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Kevin Knuth on Measuring July 8, 2007 (20)

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Kevin H. Knuth Department of Physics University at Albany Measuring

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Kevin H Knuth 8 July 2007MaxEnt 2007 Familiarity Breeds the Illusion of Understanding Anonymous

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Kevin H Knuth 8 July 2007MaxEnt 2007 We are all Familiar with Measuring

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Kevin H Knuth 8 July 2007MaxEnt 2007 We Measure All Sorts of Things

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Kevin H Knuth 8 July 2007MaxEnt 2007 Measuring Things We Don’t Understand

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Kevin H Knuth 8 July 2007MaxEnt 2007 But Not Everything

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Kevin H Knuth 8 July 2007MaxEnt 2007 Unification of Ideas

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Counting

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Kevin H Knuth 8 July 2007MaxEnt 2007 In the Beginning…

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Kevin H Knuth 8 July 2007MaxEnt 2007 A Caveman had a Collection of Rocks

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Kevin H Knuth 8 July 2007MaxEnt 2007 An Equivalence Class of Rocks If the Caveman likes, he can treat each rock as being equivalent to any other rock in his collection. He then has an Equivalence Class of Rocks

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Kevin H Knuth 8 July 2007MaxEnt 2007 Comparing Collections of Rocks Two Cavemen might like to compare their collections.

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Kevin H Knuth 8 July 2007MaxEnt 2007 One-to-One Correspondence If they treat the rocks as belonging to an equivalence class, they can attempt to make a one-to-one correspondence One Caveman has more rocks than the other since there is not a one-to-one mapping between the two sets of rocks

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Kevin H Knuth 8 July 2007MaxEnt 2007 Why Carry Your Rocks Around? Every time they want to compare their collections, Thog has to carry his rocks over a big hill to meet Bok who carried his rocks across a raging river. There must be an easier way to make comparisons

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Kevin H Knuth 8 July 2007MaxEnt 2007 Sticks? Thog has an idea, he suggests that for every rock, he should pick up a stick. Carrying sticks is easier! Bok thinks this over…

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Kevin H Knuth 8 July 2007MaxEnt 2007 One Stick Instead of a lot of sticks, Bok suggests making a mark on a stick for each rock.

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Kevin H Knuth 8 July 2007MaxEnt 2007 Natural Numbers Thog, who wants to impress his smart friend agrees and further suggests that they should make fancy marks rather than a lot of them

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Kevin H Knuth 8 July 2007MaxEnt 2007 One-to-One Correspondence Bok gets to work… 1 2 3

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Kevin H Knuth 8 July 2007MaxEnt 2007 At Their Next Meeting… 2 3 The two gentlemen compare their markings and enjoy a relaxing afternoon free from carrying rocks. Comparing their markings, they find that Bok has more rocks than Thog.

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Kevin H Knuth 8 July 2007MaxEnt 2007 Ordering

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Kevin H Knuth 8 July 2007MaxEnt 2007 Not All Rocks Are The Same Bok decides that some of his rocks are more impressive than others. One of them is HUGE!!! From this point on he no longer treats the rocks as belonging to an equivalence class

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Kevin H Knuth 8 July 2007MaxEnt 2007 Lifting Rocks Bok finds it easy to order these rocks in terms of how heavy they are to lift.

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Kevin H Knuth 8 July 2007MaxEnt 2007 Binary Ordering Relation Rock 1 is no heavier than Rock 2 Rock 1Rock 2 R1 R2 Bok lifts one rock and compares its weight to another…one pair at a time.

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Kevin H Knuth 8 July 2007MaxEnt 2007 Binary Ordering Relation Rock 2 is no heavier than Rock 3 Rock 2Rock 3 R2 R3

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Kevin H Knuth 8 July 2007MaxEnt 2007 Transitivity Rock 1 is no heavier than Rock 3 Rock 1Rock 3 R1 R2 and R2 R3 implies that R1 R3

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Kevin H Knuth 8 July 2007MaxEnt 2007 Diagram Representing the Ordering Heavier Bok uses his results to order his rocks

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Kevin H Knuth 8 July 2007MaxEnt 2007 Ranking Heavier Bok then realizes that he can use the same marks to compare one rock to another

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Kevin H Knuth 8 July 2007MaxEnt 2007 Measuring

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Kevin H Knuth 8 July 2007MaxEnt 2007 Rock Hunting Bok has become quite sophisticated, and now that his collection is finally in order, he decides to go hunting for another rock.

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Kevin H Knuth 8 July 2007MaxEnt 2007 Sophisticated Rock Hunting

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Kevin H Knuth 8 July 2007MaxEnt 2007 Ranking Heavier Bok’s new rock messes up his ranking! 1 2 3

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Kevin H Knuth 8 July 2007MaxEnt 2007 Generalizing Ranking Heavier Bok’s realizes that he can generalize his concept of heavier, to degrees of heaviness 1 2 3

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Kevin H Knuth 8 July 2007MaxEnt 2007 Measuring Heavier He now measures his rocks against his standard 1 LR (Little Rock) 2.6 LR 4.2 LR 1.8 LR

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Kevin H Knuth 8 July 2007MaxEnt 2007 Measuring Meanwhile Thog, who has sharpened up a bit, came up with a similar methodology Lighter 1 Easy 2.3 Easy 1.5 Easy

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Kevin H Knuth 8 July 2007MaxEnt 2007 Next Meeting Thog, who is proud of his accomplishments, brings his ideas to Bok along with a sheet of tree bark with numbers describing his collection. 1 Easy 2.3 Easy 1.5 Easy

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Kevin H Knuth 8 July 2007MaxEnt 2007 Not Easy to Compare Lighter 1 Easy 2.3 Easy 1.5 Easy Heavier 1 LR 2.6 LR 4.2 LR 1.8 LR

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Kevin H Knuth 8 July 2007MaxEnt 2007 An Agreement Thog likes how Bok has ordered his collection, as he too is proud of his biggest rock. To compromise, Bok agrees to use Thog’s smallest rock as a unit of weight. Thog brings it to Bok so he can begin comparing.

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Kevin H Knuth 8 July 2007MaxEnt 2007 Not Easy to Compare Lighter 1 Easy 2.3 Easy 1.5 Easy Heavier 1 LR 2.6 LR 4.2 LR 1.8 LR

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Kevin H Knuth 8 July 2007MaxEnt 2007 Adopt a Standard Ordering Heavier 2.3 Rocks 1.5 Rocks Heavier 1.0 LR 2.6 LR 4.2 LR 1.8 LR 1.0 Rock

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Kevin H Knuth 8 July 2007MaxEnt 2007 Perform a Regraduation Heavier 2.3 Rocks 1.5 Rocks Heavier 0.5 Rock 1.3 Rocks 2.1 Rocks 0.9 Rock 1.0 Rock

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Kevin H Knuth 8 July 2007MaxEnt 2007 On Their Next Meeting… Thog is pleased that his little rock is a unit of measure. Bok is surprised to find that Thog’s rock is actually heavier than his… by 0.2 Rocks!

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Kevin H Knuth 8 July 2007MaxEnt 2007 An Idea… I was thinking… Maybe we could use these ideas to describe things other than rocks? I don’t see why not! We might be able to describe anything we can order.

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Kevin H Knuth 8 July 2007MaxEnt 2007 A Set of Apples Here is a set of apples

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Kevin H Knuth 8 July 2007MaxEnt 2007 Ordering Apples We can order them according to weight Heavier

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Kevin H Knuth 8 July 2007MaxEnt 2007 Partially Ordered Set We can order them according to weight Heavier A partially ordered set is a set along with a binary ordering relation Called a Poset for short Apple 1 is no HEAVIER than Apple 2

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Kevin H Knuth 8 July 2007MaxEnt 2007 Posets of Apples We can order them according to sweetness Sweeter Apple 1 is no SWEETER than Apple 2

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Kevin H Knuth 8 July 2007MaxEnt 2007 Posets of Apples We can order them according to how ripe they are More Ripe Apple 1 is no RIPER than Apple 2

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Kevin H Knuth 8 July 2007MaxEnt 2007 Posets of Apples This configuration is called a CHAIN More Ripe

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Kevin H Knuth 8 July 2007MaxEnt 2007 Isomorphisms More Ripe Is Greater than or Equal to Divides

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Kevin H Knuth 8 July 2007MaxEnt 2007 A Set of Fruit Here is a set of apples along with an orange

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Kevin H Knuth 8 July 2007MaxEnt 2007 Apples and Oranges With the binary relation “is equally or less HEAVY than” these two set elements can be compared.

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Kevin H Knuth 8 July 2007MaxEnt 2007 Incomparable Sometimes two elements of a set cannot be compared using the binary ordering relation. It may not be possible to tell which one is SWEETER…they taste different. Such pairs of elements are called incomparable

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Kevin H Knuth 8 July 2007MaxEnt 2007 Antichains Say that these three elements are incomparable under our binary ordering relation

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Kevin H Knuth 8 July 2007MaxEnt 2007 Antichains These elements are incomparable under our binary ordering relation They form a poset called an ANTICHAIN

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Kevin H Knuth 8 July 2007MaxEnt 2007 Other Examples

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Kevin H Knuth 8 July 2007MaxEnt 2007 Partitioning Consider arrangements of fruit on a table. I could just set the fruit on the table Or I could put the apple and banana on a plate first and then set it on the table Either way, the table “contains” the fruit

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Kevin H Knuth 8 July 2007MaxEnt 2007 Partitioning A set of elements together with a binary ordering relation based on the notion of containing

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Kevin H Knuth 8 July 2007MaxEnt 2007 Two Posets with Integers Is Greater than or Equal to Divides

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Kevin H Knuth 8 July 2007MaxEnt 2007 Subsets Consider the powerset of the set S = { a, b, c } This is the set of all possible subsets of S: A natural ordering is the relation “is a subset of”,

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Kevin H Knuth 8 July 2007MaxEnt 2007 First we note that No element of the set comes between the two, so we say that covers, denoted So we draw above and connect them with a line. The First Level

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Kevin H Knuth 8 July 2007MaxEnt 2007 Completing the First Level It is also true that and so we draw them above as well and connect them with lines. However, as neither one is the subset of the other. In addition, and. So we draw them on the same level and do not connect them.

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Second Level Now we note that is covered by two elements and.

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Kevin H Knuth 8 July 2007MaxEnt 2007 These elements also cover and The Second Level

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Kevin H Knuth 8 July 2007MaxEnt 2007 Completing the Second Level Now also covers and, but these top elements are also incomparable.

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Third Level Finally covers all three two- element subsets.

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Powerset of { a, b, c } Is a subset of

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Kevin H Knuth 8 July 2007MaxEnt 2007 Posets A partially ordered set (poset) is a set of elements together with a binary ordering relation. includes covers

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Kevin H Knuth 8 July 2007MaxEnt 2007 Lattices A lattice is a poset P where every pair of elements x and y has a least upper bound called the join a greatest lower bound called the meet

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Kevin H Knuth 8 July 2007MaxEnt 2007 Lattices A lattice is a poset P where every pair of elements x and y has a least upper bound called the join a greatest lower bound called the meet Similarly The green elements are upper bounds of the blue circled pair. The green circled element is their least upper bound or their join.

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Kevin H Knuth 8 July 2007MaxEnt 2007 Lattice of Sets Note that in this example Order Theoretic Notation Set Theoretic Notation

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Lattice Identities L1. Idempotent L2. Commutative L3. Associative L4. Absorption If the meet and join follow the Consistency Relations C1. ( x is the greatest lower bound of x and y ) C2. ( y is the least upper bound of x and y ) Lattice Identities

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Kevin H Knuth 8 July 2007MaxEnt 2007 Lattices are Algebras Structural Viewpoint Operational Viewpoint

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Kevin H Knuth 8 July 2007MaxEnt 2007 Lattices are Algebras Structural Viewpoint Operational Viewpoint Structural Viewpoint Operational Viewpoint Lattices in General Sets with

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Kevin H Knuth 8 July 2007MaxEnt 2007 Lattices are Algebras Structural Viewpoint Operational Viewpoint Structural Viewpoint Operational Viewpoint Lattices in GeneralLogical Statements with Implication

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Kevin H Knuth 8 July 2007MaxEnt 2007 Lattices are Algebras Structural Viewpoint Operational Viewpoint Structural Viewpoint Operational Viewpoint Lattices in General Integers with

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Kevin H Knuth 8 July 2007MaxEnt 2007 Lattices are Algebras Structural Viewpoint Operational Viewpoint Structural Viewpoint Operational Viewpoint Lattices in GeneralPositive Integers with Divides

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Kevin H Knuth 8 July 2007MaxEnt 2007 Lattices are Algebras Structural Viewpoint Operational Viewpoint Assertions, Implies Sets, Is a subset of Positive Integers, Divides Integers, Is less than or equal to

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Hypothesis Space (Our States of Knowledge)

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Boolean Hypothesis Space The atoms are mutually exclusive and exhaustive logical statements a = ‘It is an Apple!’ b = ‘It is a Banana!’ c = ‘It is a Citrus Fruit!’

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Boolean Hypothesis Space abc The atoms are mutually exclusive and exhaustive logical statements a = ‘It is an apple!’ b = ‘It is a banana!’ c = ‘It is a citrus fruit!’

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Boolean Hypothesis Space abc The meet of any two atoms is the absurdity: a b = We do not allow our state of knowledge to include: ‘The fruit is an apple AND a banana!’

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Boolean Hypothesis Space abc

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Boolean Hypothesis Space abc a b The join of any two elements represents a logical OR: a b

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Boolean Hypothesis Space abc a ba cb c The join of any two elements represents a logical OR: a b

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Boolean Hypothesis Space abc a ba cb c The final join gives us the TOP element, which is the TRUISM = a b c “It is an Apple or a Banana or an Orange!”

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Boolean Hypothesis Space abc a ba cb c The lattice is ordered according to the ordering relation “implies”. implies

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Boolean Hypothesis Space abc a ba cb c The lattice is ordered according to the ordering relation “implies”. implies Meaning is Explicit

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Hypothesis Space This is a HYPOTHESIS SPACE!!! It consists of all the statements that can be constructed from a set of mutually exclusive exhaustive statements. The space is ordered by the ordering relation “implies” We allow concepts like: ‘The fruit is an apple OR a banana!’ while we disallow concepts like: ‘The fruit is an apple AND a banana!’ abc a ba cb c

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Kevin H Knuth 8 July 2007MaxEnt 2007 Superpositions of States??? Consider our piece of fruit… abc a ba cb c

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Kevin H Knuth 8 July 2007MaxEnt 2007 Superpositions of States??? Consider our piece of fruit… abc a ba cb c

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Kevin H Knuth 8 July 2007MaxEnt 2007 Superpositions of States??? Physically, we might agree that the fruit can only be in one of the states represented by the atomic statements. But each of these logical statements can describe my possible state of knowledge about the fruit. WHAT DOES IT MEAN??? abc a ba cb c

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Kevin H Knuth 8 July 2007MaxEnt 2007 What is your State of Knowledge?

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Kevin H Knuth 8 July 2007MaxEnt 2007 Collapse! If you learn that the fruit is not a Banana…

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Kevin H Knuth 8 July 2007MaxEnt 2007 Collapse!

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Kevin H Knuth 8 July 2007MaxEnt 2007 Collapse!

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Kevin H Knuth 8 July 2007MaxEnt 2007 Collapse!

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Kevin H Knuth 8 July 2007MaxEnt 2007 Collapse!

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Kevin H Knuth 8 July 2007MaxEnt 2007 Collapse!

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Kevin H Knuth 8 July 2007MaxEnt 2007 Collapse!

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Kevin H Knuth 8 July 2007MaxEnt 2007 The New Hypothesis Space

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Kevin H Knuth 8 July 2007MaxEnt 2007 It is an Orange!

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Kevin H Knuth 8 July 2007MaxEnt 2007 It is an Orange!

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Kevin H Knuth 8 July 2007MaxEnt 2007 It is an Orange!

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Kevin H Knuth 8 July 2007MaxEnt 2007 It is an Orange!

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Kevin H Knuth 8 July 2007MaxEnt 2007 It is an Orange!

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Kevin H Knuth 8 July 2007MaxEnt 2007 It is an Orange!

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Kevin H Knuth 8 July 2007MaxEnt 2007 It is an Orange!

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Kevin H Knuth 8 July 2007MaxEnt 2007 It is an Orange!

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Final Hypothesis Space

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Kevin H Knuth 8 July 2007MaxEnt 2007 The State of the Fruit This was your initial state of knowledge. The fruit was never in this state! This was the state of the fruit.

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Kevin H Knuth 8 July 2007MaxEnt 2007 Two Spaces State Space of the Fruit State Space of Your Knowledge about the State of the Fruit

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Kevin H Knuth 8 July 2007MaxEnt 2007 Microstates Consider a system with 3 microstates i each of energy E. 11 22 33

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Kevin H Knuth 8 July 2007MaxEnt 2007 State of Knowledge about Microstates Consider a system with 3 microstates i each of energy E. We can’t know which microstate the system is in. We only know that it is in one of the 3 microstates. 11 22 33 1 21 2 1 31 3 2 32 3 1 2 31 2 3

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Kevin H Knuth 8 July 2007MaxEnt 2007 Macrostates This is OUR STATE OF KNOWLEDGE! 11 22 33 1 21 2 1 31 3 2 32 3 1 2 31 2 3

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Kevin H Knuth 8 July 2007MaxEnt 2007 Macrostates This is what we call the MACROSTATE It is an equivalence class of microstates But if we agree that microstates are the physical states of the system, then we must admit that the system is never in the macrostate. Why? These are two separate spaces! 11 22 33 1 21 2 1 31 3 2 32 3 1 2 31 2 3

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Kevin H Knuth 8 July 2007MaxEnt 2007 States of Knowledge 11 22 33 1 21 2 1 31 3 2 32 3 1 2 31 2 3 Statistical Mechanics works in the space representing our STATE OF KNOWLEDGE NOT the state of the system! This is why it is an inferential theory that depends on measures such as probability and entropy.

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Kevin H Knuth 8 July 2007MaxEnt 2007 Mind Projection Fallacy abc a ba cb c Jaynes’ Mind Projection Fallacy warns that it is fallacious to view our state of knowledge about a system as a property possessed by the system. Which begs the question: Since we work with hypothesis spaces, is there such a thing as a STATE SPACE?

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Kevin H Knuth 8 July 2007MaxEnt 2007 Mind Projection Fallacy abc a ba cb c Jaynes’ Mind Projection Fallacy warns that it is fallacious to view our state of knowledge about a system as a property possessed by the system. Which begs the question: Since we work with hypothesis spaces, is there such a thing as a STATE SPACE? ARE THE LAWS OF PHYSICS ACTUALLY RULES OF INFERENCE?

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Kevin H Knuth 8 July 2007MaxEnt 2007 Mind Projection Fallacy abc a ba cb c Jaynes’ Mind Projection Fallacy warns that it is fallacious to view our state of knowledge about a system as a property possessed by the system. Which begs the question: Since we work with hypothesis spaces, is there such a thing as a STATE SPACE? ARE THE LAWS OF PHYSICS ACTUALLY RULES OF INFERENCE? As Carlos Rodriguez once asked: “ARE WE CRUISING A HYPOTHESIS SPACE?”

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Kevin H Knuth 8 July 2007MaxEnt 2007 Two Spaces State Space of the Fruit State Space of Your Knowledge about the State of the Fruit The State Space is a MODEL that generates the HYPOTHESIS SPACE. We make inferences in the HYPOTHESIS SPACE

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Kevin H Knuth 8 July 2007MaxEnt 2007 Intermission

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Measures

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Kevin H Knuth 8 July 2007MaxEnt 2007 Inclusion and the Zeta Function The Zeta function encodes inclusion on the lattice. abc a ba cb c

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Kevin H Knuth 8 July 2007MaxEnt 2007 Inclusion and the Zeta Function abc a ba cb c since The Zeta function encodes inclusion on the lattice.

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Zeta Function abc a ba cb c abc avbavbavcavcbvcbvc T a b c avbavb avcavc bvcbvc T

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Kevin H Knuth 8 July 2007MaxEnt 2007 Inclusion and the Zeta Function The Zeta function encodes inclusion on the lattice. We can define its dual by flipping around the ordering relation

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Kevin H Knuth 8 July 2007MaxEnt 2007 Degrees of Inclusion and Z We generalize the dual of the Zeta function to the function z

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Kevin H Knuth 8 July 2007MaxEnt 2007 Z The function z Continues to encode inclusion, but has generalized the concept to degrees of inclusion. In the lattice of logical statements ordered by implies, this function describes degrees of implication.

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Kevin H Knuth 8 July 2007MaxEnt 2007 How do we Assign Values to z? Are all of the values of the function z arbitrary? Or are there constraints? Here there be monsters… abc avbavbavcavcbvcbvc T a 1100??0? b 1010?0?? c 10010??? avbavb 11101??? avcavc 1101?1?? bvcbvc 1011??1? T

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Kevin H Knuth 8 July 2007MaxEnt 2007 Lattice Structure Imposes Constraints Consider Associativity of the Join We begin by considering the special case where Instead of talking about the function z, consider a related valuation Consistency would require that the valuation of the join of two elements be related to the valuations of the individual elements: Following the inspiration of R.T. Cox (1947) and A. Caticha (1998):

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Kevin H Knuth 8 July 2007MaxEnt 2007 Lattice Structure Imposes Constraints Consider Associativity of the Join We begin by considering the special case where Instead of talking about the function z, consider a related valuation Consistency would require that valuation of the join of two elements be related to the valuations of the individual elements via some unknown function S

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Kevin H Knuth 8 July 2007MaxEnt 2007 Consistency Now join a new element c to the previous pair: But by Associativity, this is also equal to So

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Kevin H Knuth 8 July 2007MaxEnt 2007 Our functional equation for S becomes: This is known as the Associativity Equation Simplify Notation Let

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Kevin H Knuth 8 July 2007MaxEnt 2007 Solution The general solution (Aczel 1966) is: For arbitrary function f which gives The fact that f is arbitrary, suggests there is a convenient representation (Caticha 1998). So define:

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Associativity Constraint The Associativity Constraint requires that when In general, I have shown that

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Kevin H Knuth 8 July 2007MaxEnt 2007 Inclusion-Exclusion (The Sum Rule) The Sum Rule for Lattices

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Kevin H Knuth 8 July 2007MaxEnt 2007 Inclusion-Exclusion (The Sum Rule) The Sum Rule for Probability

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Kevin H Knuth 8 July 2007MaxEnt 2007 Inclusion-Exclusion (The Sum Rule) Definition of Mutual Information

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Kevin H Knuth 8 July 2007MaxEnt 2007 Inclusion-Exclusion (The Sum Rule) Polya’s Min-Max Rule for Integers

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Kevin H Knuth 8 July 2007MaxEnt 2007 Inclusion-Exclusion (The Sum Rule) This is intimately related to the Möbius function for the lattice, which is related to the Zeta function.

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Kevin H Knuth 8 July 2007MaxEnt 2007 Lattice Structure Imposes Constraints I showed that in “general”: Associativity leads to a Sum Rule… Distributivity leads to a Product Rule… Commutivity leads to Bayes Theorem…

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Kevin H Knuth 8 July 2007MaxEnt 2007 Unification The Sum Rule, Product Rule and Bayes Theorem are CONSTRAINT EQUATIONS These are UNIFYING concepts whereas axiomatics are obfuscating

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Kevin H Knuth 8 July 2007MaxEnt 2007 Probability Changing notation The MEANING of p(x|y) is made explicit via the Zeta function, which encodes implication. These are degrees of implication! The meaning is imposed by the ordering relation. - No Guesswork - No Confusion

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Kevin H Knuth 8 July 2007MaxEnt 2007 Priors The prior probabilities are the Zeta function values that remain unconstrained after we take into account the lattice structure. How to assign them??? Find other constraints relevant to the application. It is a good thing that these priors are unconstrained by the lattice structure. If they weren’t, this formalism would not be useful.

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Kevin H Knuth 8 July 2007MaxEnt 2007 How to Derive a Measure Knuth K.H Deriving laws from ordering relations. 1.Define the objects that you want to measure 2.Select the appropriate ordering relation 3.Determine the algebra from the lattice/poset structure 4.Use the constraints from the algebra to derive a calculus 5.The meaning of the measure is imposed by the ordering relation. Skipping steps by relying on intuition will most likely result in errors and waste time and effort!!!

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Kevin H Knuth 8 July 2007MaxEnt 2007 Geometric Probability Many geometric laws can be derived from order-theoretic considerations. Geometric objects can be ordered, conjoined and disjoined often resulting in a distributive lattice structure. Valuations are assigned, which are invariant with respect to Euclidean translations and rotations. Gian-Carlo Rota

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Kevin H Knuth 8 July 2007MaxEnt 2007 Joining Parallelotopes

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Kevin H Knuth 8 July 2007MaxEnt 2007 These Valuations have a Basis! For three-dimensional Euclidean geometry all invariant valuations can be written as a linear combination of 4 basis valuations. V = volume A = surface area W = mean width = Euler characteristic = aV + bA + cW + d

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Euler characteristic is a valuation. For a 3D tetrahedron it is found by Euler Characteristic

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Kevin H Knuth 8 July 2007MaxEnt 2007 What is the Ordering Behind Quantum Mechanics?

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Kevin H Knuth 8 July 2007MaxEnt 2007 Ariel Caticha Ariel has also developed a very interesting derivation of quantum mechanics using Cox’s method applied to experimental setups rather than logical statements. The concept of consistency with the order-theoretic structure is central here as well.

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Kevin H Knuth 8 July 2007MaxEnt 2007 Particles and Motion time A particle moves from x i to x f

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Kevin H Knuth 8 July 2007MaxEnt 2007 A Little More Complex time A particle moves from x i to x 1 and then from x 1 to x f

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Kevin H Knuth 8 July 2007MaxEnt 2007 A Little More Complex time A particle goes from x i to x f via x 1 or x’ 1

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Kevin H Knuth 8 July 2007MaxEnt 2007 Experimental Setups time We can look at this experimental setup as

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Meet Operation time We can look at this experimental setup as being a combination of two setups…

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Meet Operation time We can look at this experimental setup as being a combination of two setups…

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Join Operation time This is a different way to combine setups

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Join Operation

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Kevin H Knuth 8 July 2007MaxEnt 2007 NOT a Lattice Structure What is interesting about setups is that because not all meets and joins exist, setups do not form a lattice structure. They do form something like a poset however. As the measure we will define is not probability, Ariel represented it with rather than So lets continue…

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Kevin H Knuth 8 July 2007MaxEnt 2007 Sum and Product Rules Again Caticha showed that the Sum Rule is derived from Associativity of the Join. Product Rule from Distributivity. Feynman Path Integrals arise from the Sum Rule BUT WHY COMPLEX NUMBERS?!?!?!

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Join Operation time This is a different way to combine setups

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Join Operation Double-Slit Single-Slit

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Top Clearly, if we keep joining slit experiments together this way, we will eventually, have one giant slit. This is the Top element of the poset. It represents the free particle traveling from x i to x f.

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Bottom The Bottom is the completely obstructed particle.

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Kevin H Knuth 8 July 2007MaxEnt 2007 The Atomic Elements The Atomic Elements are simple paths.

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Kevin H Knuth 8 July 2007MaxEnt 2007 A Peek at the “Poset” This is merely a portion of the “poset” of experimental setups for the slit experiment case. It is NOT a Lattice! It is NOT Boolean! It is NOT Commutative! I have said nothing about the necessity of complex numbers

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Kevin H Knuth 8 July 2007MaxEnt 2007 Some Lessons Volumes are NOT Surface Areas Even though they are measures on the same set! There may be multiple distinct measures for any given poset. Measures on Different Posets are NOT the Same Quantum Mechanics is NOT Probability Theory! First order of business… Get Your Ducks in a Row!!!

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Special Thanks to: John Skilling Ariel Caticha Philip Goyal Steve Gull Carlos Rodriguez for many insightful discussions and comments And to: For all sorts of images

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