Table of Contents
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- 1 Deductive and inductive posets
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- 1.1 Definition rlm@2:
- 1.2 A canonical map from deductive posets to inductive posets rlm@2:
rlm@2: - 2 Assigning plausibilities to inductive posets
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- 2.1 Consistent reasoning as a commutative diagram rlm@2:
- 2.2 ``Multiplicity'' is reciprocal probability rlm@2:
- 2.3 Laws for multiplicity rlm@2:
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1 Deductive and inductive posets
rlm@2:1.1 Definition
rlm@2:If you have a collection \(P\) of logical propositions, you can order them by rlm@2: implication: \(a\) precedes \(b\) if and only if \(a\) implies rlm@2: \(b\). This makes \(P\) into a poset. Since the ordering arose from rlm@2: deductive implication, we'll call this a deductive poset. rlm@2:
rlm@2:rlm@2: If you have a deductive poset \(P\), you can create a related poset \(P^*\) rlm@2: as follows: the underlying set is the same, and for any two rlm@2: propositions \(a\) and \(b\) in \(P\), \(a\) precedes rlm@2: \(ab\) in \(P^*\). We'll call this an inductive poset. rlm@2:
rlm@2:1.2 A canonical map from deductive posets to inductive posets
rlm@2:Each poset corresponds with a poset-category, that is a category with rlm@2: at most one arrow between any two objects. Considered as categories, rlm@2: inductive and deuctive posets are related as follows: there is a map rlm@2: \(\mathscr{F}\) which sends each arrow \(a\rightarrow b\) in \(P\) to rlm@2: the arrow \(a\rightarrow ab\) in \(P^*\). In fact, since \(a\) implies rlm@2: \(b\) if and only if \(a = ab\), \(\mathscr{F}\) sends each arrow in \(P\) to rlm@2: an identity arrow in \(P^*\) (specifically, it sends the arrow rlm@2: \(a\rightarrow b\) to the identity arrow \(a\rightarrow a\)). rlm@2:
rlm@2: rlm@2:2 Assigning plausibilities to inductive posets
rlm@2:rlm@2: Inductive posets encode the relative (qualitative) plausibilities of its rlm@2: propositions: there exists an arrow \(x\rightarrow y\) only if \(x\) rlm@2: is at least as plausible as \(y\). rlm@2:
rlm@2: rlm@2:2.1 Consistent reasoning as a commutative diagram
rlm@2:Inductive categories enable the following neat trick: we can interpret rlm@2: the objects of \(P^*\) as states of given information and interpret rlm@2: each arrow \(a\rightarrow ab\) in \(P^*\) as an inductive inference: the arrow rlm@2: \(a\rightarrow ab\) represents an inferential leap from the state of rlm@2: knowledge where only \(a\) is given to the state of knowledge where rlm@2: both \(a\) and \(b\) are given— in this way, it represents rlm@2: the process of inferring \(b\) when given \(a\), and we label the rlm@2: arrow with \((b|a)\). rlm@2:
rlm@2:rlm@2: This trick has several important features that suggest its usefulness, rlm@2: namely rlm@2:
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- Composition of arrows corresponds to compound inference. rlm@2: rlm@2:
- In the special case of deductive inference, the inferential arrow is an rlm@2: identity; the source and destination states of knowledge are the same. rlm@2: rlm@2:
- One aspect of the consistency requirement of Jaynes1 takes the form of a rlm@2: commutative square: \(x\rightarrow ax \rightarrow abx\) = rlm@2: \(x\rightarrow bx \rightarrow abx\) is the categorified version of rlm@2: \((AB|X)=(A|X)\cdot(B|AX)=(B|X)\cdot(A|BX)\). rlm@2: rlm@2:
- We can make plausibility assignments by enriching the inductive rlm@2: category \(P^*\) over some monoidal category, e.g. the set of real numbers rlm@2: (considered as a category) with its usual multiplication. When we do, rlm@2: the identity arrows of \(P^*\) —corresponding to rlm@2: deductive inferences— are assigned a value of certainty automatically. rlm@2: rlm@2:
2.2 ``Multiplicity'' is reciprocal probability
rlm@2:The natural numbers have a comparatively concrete origin: they are the rlm@2: result of decategorifying the category of finite sets2, or the rlm@2: coequalizer of the arrows from a one-object category to a two-object rlm@2: category with a single nonidentity arrow. Extensions of the set of rlm@2: natural numbers— such as rlm@2: the set of integers or rational numbers or real numbers— strike rlm@2: me as being somewhat more abstract (however, see the Eudoxus rlm@2: construction of the real numbers). rlm@2:
rlm@2:rlm@2: Jaynes points out that our existing choice of scale for probabilities rlm@2: (i.e., the scale from 0 for impossibility to 1 for rlm@2: certainty) has a degree of freedom: any monotonic function of rlm@2: probability encodes the same information that probability does. Though rlm@2: the resulting laws for compound probability and so on change in form rlm@2: when probabilities are changed, they do not change in content. rlm@2:
rlm@2:rlm@2: With this in mind, it seems natural and permissible to use not probability but rlm@2: reciprocal probability instead. This scale, which we might rlm@2: call multiplicity, ranges from 1 (certainty) to rlm@2: positive infinity (impossibility); higher numbers are ascribed to rlm@2: less-plausible events. rlm@2:
rlm@2:rlm@2: In this way, the ``probability'' rlm@2: associated with choosing one out of \(n\) indistinguishable choices rlm@2: becomes identified with \(n\). rlm@2:
rlm@2:2.3 Laws for multiplicity
rlm@2:Jaynes derives laws of probability; either his method or his results rlm@2: can be used to obtain laws for multiplicities. rlm@2:
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- product rule
- The product rule is unchanged: \(\xi(AB|X)=\xi(A|X)\cdot rlm@2: \xi(B|AX) = \xi(B|X)\cdot \xi(A|BX)\) rlm@2: rlm@2:
- certainty
- States of absolute certainty are assigned a multiplicity rlm@2: of 1. States of absolute impossibility are assigned a rlm@2: multiplicity of positive infinity. rlm@2: rlm@2:
- entropy
- In terms of probability, entropy has the form \(S=-\sum_i
rlm@2: p_i \ln{p_i} = \sum_i p_i (-\ln{p_i}) = \sum_i p_i \ln{(1/p_i)}
rlm@2: \). Hence, in terms of multiplicity, entropy
rlm@2: has the form \(S = \sum_i \frac{\ln{\xi_i}}{\xi_i} \).
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rlm@2: Another interesting quantity is \(\exp{S}\), which behaves rlm@2: multiplicitively rather than additively. \(\exp{S} = rlm@2: \prod_i \exp{\frac{\ln{\xi_i}}{\xi_i}} = rlm@2: \left(\exp{\ln{\xi_i}}\right)^{1/\xi_i} = \prod_i \xi_i^{1/\xi_i} \) rlm@2:
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Date: 2011-07-09 14:19:42 EDT
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