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author Robert McIntyre <rlm@mit.edu>
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rlm@2 7 <title>Categorification of Plausible Reasoning</title>
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rlm@2 122 <div id="content">
rlm@2 123
rlm@2 124
rlm@2 125
rlm@2 126
rlm@2 127 <div id="table-of-contents">
rlm@2 128 <h2>Table of Contents</h2>
rlm@2 129 <div id="text-table-of-contents">
rlm@2 130 <ul>
rlm@2 131 <li><a href="#sec-1">1 Deductive and inductive posets </a>
rlm@2 132 <ul>
rlm@2 133 <li><a href="#sec-1-1">1.1 Definition </a></li>
rlm@2 134 <li><a href="#sec-1-2">1.2 A canonical map from deductive posets to inductive posets </a></li>
rlm@2 135 </ul>
rlm@2 136 </li>
rlm@2 137 <li><a href="#sec-2">2 Assigning plausibilities to inductive posets </a>
rlm@2 138 <ul>
rlm@2 139 <li><a href="#sec-2-1">2.1 Consistent reasoning as a commutative diagram </a></li>
rlm@2 140 <li><a href="#sec-2-2">2.2 ``Multiplicity'' is reciprocal probability </a></li>
rlm@2 141 <li><a href="#sec-2-3">2.3 Laws for multiplicity </a></li>
rlm@2 142 </ul>
rlm@2 143 </li>
rlm@2 144 </ul>
rlm@2 145 </div>
rlm@2 146 </div>
rlm@2 147
rlm@2 148 <div id="outline-container-1" class="outline-2">
rlm@2 149 <h2 id="sec-1"><span class="section-number-2">1</span> Deductive and inductive posets </h2>
rlm@2 150 <div class="outline-text-2" id="text-1">
rlm@2 151
rlm@2 152
rlm@2 153
rlm@2 154 </div>
rlm@2 155
rlm@2 156 <div id="outline-container-1-1" class="outline-3">
rlm@2 157 <h3 id="sec-1-1"><span class="section-number-3">1.1</span> Definition </h3>
rlm@2 158 <div class="outline-text-3" id="text-1-1">
rlm@2 159
rlm@2 160 <p>If you have a collection \(P\) of logical propositions, you can order them by
rlm@2 161 implication: \(a\) precedes \(b\) if and only if \(a\) implies
rlm@2 162 \(b\). This makes \(P\) into a poset. Since the ordering arose from
rlm@2 163 deductive implication, we'll call this a <i>deductive poset</i>.
rlm@2 164 </p>
rlm@2 165 <p>
rlm@2 166 If you have a deductive poset \(P\), you can create a related poset \(P^*\)
rlm@2 167 as follows: the underlying set is the same, and for any two
rlm@2 168 propositions \(a\) and \(b\) in \(P\), \(a\) precedes
rlm@2 169 \(ab\) in \(P^*\). We'll call this an <i>inductive poset</i>.
rlm@2 170 </p>
rlm@2 171 </div>
rlm@2 172
rlm@2 173 </div>
rlm@2 174
rlm@2 175 <div id="outline-container-1-2" class="outline-3">
rlm@2 176 <h3 id="sec-1-2"><span class="section-number-3">1.2</span> A canonical map from deductive posets to inductive posets </h3>
rlm@2 177 <div class="outline-text-3" id="text-1-2">
rlm@2 178
rlm@2 179 <p>Each poset corresponds with a poset-category, that is a category with
rlm@2 180 at most one arrow between any two objects. Considered as categories,
rlm@2 181 inductive and deuctive posets are related as follows: there is a map
rlm@2 182 \(\mathscr{F}\) which sends each arrow \(a\rightarrow b\) in \(P\) to
rlm@2 183 the arrow \(a\rightarrow ab\) in \(P^*\). In fact, since \(a\) implies
rlm@2 184 \(b\) if and only if \(a = ab\), \(\mathscr{F}\) sends each arrow in \(P\) to
rlm@2 185 an identity arrow in \(P^*\) (specifically, it sends the arrow
rlm@2 186 \(a\rightarrow b\) to the identity arrow \(a\rightarrow a\)).
rlm@2 187 </p>
rlm@2 188
rlm@2 189 </div>
rlm@2 190 </div>
rlm@2 191
rlm@2 192 </div>
rlm@2 193
rlm@2 194 <div id="outline-container-2" class="outline-2">
rlm@2 195 <h2 id="sec-2"><span class="section-number-2">2</span> Assigning plausibilities to inductive posets </h2>
rlm@2 196 <div class="outline-text-2" id="text-2">
rlm@2 197
rlm@2 198
rlm@2 199 <p>
rlm@2 200 Inductive posets encode the relative (<i>qualitative</i>) plausibilities of its
rlm@2 201 propositions: there exists an arrow \(x\rightarrow y\) only if \(x\)
rlm@2 202 is at least as plausible as \(y\).
rlm@2 203 </p>
rlm@2 204
rlm@2 205 </div>
rlm@2 206
rlm@2 207 <div id="outline-container-2-1" class="outline-3">
rlm@2 208 <h3 id="sec-2-1"><span class="section-number-3">2.1</span> Consistent reasoning as a commutative diagram </h3>
rlm@2 209 <div class="outline-text-3" id="text-2-1">
rlm@2 210
rlm@2 211 <p>Inductive categories enable the following neat trick: we can interpret
rlm@2 212 the objects of \(P^*\) as states of given information and interpret
rlm@2 213 each arrow \(a\rightarrow ab\) in \(P^*\) as an inductive inference: the arrow
rlm@2 214 \(a\rightarrow ab\) represents an inferential leap from the state of
rlm@2 215 knowledge where only \(a\) is given to the state of knowledge where
rlm@2 216 both \(a\) and \(b\) are given&mdash; in this way, it represents
rlm@2 217 the process of inferring \(b\) when given \(a\), and we label the
rlm@2 218 arrow with \((b|a)\).
rlm@2 219 </p>
rlm@2 220 <p>
rlm@2 221 This trick has several important features that suggest its usefulness,
rlm@2 222 namely
rlm@2 223 </p><ul>
rlm@2 224 <li>Composition of arrows corresponds to compound inference.
rlm@2 225 </li>
rlm@2 226 <li>In the special case of deductive inference, the inferential arrow is an
rlm@2 227 identity; the source and destination states of knowledge are the same.
rlm@2 228 </li>
rlm@2 229 <li>One aspect of the consistency requirement of Jaynes<sup><a class="footref" name="fnr.1" href="#fn.1">1</a></sup> takes the form of a
rlm@2 230 commutative square: \(x\rightarrow ax \rightarrow abx\) =
rlm@2 231 \(x\rightarrow bx \rightarrow abx\) is the categorified version of
rlm@2 232 \((AB|X)=(A|X)\cdot(B|AX)=(B|X)\cdot(A|BX)\).
rlm@2 233 </li>
rlm@2 234 <li>We can make plausibility assignments by enriching the inductive
rlm@2 235 category \(P^*\) over some monoidal category, e.g. the set of real numbers
rlm@2 236 (considered as a category) with its usual multiplication. <i>When we do</i>,
rlm@2 237 the identity arrows of \(P^*\) &mdash;corresponding to
rlm@2 238 deductive inferences&mdash; are assigned a value of certainty automatically.
rlm@2 239 </li>
rlm@2 240 </ul>
rlm@2 241
rlm@2 242
rlm@2 243 </div>
rlm@2 244
rlm@2 245 </div>
rlm@2 246
rlm@2 247 <div id="outline-container-2-2" class="outline-3">
rlm@2 248 <h3 id="sec-2-2"><span class="section-number-3">2.2</span> ``Multiplicity'' is reciprocal probability </h3>
rlm@2 249 <div class="outline-text-3" id="text-2-2">
rlm@2 250
rlm@2 251 <p>The natural numbers have a comparatively concrete origin: they are the
rlm@2 252 result of decategorifying the category of finite sets<sup><a class="footref" name="fnr.2" href="#fn.2">2</a></sup>, or the
rlm@2 253 coequalizer of the arrows from a one-object category to a two-object
rlm@2 254 category with a single nonidentity arrow. Extensions of the set of
rlm@2 255 natural numbers&mdash; such as
rlm@2 256 the set of integers or rational numbers or real numbers&mdash; strike
rlm@2 257 me as being somewhat more abstract (however, see the Eudoxus
rlm@2 258 construction of the real numbers).
rlm@2 259 </p>
rlm@2 260 <p>
rlm@2 261 Jaynes points out that our existing choice of scale for probabilities
rlm@2 262 (i.e., the scale from 0 for impossibility to 1 for
rlm@2 263 certainty) has a degree of freedom: any monotonic function of
rlm@2 264 probability encodes the same information that probability does. Though
rlm@2 265 the resulting laws for compound probability and so on change in form
rlm@2 266 when probabilities are changed, they do not change in content.
rlm@2 267 </p>
rlm@2 268 <p>
rlm@2 269 With this in mind, it seems natural and permissible to use not <i>probability</i> but
rlm@2 270 <i>reciprocal probability</i> instead. This scale, which we might
rlm@2 271 call <i>multiplicity</i>, ranges from 1 (certainty) to
rlm@2 272 positive infinity (impossibility); higher numbers are ascribed to
rlm@2 273 less-plausible events.
rlm@2 274 </p>
rlm@2 275 <p>
rlm@2 276 In this way, the ``probability''
rlm@2 277 associated with choosing one out of \(n\) indistinguishable choices
rlm@2 278 becomes identified with \(n\).
rlm@2 279 </p>
rlm@2 280 </div>
rlm@2 281
rlm@2 282 </div>
rlm@2 283
rlm@2 284 <div id="outline-container-2-3" class="outline-3">
rlm@2 285 <h3 id="sec-2-3"><span class="section-number-3">2.3</span> Laws for multiplicity </h3>
rlm@2 286 <div class="outline-text-3" id="text-2-3">
rlm@2 287
rlm@2 288 <p>Jaynes derives laws of probability; either his method or his results
rlm@2 289 can be used to obtain laws for multiplicities.
rlm@2 290 </p>
rlm@2 291 <dl>
rlm@2 292 <dt>product rule</dt><dd>The product rule is unchanged: \(\xi(AB|X)=\xi(A|X)\cdot
rlm@2 293 \xi(B|AX) = \xi(B|X)\cdot \xi(A|BX)\)
rlm@2 294 </dd>
rlm@2 295 <dt>certainty</dt><dd>States of absolute certainty are assigned a multiplicity
rlm@2 296 of 1. States of absolute impossibility are assigned a
rlm@2 297 multiplicity of positive infinity.
rlm@2 298 </dd>
rlm@2 299 <dt>entropy</dt><dd>In terms of probability, entropy has the form \(S=-\sum_i
rlm@2 300 p_i \ln{p_i} = \sum_i p_i (-\ln{p_i}) = \sum_i p_i \ln{(1/p_i)}
rlm@2 301 \). Hence, in terms of multiplicity, entropy
rlm@2 302 has the form \(S = \sum_i \frac{\ln{\xi_i}}{\xi_i} \).
rlm@2 303
rlm@2 304 <p>
rlm@2 305 Another interesting quantity is \(\exp{S}\), which behaves
rlm@2 306 multiplicitively rather than additively. \(\exp{S} =
rlm@2 307 \prod_i \exp{\frac{\ln{\xi_i}}{\xi_i}} =
rlm@2 308 \left(\exp{\ln{\xi_i}}\right)^{1/\xi_i} = \prod_i \xi_i^{1/\xi_i} \)
rlm@2 309 </p></dd>
rlm@2 310 </dl>
rlm@2 311
rlm@2 312
rlm@2 313
rlm@2 314 <div id="footnotes">
rlm@2 315 <h2 class="footnotes">Footnotes: </h2>
rlm@2 316 <div id="text-footnotes">
rlm@2 317 <p class="footnote"><sup><a class="footnum" name="fn.1" href="#fnr.1">1</a></sup> <i>(IIIa) If a conclusion can be reasoned out in more than one way, then every possible way must lead to the same result.</i>
rlm@2 318 </p>
rlm@2 319
rlm@2 320 <p class="footnote"><sup><a class="footnum" name="fn.2" href="#fnr.2">2</a></sup> As Baez explains.
rlm@2 321 </p>
rlm@2 322 </div>
rlm@2 323 </div>
rlm@2 324 </div>
rlm@2 325
rlm@2 326 </div>
rlm@2 327 </div>
rlm@2 328 <div id="postamble">
rlm@2 329 <p class="date">Date: 2011-07-09 14:19:42 EDT</p>
rlm@2 330 <p class="author">Author: Dylan Holmes</p>
rlm@2 331 <p class="creator">Org version 7.6 with Emacs version 23</p>
rlm@2 332 <a href="http://validator.w3.org/check?uri=referer">Validate XHTML 1.0</a>
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