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author Robert McIntyre <rlm@mit.edu>
date Fri, 28 Oct 2011 00:03:05 -0700
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7 <title>Categorification of Plausible Reasoning</title>
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120 </head>
121 <body>
122 <div id="content">
123
124
125
126
127 <div id="table-of-contents">
128 <h2>Table of Contents</h2>
129 <div id="text-table-of-contents">
130 <ul>
131 <li><a href="#sec-1">1 Deductive and inductive posets </a>
132 <ul>
133 <li><a href="#sec-1-1">1.1 Definition </a></li>
134 <li><a href="#sec-1-2">1.2 A canonical map from deductive posets to inductive posets </a></li>
135 </ul>
136 </li>
137 <li><a href="#sec-2">2 Assigning plausibilities to inductive posets </a>
138 <ul>
139 <li><a href="#sec-2-1">2.1 Consistent reasoning as a commutative diagram </a></li>
140 <li><a href="#sec-2-2">2.2 ``Multiplicity'' is reciprocal probability </a></li>
141 <li><a href="#sec-2-3">2.3 Laws for multiplicity </a></li>
142 </ul>
143 </li>
144 </ul>
145 </div>
146 </div>
147
148 <div id="outline-container-1" class="outline-2">
149 <h2 id="sec-1"><span class="section-number-2">1</span> Deductive and inductive posets </h2>
150 <div class="outline-text-2" id="text-1">
151
152
153
154 </div>
155
156 <div id="outline-container-1-1" class="outline-3">
157 <h3 id="sec-1-1"><span class="section-number-3">1.1</span> Definition </h3>
158 <div class="outline-text-3" id="text-1-1">
159
160 <p>If you have a collection \(P\) of logical propositions, you can order them by
161 implication: \(a\) precedes \(b\) if and only if \(a\) implies
162 \(b\). This makes \(P\) into a poset. Since the ordering arose from
163 deductive implication, we'll call this a <i>deductive poset</i>.
164 </p>
165 <p>
166 If you have a deductive poset \(P\), you can create a related poset \(P^*\)
167 as follows: the underlying set is the same, and for any two
168 propositions \(a\) and \(b\) in \(P\), \(a\) precedes
169 \(ab\) in \(P^*\). We'll call this an <i>inductive poset</i>.
170 </p>
171 </div>
172
173 </div>
174
175 <div id="outline-container-1-2" class="outline-3">
176 <h3 id="sec-1-2"><span class="section-number-3">1.2</span> A canonical map from deductive posets to inductive posets </h3>
177 <div class="outline-text-3" id="text-1-2">
178
179 <p>Each poset corresponds with a poset-category, that is a category with
180 at most one arrow between any two objects. Considered as categories,
181 inductive and deuctive posets are related as follows: there is a map
182 \(\mathscr{F}\) which sends each arrow \(a\rightarrow b\) in \(P\) to
183 the arrow \(a\rightarrow ab\) in \(P^*\). In fact, since \(a\) implies
184 \(b\) if and only if \(a = ab\), \(\mathscr{F}\) sends each arrow in \(P\) to
185 an identity arrow in \(P^*\) (specifically, it sends the arrow
186 \(a\rightarrow b\) to the identity arrow \(a\rightarrow a\)).
187 </p>
188
189 </div>
190 </div>
191
192 </div>
193
194 <div id="outline-container-2" class="outline-2">
195 <h2 id="sec-2"><span class="section-number-2">2</span> Assigning plausibilities to inductive posets </h2>
196 <div class="outline-text-2" id="text-2">
197
198
199 <p>
200 Inductive posets encode the relative (<i>qualitative</i>) plausibilities of its
201 propositions: there exists an arrow \(x\rightarrow y\) only if \(x\)
202 is at least as plausible as \(y\).
203 </p>
204
205 </div>
206
207 <div id="outline-container-2-1" class="outline-3">
208 <h3 id="sec-2-1"><span class="section-number-3">2.1</span> Consistent reasoning as a commutative diagram </h3>
209 <div class="outline-text-3" id="text-2-1">
210
211 <p>Inductive categories enable the following neat trick: we can interpret
212 the objects of \(P^*\) as states of given information and interpret
213 each arrow \(a\rightarrow ab\) in \(P^*\) as an inductive inference: the arrow
214 \(a\rightarrow ab\) represents an inferential leap from the state of
215 knowledge where only \(a\) is given to the state of knowledge where
216 both \(a\) and \(b\) are given&mdash; in this way, it represents
217 the process of inferring \(b\) when given \(a\), and we label the
218 arrow with \((b|a)\).
219 </p>
220 <p>
221 This trick has several important features that suggest its usefulness,
222 namely
223 </p><ul>
224 <li>Composition of arrows corresponds to compound inference.
225 </li>
226 <li>In the special case of deductive inference, the inferential arrow is an
227 identity; the source and destination states of knowledge are the same.
228 </li>
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
230 commutative square: \(x\rightarrow ax \rightarrow abx\) =
231 \(x\rightarrow bx \rightarrow abx\) is the categorified version of
232 \((AB|X)=(A|X)\cdot(B|AX)=(B|X)\cdot(A|BX)\).
233 </li>
234 <li>We can make plausibility assignments by enriching the inductive
235 category \(P^*\) over some monoidal category, e.g. the set of real numbers
236 (considered as a category) with its usual multiplication. <i>When we do</i>,
237 the identity arrows of \(P^*\) &mdash;corresponding to
238 deductive inferences&mdash; are assigned a value of certainty automatically.
239 </li>
240 </ul>
241
242
243 </div>
244
245 </div>
246
247 <div id="outline-container-2-2" class="outline-3">
248 <h3 id="sec-2-2"><span class="section-number-3">2.2</span> ``Multiplicity'' is reciprocal probability </h3>
249 <div class="outline-text-3" id="text-2-2">
250
251 <p>The natural numbers have a comparatively concrete origin: they are the
252 result of decategorifying the category of finite sets<sup><a class="footref" name="fnr.2" href="#fn.2">2</a></sup>, or the
253 coequalizer of the arrows from a one-object category to a two-object
254 category with a single nonidentity arrow. Extensions of the set of
255 natural numbers&mdash; such as
256 the set of integers or rational numbers or real numbers&mdash; strike
257 me as being somewhat more abstract (however, see the Eudoxus
258 construction of the real numbers).
259 </p>
260 <p>
261 Jaynes points out that our existing choice of scale for probabilities
262 (i.e., the scale from 0 for impossibility to 1 for
263 certainty) has a degree of freedom: any monotonic function of
264 probability encodes the same information that probability does. Though
265 the resulting laws for compound probability and so on change in form
266 when probabilities are changed, they do not change in content.
267 </p>
268 <p>
269 With this in mind, it seems natural and permissible to use not <i>probability</i> but
270 <i>reciprocal probability</i> instead. This scale, which we might
271 call <i>multiplicity</i>, ranges from 1 (certainty) to
272 positive infinity (impossibility); higher numbers are ascribed to
273 less-plausible events.
274 </p>
275 <p>
276 In this way, the ``probability''
277 associated with choosing one out of \(n\) indistinguishable choices
278 becomes identified with \(n\).
279 </p>
280 </div>
281
282 </div>
283
284 <div id="outline-container-2-3" class="outline-3">
285 <h3 id="sec-2-3"><span class="section-number-3">2.3</span> Laws for multiplicity </h3>
286 <div class="outline-text-3" id="text-2-3">
287
288 <p>Jaynes derives laws of probability; either his method or his results
289 can be used to obtain laws for multiplicities.
290 </p>
291 <dl>
292 <dt>product rule</dt><dd>The product rule is unchanged: \(\xi(AB|X)=\xi(A|X)\cdot
293 \xi(B|AX) = \xi(B|X)\cdot \xi(A|BX)\)
294 </dd>
295 <dt>certainty</dt><dd>States of absolute certainty are assigned a multiplicity
296 of 1. States of absolute impossibility are assigned a
297 multiplicity of positive infinity.
298 </dd>
299 <dt>entropy</dt><dd>In terms of probability, entropy has the form \(S=-\sum_i
300 p_i \ln{p_i} = \sum_i p_i (-\ln{p_i}) = \sum_i p_i \ln{(1/p_i)}
301 \). Hence, in terms of multiplicity, entropy
302 has the form \(S = \sum_i \frac{\ln{\xi_i}}{\xi_i} \).
303
304 <p>
305 Another interesting quantity is \(\exp{S}\), which behaves
306 multiplicitively rather than additively. \(\exp{S} =
307 \prod_i \exp{\frac{\ln{\xi_i}}{\xi_i}} =
308 \left(\exp{\ln{\xi_i}}\right)^{1/\xi_i} = \prod_i \xi_i^{1/\xi_i} \)
309 </p></dd>
310 </dl>
311
312
313
314 <div id="footnotes">
315 <h2 class="footnotes">Footnotes: </h2>
316 <div id="text-footnotes">
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>
318 </p>
319
320 <p class="footnote"><sup><a class="footnum" name="fn.2" href="#fnr.2">2</a></sup> As Baez explains.
321 </p>
322 </div>
323 </div>
324 </div>
325
326 </div>
327 </div>
328 <div id="postamble">
329 <p class="date">Date: 2011-07-09 14:19:42 EDT</p>
330 <p class="author">Author: Dylan Holmes</p>
331 <p class="creator">Org version 7.6 with Emacs version 23</p>
332 <a href="http://validator.w3.org/check?uri=referer">Validate XHTML 1.0</a>
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