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author Robert McIntyre <rlm@mit.edu>
date Fri, 28 Oct 2011 00:03:05 -0700
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1 #+TITLE: Categorification of Plausible Reasoning
2 #+AUTHOR: Dylan Holmes
3 #+MATHJAX: align:"left" mathml:t path:"../MathJax/MathJax.js"
4 * COMMENT #+OPTIONS: LaTeX:dvipng
5
6 * Deductive and inductive posets
7
8 ** Definition
9 If you have a collection \(P\) of logical propositions, you can order them by
10 implication: \(a\) precedes \(b\) if and only if \(a\) implies
11 \(b\). This makes \(P\) into a poset. Since the ordering arose from
12 deductive implication, we'll call this a /deductive poset/.
13
14 If you have a deductive poset \(P\), you can create a related poset \(P^*\)
15 as follows: the underlying set is the same, and for any two
16 propositions \(a\) and \(b\) in \(P\), \(a\) precedes
17 \(ab\) in \(P^*\). We'll call this an /inductive poset/.
18
19 ** A canonical map from deductive posets to inductive posets
20 Each poset corresponds with a poset-category, that is a category with
21 at most one arrow between any two objects. Considered as categories,
22 inductive and deuctive posets are related as follows: there is a map
23 \(\mathscr{F}\) which sends each arrow \(a\rightarrow b\) in \(P\) to
24 the arrow \(a\rightarrow ab\) in \(P^*\). In fact, since \(a\) implies
25 \(b\) if and only if \(a = ab\), \(\mathscr{F}\) sends each arrow in \(P\) to
26 an identity arrow in \(P^*\) (specifically, it sends the arrow
27 \(a\rightarrow b\) to the identity arrow \(a\rightarrow a\)).
28
29
30 ** Assigning plausibilities to inductive posets
31
32 Inductive posets encode the relative (/qualitative/) plausibilities of its
33 propositions: there exists an arrow \(x\rightarrow y\) only if \(x\)
34 is at least as plausible as \(y\).
35
36 *** Consistent reasoning as a commutative diagram
37 Inductive categories enable the following neat trick: we can interpret
38 the objects of \(P^*\) as states of given information and interpret
39 each arrow \(a\rightarrow ab\) in \(P^*\) as an inductive inference: the arrow
40 \(a\rightarrow ab\) represents an inferential leap from the state of
41 knowledge where only \(a\) is given to the state of knowledge where
42 both \(a\) and \(b\) are given\mdash{} in this way, it represents
43 the process of inferring \(b\) when given \(a\), and we label the
44 arrow with \((b|a)\).
45
46 This trick has several important features that suggest its usefulness,
47 namely
48 - Composition of arrows corresponds to compound inference.
49 - In the special case of deductive inference, the inferential arrow is an
50 identity; the source and destination states of knowledge are the same.
51 - One aspect of the consistency requirement of Jaynes[fn:1] takes the form of a
52 commutative square: \(x\rightarrow ax \rightarrow abx\) =
53 \(x\rightarrow bx \rightarrow abx\) is the categorified version of
54 \((AB|X)=(A|X)\cdot(B|AX)=(B|X)\cdot(A|BX)\).
55 - We can make plausibility assignments by enriching the inductive
56 category \(P^*\) over some monoidal category, e.g. the set of real numbers
57 (considered as a category) with its usual multiplication. /When we do/,
58 the identity arrows of \(P^*\) \mdash{}corresponding to
59 deductive inferences\mdash{} are assigned a value of certainty automatically.
60
61 [fn:1] /(IIIa) If a conclusion can be reasoned out in more than one
62 way, then every possible way must lead to the same result./
63
64
65 *** Reciprocal probabilities
66 The natural numbers have a comparatively concrete origin: they are the
67 result of decategorifying the category of finite sets[fn:2], or the
68 coequalizer of the arrows from a one-object category to a two-object
69 category with a single nonidentity arrow. Extensions of the set of
70 natural numbers\mdash{} such as
71 the set of integers or rational numbers or real numbers\mdash{} strike
72 me as being somewhat more abstract.
73
74 Jaynes points out that our existing choice of scale for probabilities
75 (i.e., the scale from 0 for impossibility to 1 for
76 certainty) has a degree of freedom: any monotonic function of
77 probability encodes the same information that probability does.
78
79 With this in mind, it seems useful to use not /probability/ but
80 /reciprocal probability/ instead. This scale, which we might
81 tentatively call freeness, is a scale ranging 1 (certainty) to
82 positive infinity (impossibility).
83
84 In this way, the ``probability''
85 associated with choosing one out of \(n\) indistinguishable choices
86 becomes identified with \(n\).
87
88 The entropy
89
90 [fn:2] As Baez says.
91
92
93
94 ** self-questions
95
96 What circumstances would make \(\mathscr{F}\) an injection?
97
98 What if \(P=\{\top,\bot\}\)?
99
100
101
102 ** COMMENT
103 Inductive and deductive posets are related as follows: there is a monotone
104 inclusion map \(\mathscr{i}:P^*\hookrightarrow P\) which\mdash{} since \(a\)
105 implies \(b\) is equivalent to \(a=ab\)\mdash{} sends comparable
106 propositions in \(P\) to the same proposition in \(P^*\). Conversely,
107 only comparable propositions in \(P\) are sent to the same proposition
108 in \(P^*\).
109
110
111
112 ** Inductive posets and plausibility
113
114 * Inverse Probability