Probability and Statistics

[Boof-head]

Does anyone here understand what "fair or balanced" means?

Why is there a difference, and how is this determined, or why?

[freeztar]

Does anyone here understand what "fair or balanced" means?

Why is there a difference, and how is this determined, or why?

 

Can you elaborate please? I'm having a hard time relating your post to the subject title.

[Boof-head]

My grasp of it is:

 

A fair sample (statistical) corresponds to a balanced probabiity of finding the sample.

 

Or, a fair coin is one with two different faces; an unfair coin might have two identical faces.

A balanced probability then exists, that tossing the 'fair' coin (some arbitrary number of times) will give a distribution over the two possible outcomes (sampling the coin).

 

Or, maybe I have this wrong...

[Boof-head]

What this thread is aiming at, is in fact the Hadamard transform of Pauli operators.

 

That's in quantum circuits.

 

I can explain this in terms of matrix algebra, but want a 'deeper' grasp of what it does and why.

[CraigD]

Does anyone here understand what "fair or balanced" means?

Why is there a difference, and how is this determined, or why?

In statistic, I believe “fair” is an informal term that is applied as an adjective to a real-world observable to mean “matches a given distribution function satisfactorily”. For example, a “fair coin” means a physical object and a technique of using it that matches a discrete distribution {0:0.5, 1:0.5} such that, for some use, bias is not detectable.

 

I’ve not heard the term “balanced” used in statistics as a synonym for “fair”. It’s sometimes used in a very vague way, as an anecdotal urging to design sampling techniques well via such methods as randomization or stratification. For example, we would say that a survey designed to estimate the number of people in a country who approve of an elected leader that surveys only members of that leaders political campaign staff, or only members of an opponent’s staff, is not “balanced”. Balance has a further nuance, I believe, that hints at the idea of sampling from “extreme” sub-populations as in the last example, which one could consider a very crude form of stratified sampling.

 

Graph theory uses the term “balanced” a lot to categorize various sorts of graphs (eg: “a balanced b-tree”), but I’m guessing from the thread’s title and a later post that this isn’t the sense Boof-head means.

What this thread is aiming at, is in fact the Hadamard transform of Pauli operators.

 

That's in quantum circuits.

 

I can explain this in terms of matrix algebra, but want a 'deeper' grasp of what it does and why.

I’d love even a superficial introduction to “Hadamard transform of Pauli operators” and quantum circuits, so please expand on these tantalizing tid-bits! :)

[Boof-head]

There's a online course or three; Seth Lloyd at MIT has a good one, it's easy to follow, and as he says, it's "just matrix multiplication and addition".

 

The way I visualise it is the Hadamard is 'outside' the unit circle; (1,1) and (-1,1) are like orthogonal axes for the transform, which is equivalent to a Euclidean rotation (in the circle).

[CraigD]

There's a online course or three; Seth Lloyd at MIT has a good one, it's easy to follow, and as he says, it's "just matrix multiplication and addition".

Got a URL (a link) for this, Boof-head? :shrug: I wasn’t able to turn up one with a few minutes of searching. :phones:

 

As a weird aside, “Seth Lloyd”, who I’m vaguely aware of from reading reviews and discussions of his 2006 book “Programming the Universe”, is almost my age (b 8/2/1960 vs. my 4/26/1960), and went to my sister high school. The similarities seem to end there, as about the time I embarked on a series of career-ruinous misadventures, he established a good academic career, after which the similarities increasingly diverged. Though I’m an IT backwater procedural coder vs. Loyd’s quatum computing guruhood, I’m of a like mind, I think, about the universe being a computation, though far less optimistic that quantum computers have the practical potential Loyd believes they do.

[Boof-head]

Lloyd and Shor's online stuff:

2.111 Quantum Computation

18.435 - Quantum Computation

[Boof-head]

OK, on to the guts, as it were.

The probability/statistics in quantum systems is quite different to classical.

Lec. 8 in Dr, Lloyd's notes, introduces the concept using the usual coin-tossing model - this is an obvious kind of connection to a spin-1/2 particle, since we know how to 'flip' their spin states or prepare them in a superposition.

 

QM superposition is fundamentally different to classical discrete 'bits' (or coins); a quantum coin can have 'inner' values anywhere from 0 to 1 (heads to tails), a classical coin is always one face or the other.

 

State preparation is easier with a 2-level system (two discrete qbits); the Hadamard transformation is key to understanding the rotations (Euclidean), and how quantum statistics/probability evolve.

Dr. Lloyd says: "determining if a quantum function is fair (constant) or balanced requires only one measurement, a classical function requires two".

 

This more or less encapsulates the big difference between a 2-level qbit system and classical bits (which have to be measured many times, or at least twice, depending if you are determining if a sample is fair or balanced).

[Boof-head]

Just to give a better idea of why the IS angle is a smoother ride. You need to know undergrad calculus and algebra; it borrows from QFT and QCD, in fact from any aspect of QM that means a physical system can be realised.

 

The path has been blazed already (through the theoretical forest), so there are plenty of experts and QIS is essentially the collation of those parts of QM that are in the problem domain of: "build a functional quantum device".

You can do this with two laser pointers and a screen and some polarising filters; you can put these filters which are operators that rotate a quantum state, between you and most of the visible light that you see.

There is also topology and group theory, the theory of numbers, unitary and monadic groups, yada yada.

 

But you don't need to be able to grasp it any much differently or deeply (although some deep stuff is in there) than a Rubik's cube.

 

Which means in not so many words, going from rotating the sides of a Rubik's cube (or R-cube for short), to a set of rotations that rotate a corner's colours (a map) and so reversing these rotations in a way that restores the corner - a unitary transform of the map.

 

Then this unitary operator = 'a set of geometric rotations in Euclidean space' (and time of course) is in the group of 'unitary transforms in the group of transforms' in or on the R-cube's space or surface = a manifold in Euclidean space.

Which is a classical phase space of colour maps.

 

How this ties to probabilities and statistics is: a fair R-cube is one coloured so it has a solution = all 6 sides different colours (no mixing. a pure state) = a map over the phase-space.

Balanced is, I guess, how many combinations lead to a solution against how many that don't; both are very large (but finite) numbers.

[Boof-head]

Sorry if this is like, carrying on regardless; and there is still the matter of Pauli operators, but here's the thing - I'm about to lose my broadband so I need to arrange a new provider and I might be moving places soon so who knows.

 

So, back to classical phase spaces, color-maps etc.

Rubik, by making his cube - an architectural solution to finding a spherical path 'inside' a cube for a corner of the cube (??), demonstrated that we use spherical and 'flat' geometries to build, well, anything.

If this is not the case, then we can build a device of some kind that uses a different set of curves, and in a non-Euclidean space, but let's not go there just now.

 

An R-cube, in its default or original version, is 3x3 sub-squares per face = 2x2x2 sub-cubes as 2x2 per face, each sub-cube c, over the n-cube C has 2 hidden colors if only one face is viewed, where 'view' is a Euclidean rotation in 6 degrees of freedom.

 

An R-cube is a 3-cube, which implies an n-cube (or R(N)-cube) where any sub-cube c is always 3-colored.

There are a finite number of color schemes, if 6 constant colors or hues h1, h2, ..., h6 are plied or mapped = a graph in logic-space L(h); L also contains all rotations available in the color phase-space, including those which have no conventional solution as a 'constant-color' map over the space.

The R(N)-cube implies a minimum N, which must be positive.

A 1-cube can only have fixed maps in L(h), but has no dynamic rotations in C, the color-space, they are a 'Laplacian' in the space.

The minimum cube in R is a 2-cube; this is a tensor in the color-map phase-space.

 

The logic is complete or affine - the only way to rotate a corner without rotating its opposite (a 'hidden' color) is to dismantle a pure cube or color it in a 'non-affine' way.

A 2-cube is a sigma-complete algebra, it represents a Pauli operator, when only 2 color charges are applied.

 

So this from CraigD is actually quite relevant::

Balance has a further nuance, I believe, that hints at the idea of sampling from “extreme” sub-populations as in the last example, which one could consider a very crude form of stratified sampling.

 

Graph theory uses the term “balanced” a lot to categorize various sorts of graphs (eg: “a balanced b-tree”),

[Boof-head]

So now, when your kid asks what probility and stistics have to do with logic, you can pick up a toy and explain it; you can also pick up a spherical one, the color phase-space is different on it, it explains a few more things; in this case they're tied to the geometry in the inner surface below the color one (it's color-independent) which is a lattice over a sphere; in the cube puzzle it's joined to the pieces, so is color-dependent but 'invisible'.

 

In the 'round' puzzle, the pieces are vectors over the color space, the numbers have an orientation, in the cube the orientation is corner-dependent, on the sphere you have to 'make' a corner by removing a piece - you 'puncture' the logic-space L(h).

[Boof-head]

So, on to the math, the language kids have to learn to be able to play with bigger toys (like say, the LHC).

The first obvious thing about the two Rubik toys is the cube is made of planar sections through a sphere; the R-sphere is made of triangular sections over a spher(ical lattic)e, each side of which equilateral figure or piece is a geodesic over G, which is in S^2, where G is the rotation-group of color algebras over the sphere, for the default R(ubik's-rotation)-sphere this is h1,h2,h3,h4 a quadruple, geometrically there is a (3,2) rotation in that the hole can have 3 colors rotated into it and each piece can rotate 2 ways; the hole has to choose 1 of 2 x 3 color-sides. The color-map is plied according to a different L(h) in each case, and a map exists from L{R(4)} to L{R(6)} in C, the color-space, which is therefore universal for the cube and sphere.

 

There is a fiber for the space, of phase rotations; the algebras correspond to a general space of vectors; the simplest is a 2-color phase-space. So the 2x2 cube would have a color-phase described by 2 independent (i.e. polarized, or orthogonal in the space) colors c, or C(a,:), which have a phase p = a|0> + b|1>, a sigma-finite algebra, which is complete if it describes a set of affine operations (i.e. color-preserving), which are Abelian, or the fiber has a different curvature.

 

This is tied to chaos and the way combinations of 6 colors over 2x2, 3x3, ..., nxn cubes in R can diverge (and be outside) over a geometric surface as sets of affine rotations or operators in the color-space. It's about rotational symmetries and connected spaces. The connection which leads to a general, topologically-invariant 'fiber' is a color-algebra with a curvature for each connection.

[Boof-head]

From here, we can diverge from the game of rotating layers, or faces F of a cube and scrambling the colors (from a fixed or solved state S) over the faces, to Newtonian phase space, because he was the first to use a classical particle with mass = a planet or comet, and calculus to gauge the inertial system (our solar one which condensed from a big cloud), as a fixed, large solar mass, wrt background points of reference on the inner surface, of what he considered was a sphere in Euclidean polynomial space.

 

This space, he found, was enumerable with the algebra he constructed using Kepler's and Galileo's geometry and algebra; his phase-space was of gravitational acceleration over a time and distance surface, and he imagined fixed stars (a lattice) in a universe that ran like a clock, though could not descry where was the 'spring' or pendulum that started it, and kept it all running.

 

Newton decided this was ultimately the first cause; it was his God or Lord of the universal domain - each star maintaining its fixed position faithfully according to the Laws which His divine nature had revealed to man's mortal reason (it was the Renaissance, after all), had he known about the connection between charge, magnetism and visible radiation (and heat, of course), and been able to make the leap Planck made centuries later; well who knows.

He didn't so did the best he could given the paradigm and what he was able to publish as irrefutable logic.

 

So the charge in his space, or charges, are gravitational, in the C-space of Rubik's puzzles (games, manifolds, geometric devices with algebraic solutions), the velocities and 'stationary states' are based on a color basis. But there is a map, you can find a logic that transforms gravity into a tensor and 3 different colors into velocity (with an initial time and space vector) and momentum.

 

A phase-space is topological, it has a fiber; quantum spaces and the infinite in extent spaces of gravity and charge (and spin), are gauged by the same rotation groups, SU(2) and U(1); vector spaces are a general model of physical and logical (i.e. problem) spaces.

 

We have computers measuring distant objects to determine algorithmically if their solar orbits correspond to any future earth orbits - a solution in an inertial space of rotations R, with a group G, and an algebra A which Avogadro gauged a while ago.

 

Color relativity on puzzles and the general relativity of space and time are connected - this is the same idea as Alan Turing had, we can use a logic to build something, then use that to build something else - every design is a transformation from one algebra to another; machines build more machines; this is directly reflected in the design rule-of-thumb, that generally you use the newest design to design its successor, to improve the logic, or compress functionality into a smaller space, and make it clock faster.

[Qfwfq]

Gosh it would be hard to follow your gymnastics, even if I had more time!

 

How this ties to probabilities and statistics is: a fair R-cube is one coloured so it has a solution = all 6 sides different colours (no mixing. a pure state) = a map over the phase-space.

Balanced is, I guess, how many combinations lead to a solution against how many that don't; both are very large (but finite) numbers.

Also, back when I was playing with the cube, I realised a cube could be unfair after having been taken apart and wrongly put back together.

 

A fair die is one for wich equiprobability applies, a weighted die is one with which numbers have different odds (instead of 1 in 6 each).

[CraigD]

So now, when your kid asks what probility and stistics have to do with logic, you can pick up a toy and explain it; you can also pick up a spherical one, the color phase-space is different on it

…

in the cube puzzle it's joined to the pieces, so is color-dependent but 'invisible'.

Toys can be very useful for explaining math. However, I don’t think Rubic’s cubes and similar puzzles are as useful for explaining probability and statistics as are dice and coins.

 

Probability is meaningful when something about a system is unknown – in game theory, a condition known as imperfect knowledge. Imperfect knowledge applies to the most common game people play with puzzles like Rubic’s cubes, “solve the cube”, in which a “pristine” (all labels on the same face the same, no labels on different faces the same) cube is “scrambled” by moving the faces a few times in a hard-to-predict way without allowing the player to see the moves, then requiring the player to move the faces to change it back to a pristine cube, because if the player had perfect knowledge – that is, was allowed carefully watch and note how the cube was scrambled, he could solve it in a few moves by performing the scrambling moves in reverse. Because the player lacks this knowledge, he instead makes different moves, usually many more than were used to scramble the cube, to solve it.

 

Even though knowledge is imperfect in this game, probability isn’t very useful, because the probabilities of each face being rotated during scrambling are about equal, and the number of ways in which they can be picked is very large. Even knowing the exact probability of a given face rotation occurring during scrambling, if the player were to precisely calculate a guess at the most probable way the cube was scrambled, and reverse it, the probability of that solving the cube is low. Probability, therefore, isn’t a very useful idea for playing “solve the cube”.

 

Statistics are data about a system less than all of the data about the system. Several statistics about a cube are commonly used, “flippancy” being in my experience most common. A pristine cube has a corner flippancy of zero. A cube with a single corner unsolved is has a flippancy of plus or minus 1. “Solve the cube” is almost always played with a cube with both corner and edge flippancy zero. Other flippancies, which require either disassembling or forcing the pieces of the puzzle, are usually considered “cheating” on the part of the scrambler. Statistic, therefore, aren’t very useful in playing “solve the cube”.

 

Mathematically, there are many ways to describe a cube. My favorite is to describe it as an ordered sequence (a tuple), where each member represents on of the paper or painted labels on a physical cube puzzle. This is useful, because every possible state of an “not cheating” cube can be considered a theorem – a true statement – based on the description of the pristine cube (its identity tuple), and three given theorems describing the result of the possible physical rotations and turning of a single face of the entire pristine cube, for example, for a 2x2x2 cube (a cube with corners only), the identity 24-tuple is

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24 (identity)

the tuple representing turning a single face is

1,2,8,6,5,21,7,22,11,9,12,10,3,14,4,16,17,18,19,20,15,13,23,24 (face)

and the tuples representing turning the entire cube are

2,4,1,3,17,18,19,20,5,6,7,8,9,10,11,12,13,14,15,16,23,21,24,22 (horizontal)

20,19,18,17,7,5,8,6,1,2,3,4,14,16,13,15,24,23,22,21,9,10,11,12 (vertical).

 

Any theorem tuple can be applied to any other by rearranging the applied-to tuple in the same way the applied tuple affects the identity tuple. This can be used to generate all of the theorems of the formal system given by this identity and three theorems, which are all of the possible states of a physical 2x2x2 cube puzzle. For example, identity*face*horizontal*vertical is

16,4,14,3,19,17,20,18,2,6,1,8,9,10,11,12,13,24,15,23,5,21,7,22.

 

I find this way of describing “permutation puzzles” like the cube easier and more useful than a way using language like Boof-head’s in post #12 and 13. It has additional value in that it’s simple to use to write computer programs to play cube games, can easily be applied to many different kinds of puzzles in addition to 2x2x2, 3x3x3, 4x4x4 ect. Rubic cubes, and can be used to describe interesting puzzles that are practically impossible to physically make. A single interactive program can be used to play any cube-like puzzle, given it’s initial given theorems. Here’s the actual MUMPS code to such a program, the one I used mostly to play with cube puzzles, and an example of it being used to play a 2x2x2 cube:

n (XRUBCUBE,RC,M,W) m M=RC("M") s R="" f  x XRUBCUBE(2):R'="?",XRUBCUBE(3) q:R=""  ;XRUBCUBE(1): interactive Rubic's cube display & read command
n (RC,W) s X="" f Y=1:1:$o(W(""),-1) f  s X=$o(W(Y,X)) w:X="" ! q:X=""  w ?X-1,$e(RC,W(Y,X)) ;XRUBCUBE(2): display
n (XRUBCUBE,RC,M,R) r "XRUBCUBE>",R q:R=""  X XRUBCUBE(3,2),XRUBCUBE(3,3),XRUBCUBE(3,4) I R'?1"?".e X XRUBCUBE(3,1) f P=1:1:$l(R,",") s E=$p(R,",",P) i E]"" w:'$d(M(E)) " -",E,"?",! q:'$d(M(E))  w:P=1 ! s M=E x XRUBCUBE(4) ;XRUBCUBE(3)
x XRUBCUBE(3,1,1) i I="" s R=$tr(R,",") f I=$l(R):-1:2 s $e(R,I)=","_$e(R,I) ;XRUBCUBE(3,1): allow undelimited
s I="" f  s I=$o(M(I)) q:$l(I)>1!'$l(I) ;XRUBCUBE(3,1,1)
i R="?" w !,"This application simulates a particular kind of Rubic's cube.",!,"To perform an operation, enter one or more valid opcodes, separated by commas.",!,"Valid opcodes are: " x XRUBCUBE(3,2,1) w !,"To define or redefine an opcode, enter an opcode followed by = followed by one",!,"or more valid opcodes, separated by commas. Defined opcodes are: " x XRUBCUBE(3,2,2) w !,"To reset the cube, enter !.  To scramble it, enter *.  To exit, press Enter.",! ;XRUBCUBE(3,2)
n (M) s (I,D)="" f  s I=$o(M(I)) q:I=""  w D,I s D="," ;XRUBCUBE(3,2,1)
n (M) s (I,D)="" f  s I=$o(M(I)) q:I=""  i $d(M(I))=1,$L(M(I)) s E=I_"="_M(I) w:$l(D_E)+$x>79 ! s:'$x D="" w D,E s D="; " ;XRUBCUBE(3,2,2)
n (XRUBCUBE,RC,M,R) i R?1.e1"=".e s A=$p(R,"="),B=$p(R,"=",2,3) x XRUBCUBE(3,3,1),XRUBCUBE(3,3,2) W ! i P s M(A)=B,RC("M",A)=B,R="?" ;XRUBCUBE(3,3): define macro
n (XRUBCUBE,:evil: s R=B x XRUBCUBE(3,1) s B=R ;XRUBCUBE(3,3,1)
s:$d(M(A))>9 B=A_"=" s P=1 i B]"" f P=1:1:$l(B,",") s E=$p(B,",",P) i $s(E="":1,$d(M(E)):0,1:1) w " -",E,"?" s P=0 q  ;XRUBCUBE(3,3,2)
i R?1(1"!",1"*".e) x XRUBCUBE(3,4,2):R="!",XRUBCUBE(3,4,1):R?1"*".e ;XRUBCUBE(3,4)
n (XRUBCUBE,RC,M,R) x XRUBCUBE(3,4,1,1) s I=$e(R,2,9),R="" f I=1:1:$s(I>0:I,1:30) s R=R_$s(R="":"",1:",")_$p(A,",",$r($l(A,","))+1) ;XRUBCUBE(3,4,1): scramble
s (A,I)="" f  s I=$o(M(I)) q:I=""  s:$d(M(I))>9 A=A_$s(A="":"",1:",")_I ;XRUBCUBE(3,4,1,1)
s RC=RC(0),R="?!" w ! ;XRUBCUBE(3,4,2): clear
n (XRUBCUBE,RC,M) x XRUBCUBE(4,$s($d(M(M))<9:2,1:1)) ;XRUBCUBE(4): move (w/recursive macros)
s PC=RC,P="" f  s P=$o(M(M,P)) q:P=""  s $e(RC,M(M,P))=$e(PC,P) ;XRUBCUBE(4,1): move
n MR,P s MR=M(M) f P=1:1:$l(MR,",") s M=$p(MR,",",P) x XRUBCUBE(4) ;XRUBCUBE(4,2): macro
x XRUBCUBE2(1,1) s:$s($l($g(RC))-24:1,$tr(RC,123456)="":0,1:1) RC=RC(0) ;XRUBCUBE2(1): initialize 2x2x2 cube
n (RC) s RC(0)="" f I=1:1:6 s RC(0)=RC(0)_$tr($j("",4)," ",I) ;XRUBCUBE2(1,1)
n (XRUBCUBE2,M) s M="" f  s M=$o(XRUBCUBE2(3,1,M)) q:M=""  s A=$p(XRUBCUBE2(3,1,M)," "),(I,J)="" f P=1:1:$l(A,",") s LI=I,LJ=J,E=$p(A,",",P),I=$p(E,"-"),J=$p(E,"-",2) s:'I I=LI+1 s:'J J=LJ+1 s M(M,I)=J ;XRUBCUBE2(3): minimal moves
3-13,4-15,13-22,15-21,22-8,21-6,8-3,6-4,9-10,10-12,12-11,11-9 ;XRUBCUBE2(3,1,"+")
5-9,,,,9-13,,,,13-17,,,,17-5,,,,1-3,2-1,4-2,3-4,21-22,22-24,24-23,23-21 ;XRUBCUBE2(3,1,"H")
1-9,,,,9-21,,,,24-17,23,22,21,20-1,19,18,17,5-6,6-8,8-7,7-5,13-15,14-13,16-14,15-16 ;XRUBCUBE2(3,1,"V")
n (XRUBCUBE,XRUBCUBE2,XRUBCUBE2B,RC) x XRUBCUBE2(1),XRUBCUBE2B(2),XRUBCUBE2(3),XRUBCUBE(1) ;XRUBCUBE2B: 2x2x2 Rubic's cube simulator, basic view
n (W) s C=0 f A=104,301,304,307,310,504 f Y=A100:1:A100+1 f X=A#100:1:A#100+1 s (C,W(Y,X))=C+1 ;XRUBCUBE2B(2): display map

x XRUBCUBE2B
  11
  11
22 33 44 55
22 33 44 55
  66
  66
XRUBCUBE>*
  11
  33
36 44 15 22
22 36 44 15
  65
  65
XRUBCUBE>?
This application simulates a particular kind of Rubic's cube.
To perform an operation, enter one or more valid opcodes, separated by commas.
Valid opcodes are: +,H,V
To define or redefine an opcode, enter an opcode followed by = followed by one
or more valid opcodes, separated by commas. Defined opcodes are:
To reset the cube, enter !.  To scramble it, enter *.  To exit, press Enter.
XRUBCUBE>

PS: Boof-head, I think more people would be able to understand your posts if you linked the terms you used to standard references, and explained them more. Notice that the "Reply to Thread" feature has not only an "Insert Link" button and corresponding markup, but a "wrap [WIKI] tags around selected text" button that will link common terms to their wikipedia article.

[HydrogenBond]

Say we had a single balance dice, with six sides. All sides are equally likely and have the same odds. Before we throw that die, we have 60 people betting which number will appear on a single throw. Will we get 10 people betting on each number?

 

The answer is probably no. With the odds of the dice staying the same, we could alter the betting conditions, to shift the betting distribution. Say there is a 1% chance of eating Y causing a health problem, and a 99% chance it will not. What is the best bet using probability? Would more than 1% of the people bet on the long shot even if the math says it is a 99-1 long shot?