Foundations and Geometry

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Publication MaxEnt 2014


Foundations and Geometry


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        <identifier identifierType="DOI">10.23723/9603/11313</identifier><creators><creator><creatorName>John Skilling</creatorName></creator></creators><titles>
            <title>Foundations and Geometry</title></titles>
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	    <date dateType="Created">Sat 30 Aug 2014</date>
	    <date dateType="Updated">Mon 2 Oct 2017</date>
            <date dateType="Submitted">Sun 24 Mar 2019</date>
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            <description descriptionType="Abstract"></description>

Foundations and Geometry John Skilling ( Probability theory has a solid foundation based on elementary symmetries, nowadays refined to associativity augmented with either commutativity or order. Few workers now contest the sum and product rules of standard probability calculus (often called “Bayesian” although there’s no rational alternative and a unique form needs no adjective). The practice of inference is understood and agreed. Yet pure inference is not the end of the story. We may wish to simplify a distribution, or aggregate several into a single representative, in a minimally damaging way. For such purposes, we wish to know how far one probability distribution is from another, so that we can define what we mean by minimal damage. Over the years, many candidate distances have been proposed, used, and generalised to cover measures (distributions that can have any total) and even to positive matrices. On these distances, geometries have been constructed. Their very multiplicity, though, to say nothing of their ad hoc production, indicates that none has been found wholly satisfactory. There is a reason for that. The reason is that there is only one connection that allows data from arbitrary partitions of the coordinate space to be combined consistently. That connection is the unique information H(p; q) = pi log pi qi known to statisticians as the Kullback-Leibler. And H is asymmetric, H(p; q) = H(q; p). The asymmetry is both central and obvious. It can’t be evaded. To pass from distribution q = (1 2 , 1 2 ) to p = (1, 0) takes one bit of information (which might tell us that a coin was “heads”). But the reverse passage from (1, 0) representing a coin known to be “heads” to (1 2 , 1 2 ) is impossible because “tails” is supposedly known to be false. It follows that probability distributions do not form a metric space. Consequently, all geometries must fail. More precisely, any proposal based on a symmetric distance (p; q) = (q; p) must be opposed to the uniqueness of H and will fall foul of elementary criteria, thereby being open to counter-example. Specifically, the popular claim that the Fisher metric g = H defines generally applicable geodesic paths and lengths, with associated density √ det g, is to be resisted because it contradicts the foundation of H itself. 150 years ago, Riemann ditched Euclid. Perhaps it is time to ditch Riemann.