801 | \begin{center} |
| 801 | 802 | \begin{tikzpicture}[node distance=2.5cm] |
| 802 | - | |
| 803 | + | |
| 803 | 804 | % Nodes |
| 804 | 805 | \node[smodel, circle] (A) {A}; |
| 805 | 806 | \node[tchoice, above right of=A] (B) {B}; |
| 806 | 807 | \node[tchoice, above left of=A] (E) {E}; |
| 807 | 808 | \node[tchoice, below left of=A] (M) {M}; |
| 808 | 809 | \node[tchoice, below right of=A] (J) {J}; |
| 809 | - | |
| 810 | + | |
| 810 | 811 | % Edges |
| 811 | 812 | \draw[->] (B) to[bend left] (A) node[right,xshift=1.1cm,yshift=0.8cm] {\footnotesize{$P(B)=0.001$}} ; |
| 812 | 813 | \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 ($M$) |
| 814 | 815 | \draw[->] (A) to[bend left] (J) node[right,xshift=-0.2cm,yshift=0.7cm] {\footnotesize{$P(J|A)$}} ; |
| 815 | 816 | \end{tikzpicture} |
| 816 | 817 | \end{center} |
| 817 | - | |
| 818 | + | |
| 818 | 819 | \begin{multicols}{3} |
| 819 | - | |
| 820 | + | |
| 820 | 821 | \footnotesize{ |
| 821 | - \begin{equation*} | |
| 822 | - \begin{split} | |
| 823 | - &P(M|A)\\ | |
| 824 | - & \begin{array}{c|cc} | |
| 825 | - & m & \neg m \\ | |
| 826 | - \hline | |
| 827 | - a & 0.9 & 0.1\\ | |
| 828 | - \neg a& 0.05 & 0.95 | |
| 829 | - \end{array} | |
| 830 | - \end{split} | |
| 831 | - \end{equation*} | |
| 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*} | |
| 832 | 833 | } |
| 833 | - | |
| 834 | + | |
| 834 | 835 | \footnotesize{ |
| 835 | - \begin{equation*} | |
| 836 | - \begin{split} | |
| 837 | - &P(J|A)\\ | |
| 838 | - & \begin{array}{c|cc} | |
| 839 | - & j & \neg j \\ | |
| 840 | - \hline | |
| 841 | - a & 0.7 & 0.3\\ | |
| 842 | - \neg a& 0.01 & 0.99 | |
| 843 | - \end{array} | |
| 844 | - \end{split} | |
| 845 | - \end{equation*} | |
| 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*} | |
| 846 | 847 | } |
| 847 | 848 | \footnotesize{ |
| 848 | - \begin{equation*} | |
| 849 | - \begin{split} | |
| 850 | - P(A|B \wedge E)\\ | |
| 851 | - \begin{array}{c|c|cc} | |
| 852 | - & & a & \neg a \\ | |
| 853 | - \hline | |
| 854 | - b & e & 0.95 & 0.05\\ | |
| 855 | - b & \neg e & 0.94 & 0.06\\ | |
| 856 | - \neg b & e & 0.29 & 0.71\\ | |
| 857 | - \neg b & \neg e & 0.001 & 0.999 | |
| 858 | - \end{array} | |
| 859 | - \end{split} | |
| 860 | - \end{equation*} | |
| 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*} | |
| 861 | 862 | } |
| 862 | 863 | \end{multicols} |
| 863 | 864 | \caption{The Earthquake, Burglary, Alarm model} |
| ... | ... | @@ -869,9 +870,9 @@ Considering the probabilities given in \cref{Figure_Alarm} we obtain the followi |
| 869 | 870 | |
| 870 | 871 | \begin{equation*} |
| 871 | 872 | \begin{aligned} |
| 872 | - \probfact{0.001}{b}&,\cr | |
| 873 | - \probfact{0.002}{e}&,\cr | |
| 874 | - \end{aligned} | |
| 873 | + \probfact{0.001}{b} & ,\cr | |
| 874 | + \probfact{0.002}{e} & ,\cr | |
| 875 | + \end{aligned} | |
| 875 | 876 | \label{eq:not_so_simple_example} |
| 876 | 877 | \end{equation*} |
| 877 | 878 | |
| ... | ... | @@ -880,11 +881,11 @@ For the table giving the probability $P(M|A)$ we obtain the specification: |
| 880 | 881 | |
| 881 | 882 | \begin{equation*} |
| 882 | 883 | \begin{aligned} |
| 883 | - \probfact{0.9}{pm\_a}&,\cr | |
| 884 | - \probfact{0.05}{pm\_na}&,\cr | |
| 885 | - m & \leftarrow a, pm\_a,\cr | |
| 886 | - \neg m & \leftarrow a, \neg pm\_a. | |
| 887 | - \end{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} | |
| 888 | 889 | \end{equation*} |
| 889 | 890 | |
| 890 | 891 | 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 |
| 893 | 894 | |
| 894 | 895 | \begin{equation*} |
| 895 | 896 | \begin{aligned} |
| 896 | - &\probfact{0.7}{pj\_a},\cr | |
| 897 | - &\probfact{0.01}{pj\_na},\cr | |
| 898 | - j & \leftarrow a, pj\_a,\cr | |
| 899 | - \neg j & \leftarrow a, \neg pj\_a.\cr | |
| 900 | - \end{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} | |
| 901 | 902 | \end{equation*} |
| 902 | 903 | |
| 903 | 904 | 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 |
| 906 | 907 | |
| 907 | 908 | \begin{equation*} |
| 908 | 909 | \begin{aligned} |
| 909 | - &\probfact{0.95}{a\_be},\cr | |
| 910 | - &\probfact{0.94}{a\_bne},\cr | |
| 911 | - &\probfact{0.29}{a\_nbe},\cr | |
| 912 | - &\probfact{0.001}{a\_nbne},\cr | |
| 913 | - a & \leftarrow b, e, a\_be,\cr | |
| 914 | - \neg a & \leftarrow b,e, \neg a\_be, \cr | |
| 915 | - a & \leftarrow b,e, a\_bne,\cr | |
| 916 | - \neg a & \leftarrow b,e, \neg a\_bne, \cr | |
| 917 | - a & \leftarrow b,e, a\_nbe,\cr | |
| 918 | - \neg a & \leftarrow b,e, \neg a\_nbe, \cr | |
| 919 | - a & \leftarrow b,e, a\_nbne,\cr | |
| 920 | - \neg a & \leftarrow b,e, \neg a\_nbne. \cr | |
| 921 | - \end{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} | |
| 922 | 923 | \end{equation*} |
| 923 | 924 | |
| 924 | -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. | |
| 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. | |
| 925 | 926 | |
| 926 | 927 | |
| 927 | 928 | \section{Discussion} |
| ... | ... | @@ -934,119 +935,119 @@ One can then proceed as in the previous subsection and analyse this example. The |
| 934 | 935 | % |
| 935 | 936 | % My first guess was |
| 936 | 937 | % \begin{equation*} |
| 937 | - % \pr{W = w \given C = c} = \sum_{s \supset w}\pr{S = s \given C = c}. | |
| 938 | - % \end{equation*} | |
| 939 | - % | |
| 940 | - % $\pr{W = w \given C = c}$ already separates $\pr{W}$ into \textbf{disjoint} events! | |
| 941 | - % | |
| 942 | - % Also, I am assuming that \aclp{SM} are independent. | |
| 943 | - % | |
| 944 | - % 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 | |
| 945 | - % | |
| 946 | - % Also, if I set $p(w) = p(s_1) + p(s_2)$ and respect the laws of probability, this entails $p(s_1)p(s_2) = 0$. | |
| 947 | - % | |
| 948 | - % So, maybe what I want is (1) to define the cover $\hat{w} = \cup_{s \supset w} s$ | |
| 949 | - % | |
| 950 | - % \begin{equation*} | |
| 951 | - % \pr{W = w \given C = c} = \sum_{s \supset w}\pr{S = s \given C = c} - \pr{W = \hat{w} \given C = c}. | |
| 952 | - % \end{equation*} | |
| 953 | - % | |
| 954 | - % But this doesn't works, because we'd get $\pr{W = a \given C = a} < 1$. | |
| 955 | - % % | |
| 956 | - % | |
| 957 | - % % | |
| 958 | - % \bigskip | |
| 959 | - % \hrule | |
| 960 | - % | |
| 961 | - % INDEPENDENCE | |
| 962 | - % | |
| 963 | - %, per equation (\ref{eq:weight.class.independent}). | |
| 964 | - % | |
| 965 | - % ================================================================ | |
| 966 | - % | |
| 967 | - \begin{itemize} | |
| 968 | - \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. | |
| 969 | - \item \todo{The `up and down' choice in the equivalence relation and the possibility of describing any probability distribution.} | |
| 970 | - \item \todo{Remark that no benchmark was done with other SOTA efforts.} | |
| 971 | - \item \todo{The possibility to `import' bayesian theory and tools to this study.} | |
| 972 | - \end{itemize} | |
| 973 | - | |
| 974 | - | |
| 975 | - \subsection{Dependence} | |
| 976 | - \label{subsec:dependence} | |
| 977 | - | |
| 978 | - 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. | |
| 979 | - | |
| 980 | - 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 | |
| 981 | - $$ | |
| 982 | - \probfact{0.3}{a}, b \vee c \leftarrow a, \probfact{0.2}{d}, b \leftarrow c \wedge d. | |
| 983 | - $$ | |
| 984 | - with \aclp{SM} | |
| 985 | - $ | |
| 986 | - \co{ad}, \co{a}d, a\co{d}b, a\co{d}c, adb | |
| 987 | - $. | |
| 988 | - | |
| 989 | - | |
| 990 | - 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 $adb \subset s$. | |
| 991 | - | |
| 992 | - Following equations \eqref{eq:world.fold.stablemodel} and \eqref{eq:world.fold.independent} {\bruno What are these equations?} this entails | |
| 993 | - \begin{equation*} | |
| 994 | - \begin{cases} | |
| 995 | - \pr{W = adc \given C = ad} = 0,\cr | |
| 996 | - \pr{W = adb \given C = ad} = 1 | |
| 997 | - \end{cases} | |
| 998 | - \end{equation*} | |
| 999 | - 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: | |
| 1000 | - $$ | |
| 1001 | - \begin{array}{l|c} | |
| 1002 | - x & \pr{W = x \given C = ad}\\ | |
| 1003 | - \hline | |
| 1004 | - ad & 1 \\ | |
| 1005 | - adb & 1\\ | |
| 1006 | - adc & 0\\ | |
| 1007 | - adbc & 1 | |
| 1008 | - \end{array} | |
| 1009 | - $$ | |
| 1010 | - so, for $C = ad$, | |
| 1011 | - $$ | |
| 1012 | - \begin{aligned} | |
| 1013 | - \pr{W = b} &= \frac{2}{4} \cr | |
| 1014 | - \pr{W = c} &= \frac{1}{4} \cr | |
| 1015 | - \pr{W = bc} &= \frac{1}{4} \cr | |
| 1016 | - &\not= \pr{W = b}\pr{W = c} | |
| 1017 | - \end{aligned} | |
| 1018 | - $$ | |
| 1019 | - \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. | |
| 1020 | - | |
| 1021 | - {\bruno Why does this not contradict Assumption 1?} | |
| 1022 | - | |
| 1023 | - % | |
| 1024 | - | |
| 1025 | - % | |
| 1026 | - \hrule | |
| 1027 | - \begin{quotation}\note{Todo} | |
| 1028 | - | |
| 1029 | - Prove the four world cases (done), support the product (done) and sum (tbd) options, with the independence assumptions. | |
| 1030 | - \end{quotation} | |
| 1031 | - | |
| 1032 | - \subsection{Future Work} | |
| 1033 | - | |
| 1034 | - \todo{develop this section.} | |
| 1035 | - | |
| 1036 | - \begin{itemize} | |
| 1037 | - \item The measure of the inconsistent events doesn't need to be set to $0$ and, maybe, in some cases, it shouldn't. | |
| 1038 | - \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}. | |
| 1039 | - \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. | |
| 1040 | - \begin{equation*} | |
| 1041 | - \pw{e} := \sum_{c\in\fml{T}} \pw{e, c}\theta_c. | |
| 1042 | - \end{equation*} | |
| 1043 | - \end{itemize} | |
| 1044 | - | |
| 1045 | - | |
| 1046 | - \section*{Acknowledgements} | |
| 1047 | - | |
| 1048 | - This work is supported by NOVA\textbf{LINCS} (UIDB/04516/2020) with the financial support of FCT.IP. | |
| 1049 | - | |
| 1050 | - \printbibliography | |
| 1051 | - | |
| 1052 | - \end{document} | |
| 1053 | 938 | \ No newline at end of file |
| 939 | +% \pr{W = w \given C = c} = \sum_{s \supset w}\pr{S = s \given C = c}. | |
| 940 | +% \end{equation*} | |
| 941 | +% | |
| 942 | +% $\pr{W = w \given C = c}$ already separates $\pr{W}$ into \textbf{disjoint} events! | |
| 943 | +% | |
| 944 | +% Also, I am assuming that \aclp{SM} are independent. | |
| 945 | +% | |
| 946 | +% 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 | |
| 947 | +% | |
| 948 | +% Also, if I set $p(w) = p(s_1) + p(s_2)$ and respect the laws of probability, this entails $p(s_1)p(s_2) = 0$. | |
| 949 | +% | |
| 950 | +% So, maybe what I want is (1) to define the cover $\hat{w} = \cup_{s \supset w} s$ | |
| 951 | +% | |
| 952 | +% \begin{equation*} | |
| 953 | +% \pr{W = w \given C = c} = \sum_{s \supset w}\pr{S = s \given C = c} - \pr{W = \hat{w} \given C = c}. | |
| 954 | +% \end{equation*} | |
| 955 | +% | |
| 956 | +% But this doesn't works, because we'd get $\pr{W = a \given C = a} < 1$. | |
| 957 | +% % | |
| 958 | +% | |
| 959 | +% % | |
| 960 | +% \bigskip | |
| 961 | +% \hrule | |
| 962 | +% | |
| 963 | +% INDEPENDENCE | |
| 964 | +% | |
| 965 | +%, per equation (\ref{eq:weight.class.independent}). | |
| 966 | +% | |
| 967 | +% ================================================================ | |
| 968 | +% | |
| 969 | +\begin{itemize} | |
| 970 | + \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. | |
| 971 | + \item \todo{The `up and down' choice in the equivalence relation and the possibility of describing any probability distribution.} | |
| 972 | + \item \todo{Remark that no benchmark was done with other SOTA efforts.} | |
| 973 | + \item \todo{The possibility to `import' bayesian theory and tools to this study.} | |
| 974 | +\end{itemize} | |
| 975 | + | |
| 976 | + | |
| 977 | +\subsection{Dependence} | |
| 978 | +\label{subsec:dependence} | |
| 979 | + | |
| 980 | +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. | |
| 981 | + | |
| 982 | +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 | |
| 983 | +$$ | |
| 984 | + \probfact{0.3}{a}, b \vee c \leftarrow a, \probfact{0.2}{d}, b \leftarrow c \wedge d. | |
| 985 | +$$ | |
| 986 | +with \aclp{SM} | |
| 987 | +$ | |
| 988 | + \co{ad}, \co{a}d, a\co{d}b, a\co{d}c, adb | |
| 989 | +$. | |
| 990 | + | |
| 991 | + | |
| 992 | +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 $adb \subset s$. | |
| 993 | + | |
| 994 | +Following equations \eqref{eq:world.fold.stablemodel} and \eqref{eq:world.fold.independent} {\bruno What are these equations?} this entails | |
| 995 | +\begin{equation*} | |
| 996 | + \begin{cases} | |
| 997 | + \pr{W = adc \given C = ad} = 0,\cr | |
| 998 | + \pr{W = adb \given C = ad} = 1 | |
| 999 | + \end{cases} | |
| 1000 | +\end{equation*} | |
| 1001 | +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: | |
| 1002 | +$$ | |
| 1003 | + \begin{array}{l|c} | |
| 1004 | + x & \pr{W = x \given C = ad} \\ | |
| 1005 | + \hline | |
| 1006 | + ad & 1 \\ | |
| 1007 | + adb & 1 \\ | |
| 1008 | + adc & 0 \\ | |
| 1009 | + adbc & 1 | |
| 1010 | + \end{array} | |
| 1011 | +$$ | |
| 1012 | +so, for $C = ad$, | |
| 1013 | +$$ | |
| 1014 | + \begin{aligned} | |
| 1015 | + \pr{W = b} & = \frac{2}{4} \cr | |
| 1016 | + \pr{W = c} & = \frac{1}{4} \cr | |
| 1017 | + \pr{W = bc} & = \frac{1}{4} \cr | |
| 1018 | + & \not= \pr{W = b}\pr{W = c} | |
| 1019 | + \end{aligned} | |
| 1020 | +$$ | |
| 1021 | +\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. | |
| 1022 | + | |
| 1023 | + {\bruno Why does this not contradict Assumption 1?} | |
| 1024 | + | |
| 1025 | +% | |
| 1026 | + | |
| 1027 | +% | |
| 1028 | +\hrule | |
| 1029 | +\begin{quotation}\note{Todo} | |
| 1030 | + | |
| 1031 | + Prove the four world cases (done), support the product (done) and sum (tbd) options, with the independence assumptions. | |
| 1032 | +\end{quotation} | |
| 1033 | + | |
| 1034 | +\subsection{Future Work} | |
| 1035 | + | |
| 1036 | +\todo{develop this section.} | |
| 1037 | + | |
| 1038 | +\begin{itemize} | |
| 1039 | + \item The measure of the inconsistent events doesn't need to be set to $0$ and, maybe, in some cases, it shouldn't. | |
| 1040 | + \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}. | |
| 1041 | + \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. | |
| 1042 | + \begin{equation*} | |
| 1043 | + \pw{e} := \sum_{c\in\fml{T}} \pw{e, c}\theta_c. | |
| 1044 | + \end{equation*} | |
| 1045 | +\end{itemize} | |
| 1046 | + | |
| 1047 | + | |
| 1048 | +\section*{Acknowledgements} | |
| 1049 | + | |
| 1050 | +This work is supported by NOVA\textbf{LINCS} (UIDB/04516/2020) with the financial support of FCT.IP. | |
| 1051 | + | |
| 1052 | +\printbibliography | |
| 1053 | + | |
| 1054 | +\end{document} | |
| 1054 | 1055 | \ No newline at end of file | ... | ... |