# Mad-e-matics

Written by Leonard R. Jaffee

Monday, 07 April 2014 01:25

Copyright © 2014 by Leonard R. Jaffee: all rights reserved

Legislative committees, administrative agencies, judges, economists, political survey-takers, pundits, historians, ecologists, sociologists.........all try to support their arguments with math, or pseudo-math, whether statistics, the calculus, or aught else quantitative.

Math can be beautiful as a leaf unfolding or Bach's contrapuntal music or the Krebs cycle or a starfish or a snow flake or high Gothic architecture ( http://hum120.files.wordpress.com/2011/03/ext-cc-mannyyy.jpg ).

But, like even that nude emperor, Western physics, which hallucinated the phony "Big Bang" and cannot manage its own quantum mechanics, math, too, bears critical flaws — inconsistencies, critical insufficiencies, logic-errors that may be corrigible but persist.........

The ultimate problem is that physics & math & logic & all other "objective" fields are, at root, emotional — matters of emotional choice.

Scientists, mathematicians, and logicians choose premises, theorems, axioms (which bear no premise but desire), definitions, observations........that fit their emotional imperatives.

Emotional imperatives?

Craving to make a system "symmetrical" or "consistent" or picturing "the universe" and its phenomena in ways that satisfy a wish of controlling nature or attaining public acclaim. Below follows a demonstration: an aspect of uncertainty of commutativity or noncommutativity of multiplication. The demonstration is just one of myriad available showings.

Witness Western astrophysics's absurd assertion of the Big Bang, which astrophysics has had increasingly to modify, and will continue to need to modify, until the modification-stream shows the theory's absurdity, which simple logic ought to have proven stupidly absurd when the theory was invented.

The Big Bang theory's ultimate source is Western science's emotional need of perceiving that the universe had a beginning & has or will have an end: a limitless universe is a concept unbearable to Western science, which cannot fathom the prospect that anything is beyond its comprehension, measurement, and control.

Simple logic disproves the Western science's perception: If "the" universe suffers limit, either nothing or something is beyond. If something is beyond, limit is illusion. If nothing is beyond, limit is illusion. A nothingness bears no limit — unless a something beyond. Suppose a something beyond. If another nothing is beyond that something, either it is limitless or, if that nothing ends, another something is beyond. And so on, infinitely. Limit is illusion.

This logic works in every real or dreamed direction — for time, too.

If "the" universe had "origin," then earlier "another" universe existed. If "Big Bang" happened, it needed surrounding space — empty till eventually bearing the particles(?) of whatever "banged" (if not also something else). The exploding hyper-dense mass could not materialize magically from naught. Did an earlier contraction occur? Was the explosion the end of just one contraction of many? Did the explosion reverse just one collapse of just one slice of infinity?

If time starts or stops, only counting begins or ends, not existence.

So, lost, the Western physics asks:

Do black holes lead to other universes?

Is "our universe" a black hole of a "meta-universe"?

Do many "universes" exist (all grown from some number of discrete big bangs)?

Can we measure space and time by the speed of light, or does the speed of light fail as a standard because it is not the ultimate speed and speed may have no limit (so that the theory of special relativity is junk and the general relativity theory must be altered to the proposition that our universe's expansion moves faster than the speed of light)?

See, E.G.,

* http://www.newscientist.com/article/dn11745-could-black-holes-be-portals-to-other-universes.html

* http://www.mathpages.com/home/kmath339/kmath339.htm

* http://huffingtonpost.com/2014/03/31/multiverse-gravitational-waves_n_5062376.html?utm_hp_ref=world&ir=World

* http://english.aljazeera.net/news/europe/2011/09/2011922183317431339.html

* http://gizmodo.com/no-stephen-hawking-did-not-say-black-holes-dont-exist-1513870928

* http://www.angelfire.com/az/BIGBANGisWRONG (This source is useful much for referencing the other sources it cites or discusses.)

Those questions imply not only that Big Bang theory is false. They imply also that the universe — the one & only universe — had no beginning & has & will have no end: Though forever expanding & contracting, the universe is, has been, and will be always one, same, eternal.

For thorough deconstruction of the Big Bang theory, see Eric Lerner, The Big Bang Never Happened: A Startling Refutation of the Dominant Theory of the Origin of the Universe (1991).

Like Western physics, math cannot afford to agree completely with REAL reality, even the "reality" of "pure" logic or the "reality" of consistency. Otherwise math would crumble into oblivion.

Witness the case of 0 (zero) factorial. Math theory insists that 0! = 1, despite 0 x N = 0, and 0 x 0 = 0.

How very consistent, symmetrical! If zero factorial equaled zero (as consistency insists), the concept factorial would fail and, with it, much of math that depends on multiplication-streams.

Zero factorial is only one example of just one of many KINDS of problems of consistency of the operation of multiplication (and of math).

Once, while teaching an Evidence Law class treating the matter of whether the law ought to permit Bayesian probability calculus to determine an issue like paternity or identity or projected life-span (of a plaintiff seeking damages for an allegedly life-shortening tort), I asserted the negative (as always I do). Among my many reasons was that statisticians tend to disregard the significance of multiplication-order.

[Side note # 1: Bayesian probability calculus is a rather metamorphic, set-theoretic-cognate extension of Pascalian conditional probability calculus. End of side note # 1]

One student was much math-trained and an actuary. He insisted my argument was nonsense because (he insisted) always multiplication is a "commutative operation."

A commutative operation is one in which the operation's order does not affect the outcome. So, if multiplication is commutative, then never can multiplication order affect multiplication product: Always, A x B x C would equal B x C x A or C x A x B or B x A x C or C x B x A or A x C x B.

Eventually, I proved to my student that his position was wrong for some (maybe many) cases of Bayesian probability calculus (and I proved so without even mentioning the whole fields of substantially, or even utterly, non-commutative multiplication, like scalar multiplication, ring theory & inverse ring theory, or matrix multiplication).

He was very resistant and kept posing irrelevant math-demonstrations that seemed pertinent if perceived in a world of superficial analysis. His world treated Bayesian probability as if it were just an alternative of analytic geometry (which not Descartes, but the Chinese invented, and about 1400 years before Descartes was born).

So, when is multiplication not commutative even where pertinently expert mathematicians suppose or insist, religiously, that it is commutative?

Consider a rather common real kind of case — Bayesian calculation of the paternity probability indicated by genetic "evidence." Take the calculation through ABO blood-type and HLA (human leukocyte antigen) data.

[Side note # 2: I exclude DNA evidence only because two variables suffice for this demonstration's purpose, and a third variable would not alter the essence of the case. Still multiplication order would affect the outcome. End of side note # 2]

Assume, for simplicity, that genetic mutation does not occur: So, the true father's and the infant's HLA types must match; and the infant's ABO type must be either the true other's or the true father's.

[Side note # 3: Mutation-probability is real, but if the case accounted for mutation-probability the outcome would be essentially the same as if the case did not account for mutation-probability. With or without the accounting, multiplication order would affect the outcome. End of side note # 3.]

Suppose both the defendant's and the mother's HLA types "match" the infant's. Infant, mother, and defendant have ABO type O negative.

Apply the "principle of indifference." Near-all statisticians would. Otherwise the operation would be impossible, since a true "prior probability" — in this case, P(F), see below — would be incalculable. [The principle of indifference is a false supposition that if no "prior probability" quantity is obtainable, one may assert, arbitrarily (FALSELY), that the positive & contrapositive prior probabilities are equal (0.5 each) & hallucinate that the posterior probabilities — e.g., P(A∣F)P(H∣A&F), see below — shall have rendered the falsehood insignificant.]

So, per principle of indifference, 0.5 is the (artificial, false) prior probability of paternity and also of non-paternity.

[Side note # 4: The principle of indifference is very like the choice of making zero factorial = 1 (0! = 1). The real and logical truth would crash the system, which is, in essence, partly delusional. So, emotionally, the mathematician (or statistician) denies the true truth and supplies a falsehood that "saves" the system. (Side note # 4 continues, next ¶.)

Cf. Gödel's proof — especially its second inconsistency theorem, but also, implicitly, its first theorem, too. Gödel's inconsistency theorems showed, among else, that "1 + 1 = 2" and many other "obviously true" propositions are unprovable assertions resting on axioms incapable of proving either (a) all statements concerning natural numbers or (b) the consistency of a number-system or arithmetic-system. (Side note # 4 continues, next ¶.)

Not only Bayesian probability calculus suffers the trouble of the principle of indifference. Integral calculus suffers the identical trouble; and that, and other, troubles (like 0! = 1) explain relatively often why bridges & tunnel-ceilings collapse & many transactional outcomes are mis-predicted. (Side note # 4 continues, next ¶.)

Concerning why the principle of indifference cannot save Bayesian probability calculus (or even traditional relative frequentist statistics) from being limited to subjective guesses (excluded from the field of determining actualities or even actual probabilities), see Jaffee, Of Probativity and Probability: Statistics, Scientific Evidence, and the Calculus of Chance At Trial, 46 U. Pittsburgh L. Rev. 925 (1985), and Jaffee, Prior Probability — A Black Hole in the Mathematician's View of the Sufficiency and Weight of Evidence, 9 Cardozo L. Rev. 967 (1988). End of side note # 4.]

The "matching" HLA type occurs in one of every 1000 men. The ABO type (say O negative) occurs in thirteen of every 100.

Among men who have ABO-type O negative blood, eight of 100 have the matching HLA characteristic. Among men who have the matching HLA characteristic, seven of ten have ABO-type O negative. This is very possible. The O negative male subpopulation is immensely greater than that of men who have the matching HLA type. So, the O negative male subpopulation can include many other HLA types of varying relative frequencies. Yet, an HLA type can be very positively correlated to O negative ABO type. [Oddly, reality's like that.]

Let F be paternity.

Let A be ABO-type match of putative father and child.

Let H be HLA "match" of putative father and child.

Let P(F∣A&H) or P(F∣H&A) be probability of paternity given A&H or H&A *

* "(F∣A&H)" means "F 'given' A&H" & "(F∣H&A)" means "F 'given' H&A." "Given" equates with set theory's "in the world of" (as in "C in the world of D"), which set theory, too, calls "given" (as in "C given D"). *

Let P(F) be "prior probability" of paternity.

Let P(H∣F) be probability of HLA match given paternity.

Let P(A∣F) be probability of ABO match given paternity.

Let P(H∣A&F) be probability of HLA match given ABO match and paternity.

Let P(A∣H&F) be probability of ABO match given HLA match and paternity.

Let the contrapositive probability symbols be parallel

[Example: P(not-F)P(A∣not-F)P(H∣A & not-F)]

P(H∣A&F) is 1. Whatever defendant's ABO type, if he is the father, his HLA type must match the infant's.

P(A∣H&F) is .25. [If defendant is the father, the infant must have an HLA attribute that, alone, would make 70% the chance that she is O negative; but an infant has the mother's ABO type 75% more often, and the infant's whole HLA type also may alter the chance that she is O negative.]

[Side note # 5: (a) The mother's actual HLA and ABO types would not affect calculation — not yet. (b) I left P(A∣H&F) at .25, since that could be right and I want to limit disparity to the not-F field. End of side note # 5.]

The conditional contrapositives (the not-F quantities) are governed by random population frequencies. So, E.G., P(A∣not-F) = .13 (since .13 is the random chance of the ABO type), and P(A∣H & not-F) = .08 (since among type O blood men, 8 of 100 have the matching HLA characteristic and that matter equals the random chance of type O given the populational frequency of the HLA characteristic). Likewise, P(H∣A&F) is .7 (since among men having the HLA characteristic, 7 of 10 have type O blood).

The Bayesian equation is:

P(F∣A&H) =

P(F)P(A∣F)P(H∣A&F) ÷

[P(F)P(A∣F)P(H∣A&F) + P(not-F)P(A∣not-F)P(H∣A & not-F)].

Or, alternatively, the equation is:

P(F∣H&A) =

P(F)P(H∣F)P(A∣H&F) ÷

[P(F)P(H∣F)P(A∣H&F) + P(not-F)P(H∣not-F)P(A∣H & not-F)]

— which reverses some aspects of the multiplication order of

P(F∣A&H) =

P(F)P(A∣F)P(H∣A&F) ÷

[P(F)P(A∣F)P(H∣A&F) + P(not-F)P(A∣not-F)P(H∣A & not-F)]

[The alternative puts the P(A) aspects second instead of first.]

If the statistician accounts for ABO type before HLA type, the Bayesian fraction is:

(.5 x .25 x 1) ÷

[(.5 x .25 x 1) + (.5 x .13 x .08)]

or

.125 ÷ (.125 + .0052),

which equals 96%.

But if the statistician accounts for HLA type first, the Bayesian fraction is:

(.5 x 1 x .25) ÷

[(.5 x 1 x .25) + (.5 x .001 x .7)]

or

.125 ÷ (.125 + .00035),

which equals 99.72%.

The not-F field condition-order's reversal altered the outcome — from insignificant to significant (or marginally significant to significant, if, as is uncommon, 95% is at least marginally significant).

You may argue that the order-change changes the quantities multiplied, rather than changes the product of multiplication of the same quantities. But the argument is a matter of perspective, NOT a matter of TRUTH.

One "reason" is that the Bayesian theorem treats P(F∣A&H) as interchangeable with P(F∣H&A), since the theorem holds that H&A = A&H — apparently because (X intersection Y) = (Y intersection X).

So, per the theorem,

P(F∣A&H) =

P(F)P(A∣F)P(H∣A&F) ÷

[P(F)P(A∣F)P(H∣A&F) + P(not-F)P(A∣not-F)P(H∣A & not-F)]

is identical to

P(F∣H&A) =

P(F)P(H∣F)P(A∣H&F) ÷

[P(F)P(H∣F)P(A∣H&F) + P(not-F)P(H∣not-F)P(A∣H & not-F)]

— despite P(A∣H & not-F) does not = P(H∣A & not-F).

The (supposed) "reason" is that P(F∣A&H) = P(F∣H&A) since H&A = A&H.

Another (actual) reason is emotional: love of "symmetry" and "consistency" AND the statistician's not being able to endure (hence the statistician's delusional denial of) the system's actual failures.

Here statisticians (or Bayesianists) appear to feel "consistency" requires that "intersection" means in Bayesian calculus exactly what it means in set theory [in which (X&Y) = (Y&X) and (X&Y&Z) = (Z&Y&X) = (Z&X&Y) = (Y&X&Z) = (Y&Z&X) = (X&Z&Y)], because Bayesian calculus derives from set-theory-like reasoned extension of conditional probability calculus. In set theory, "intersections" bear the "symmetry" of commutativity. Therefore, Bayesian calculus must follow. So, P(X&Y) must equal P(Y&X) [since in set theory, (X&Y) = (Y&X)] — despite in set theory (A&B)∣B = (B&A)∣B but may not = (A&B)∣A [or (B&A)∣A].

A third (actual) reason is fear of having to choose either overwhelmingly intractable complexity or risibly arbitrary avoidance of real inconsistency. If Bayesian buffs admitted the non-commutativity trouble my hypothetical illustrates, then Bayesian calculus would need to

(a) establish a method that tries to (but cannot) account the often infinite inconsistency of consequences of the non-commutativity my hypothetical illustrates

or

(b) establish an indefensible arbitrary (and invalid, immensely unreliable, and very practically dangerous) rule like "the statistician is stuck with whatever happens to be the statistician's first choice of variables-order (a rule just a little more contemnable than the principle of indifference & 0! = 1)

or

(c) trash the system (and lose oodles of money-compensation now obtained by estimating outcome odds for big businesses or financial institutions or giving expert testimony in litigations or......).

A much simpler reason dismisses the argument that the order-change changes the quantities multiplied. The things multiplied are not mere "quantities." They are probability-indicators.

One probability-indicator is the paternity-probability-indicating bearing of the ABO type (the A∣F, and the A∣not-F). The other probability-indicator is the paternity-probability-indicating bearing of the HLA-type (the H∣F and the H∣not-F). The order-change changes only the relative "timing" of integration of the two indicators — the order of the positive or contrapositive indicators' being multiplied with each other and with the relevant prior probability [the P(F) or the P(not-F)].

But we must not miss the elusively obvious reason: If the multiplication-order-change changed the quantities being multiplied, then, FOR THAT REASON, multiplication is non-commutative in the case hypothesized and any like case. The root-and-ultimate point is that multiplication-order alters the consequence of multiplication.

Cf. quaternion multiplication: e.g., (ij = k) BUT (ji = -k). In that pair of equations, the first equation's ij must be somehow different from the ji of the 2nd, since -k cannot = k.

Ain't that fun?

Legislative committees, administrative agencies, judges, economists, political survey-takers, pundits, historians, ecologists, sociologists.........all try to support their arguments with math, or pseudo-math, whether statistics, the calculus, or aught else quantitative.

Math can be beautiful as a leaf unfolding or Bach's contrapuntal music or the Krebs cycle or a starfish or a snow flake or high Gothic architecture ( http://hum120.files.wordpress.com/2011/03/ext-cc-mannyyy.jpg ).

But, like even that nude emperor, Western physics, which hallucinated the phony "Big Bang" and cannot manage its own quantum mechanics, math, too, bears critical flaws — inconsistencies, critical insufficiencies, logic-errors that may be corrigible but persist.........

The ultimate problem is that physics & math & logic & all other "objective" fields are, at root, emotional — matters of emotional choice.

Scientists, mathematicians, and logicians choose premises, theorems, axioms (which bear no premise but desire), definitions, observations........that fit their emotional imperatives.

Emotional imperatives?

Craving to make a system "symmetrical" or "consistent" or picturing "the universe" and its phenomena in ways that satisfy a wish of controlling nature or attaining public acclaim. Below follows a demonstration: an aspect of uncertainty of commutativity or noncommutativity of multiplication. The demonstration is just one of myriad available showings.

Witness Western astrophysics's absurd assertion of the Big Bang, which astrophysics has had increasingly to modify, and will continue to need to modify, until the modification-stream shows the theory's absurdity, which simple logic ought to have proven stupidly absurd when the theory was invented.

The Big Bang theory's ultimate source is Western science's emotional need of perceiving that the universe had a beginning & has or will have an end: a limitless universe is a concept unbearable to Western science, which cannot fathom the prospect that anything is beyond its comprehension, measurement, and control.

Simple logic disproves the Western science's perception: If "the" universe suffers limit, either nothing or something is beyond. If something is beyond, limit is illusion. If nothing is beyond, limit is illusion. A nothingness bears no limit — unless a something beyond. Suppose a something beyond. If another nothing is beyond that something, either it is limitless or, if that nothing ends, another something is beyond. And so on, infinitely. Limit is illusion.

This logic works in every real or dreamed direction — for time, too.

If "the" universe had "origin," then earlier "another" universe existed. If "Big Bang" happened, it needed surrounding space — empty till eventually bearing the particles(?) of whatever "banged" (if not also something else). The exploding hyper-dense mass could not materialize magically from naught. Did an earlier contraction occur? Was the explosion the end of just one contraction of many? Did the explosion reverse just one collapse of just one slice of infinity?

If time starts or stops, only counting begins or ends, not existence.

So, lost, the Western physics asks:

Do black holes lead to other universes?

Is "our universe" a black hole of a "meta-universe"?

Do many "universes" exist (all grown from some number of discrete big bangs)?

Can we measure space and time by the speed of light, or does the speed of light fail as a standard because it is not the ultimate speed and speed may have no limit (so that the theory of special relativity is junk and the general relativity theory must be altered to the proposition that our universe's expansion moves faster than the speed of light)?

See, E.G.,

* http://www.newscientist.com/article/dn11745-could-black-holes-be-portals-to-other-universes.html

* http://www.mathpages.com/home/kmath339/kmath339.htm

* http://huffingtonpost.com/2014/03/31/multiverse-gravitational-waves_n_5062376.html?utm_hp_ref=world&ir=World

* http://english.aljazeera.net/news/europe/2011/09/2011922183317431339.html

* http://gizmodo.com/no-stephen-hawking-did-not-say-black-holes-dont-exist-1513870928

* http://www.angelfire.com/az/BIGBANGisWRONG (This source is useful much for referencing the other sources it cites or discusses.)

Those questions imply not only that Big Bang theory is false. They imply also that the universe — the one & only universe — had no beginning & has & will have no end: Though forever expanding & contracting, the universe is, has been, and will be always one, same, eternal.

For thorough deconstruction of the Big Bang theory, see Eric Lerner, The Big Bang Never Happened: A Startling Refutation of the Dominant Theory of the Origin of the Universe (1991).

Like Western physics, math cannot afford to agree completely with REAL reality, even the "reality" of "pure" logic or the "reality" of consistency. Otherwise math would crumble into oblivion.

Witness the case of 0 (zero) factorial. Math theory insists that 0! = 1, despite 0 x N = 0, and 0 x 0 = 0.

How very consistent, symmetrical! If zero factorial equaled zero (as consistency insists), the concept factorial would fail and, with it, much of math that depends on multiplication-streams.

Zero factorial is only one example of just one of many KINDS of problems of consistency of the operation of multiplication (and of math).

Once, while teaching an Evidence Law class treating the matter of whether the law ought to permit Bayesian probability calculus to determine an issue like paternity or identity or projected life-span (of a plaintiff seeking damages for an allegedly life-shortening tort), I asserted the negative (as always I do). Among my many reasons was that statisticians tend to disregard the significance of multiplication-order.

[Side note # 1: Bayesian probability calculus is a rather metamorphic, set-theoretic-cognate extension of Pascalian conditional probability calculus. End of side note # 1]

One student was much math-trained and an actuary. He insisted my argument was nonsense because (he insisted) always multiplication is a "commutative operation."

A commutative operation is one in which the operation's order does not affect the outcome. So, if multiplication is commutative, then never can multiplication order affect multiplication product: Always, A x B x C would equal B x C x A or C x A x B or B x A x C or C x B x A or A x C x B.

Eventually, I proved to my student that his position was wrong for some (maybe many) cases of Bayesian probability calculus (and I proved so without even mentioning the whole fields of substantially, or even utterly, non-commutative multiplication, like scalar multiplication, ring theory & inverse ring theory, or matrix multiplication).

He was very resistant and kept posing irrelevant math-demonstrations that seemed pertinent if perceived in a world of superficial analysis. His world treated Bayesian probability as if it were just an alternative of analytic geometry (which not Descartes, but the Chinese invented, and about 1400 years before Descartes was born).

So, when is multiplication not commutative even where pertinently expert mathematicians suppose or insist, religiously, that it is commutative?

Consider a rather common real kind of case — Bayesian calculation of the paternity probability indicated by genetic "evidence." Take the calculation through ABO blood-type and HLA (human leukocyte antigen) data.

[Side note # 2: I exclude DNA evidence only because two variables suffice for this demonstration's purpose, and a third variable would not alter the essence of the case. Still multiplication order would affect the outcome. End of side note # 2]

Assume, for simplicity, that genetic mutation does not occur: So, the true father's and the infant's HLA types must match; and the infant's ABO type must be either the true other's or the true father's.

[Side note # 3: Mutation-probability is real, but if the case accounted for mutation-probability the outcome would be essentially the same as if the case did not account for mutation-probability. With or without the accounting, multiplication order would affect the outcome. End of side note # 3.]

Suppose both the defendant's and the mother's HLA types "match" the infant's. Infant, mother, and defendant have ABO type O negative.

Apply the "principle of indifference." Near-all statisticians would. Otherwise the operation would be impossible, since a true "prior probability" — in this case, P(F), see below — would be incalculable. [The principle of indifference is a false supposition that if no "prior probability" quantity is obtainable, one may assert, arbitrarily (FALSELY), that the positive & contrapositive prior probabilities are equal (0.5 each) & hallucinate that the posterior probabilities — e.g., P(A∣F)P(H∣A&F), see below — shall have rendered the falsehood insignificant.]

So, per principle of indifference, 0.5 is the (artificial, false) prior probability of paternity and also of non-paternity.

[Side note # 4: The principle of indifference is very like the choice of making zero factorial = 1 (0! = 1). The real and logical truth would crash the system, which is, in essence, partly delusional. So, emotionally, the mathematician (or statistician) denies the true truth and supplies a falsehood that "saves" the system. (Side note # 4 continues, next ¶.)

Cf. Gödel's proof — especially its second inconsistency theorem, but also, implicitly, its first theorem, too. Gödel's inconsistency theorems showed, among else, that "1 + 1 = 2" and many other "obviously true" propositions are unprovable assertions resting on axioms incapable of proving either (a) all statements concerning natural numbers or (b) the consistency of a number-system or arithmetic-system. (Side note # 4 continues, next ¶.)

Not only Bayesian probability calculus suffers the trouble of the principle of indifference. Integral calculus suffers the identical trouble; and that, and other, troubles (like 0! = 1) explain relatively often why bridges & tunnel-ceilings collapse & many transactional outcomes are mis-predicted. (Side note # 4 continues, next ¶.)

Concerning why the principle of indifference cannot save Bayesian probability calculus (or even traditional relative frequentist statistics) from being limited to subjective guesses (excluded from the field of determining actualities or even actual probabilities), see Jaffee, Of Probativity and Probability: Statistics, Scientific Evidence, and the Calculus of Chance At Trial, 46 U. Pittsburgh L. Rev. 925 (1985), and Jaffee, Prior Probability — A Black Hole in the Mathematician's View of the Sufficiency and Weight of Evidence, 9 Cardozo L. Rev. 967 (1988). End of side note # 4.]

The "matching" HLA type occurs in one of every 1000 men. The ABO type (say O negative) occurs in thirteen of every 100.

Among men who have ABO-type O negative blood, eight of 100 have the matching HLA characteristic. Among men who have the matching HLA characteristic, seven of ten have ABO-type O negative. This is very possible. The O negative male subpopulation is immensely greater than that of men who have the matching HLA type. So, the O negative male subpopulation can include many other HLA types of varying relative frequencies. Yet, an HLA type can be very positively correlated to O negative ABO type. [Oddly, reality's like that.]

Let F be paternity.

Let A be ABO-type match of putative father and child.

Let H be HLA "match" of putative father and child.

Let P(F∣A&H) or P(F∣H&A) be probability of paternity given A&H or H&A *

* "(F∣A&H)" means "F 'given' A&H" & "(F∣H&A)" means "F 'given' H&A." "Given" equates with set theory's "in the world of" (as in "C in the world of D"), which set theory, too, calls "given" (as in "C given D"). *

Let P(F) be "prior probability" of paternity.

Let P(H∣F) be probability of HLA match given paternity.

Let P(A∣F) be probability of ABO match given paternity.

Let P(H∣A&F) be probability of HLA match given ABO match and paternity.

Let P(A∣H&F) be probability of ABO match given HLA match and paternity.

Let the contrapositive probability symbols be parallel

[Example: P(not-F)P(A∣not-F)P(H∣A & not-F)]

P(H∣A&F) is 1. Whatever defendant's ABO type, if he is the father, his HLA type must match the infant's.

P(A∣H&F) is .25. [If defendant is the father, the infant must have an HLA attribute that, alone, would make 70% the chance that she is O negative; but an infant has the mother's ABO type 75% more often, and the infant's whole HLA type also may alter the chance that she is O negative.]

[Side note # 5: (a) The mother's actual HLA and ABO types would not affect calculation — not yet. (b) I left P(A∣H&F) at .25, since that could be right and I want to limit disparity to the not-F field. End of side note # 5.]

The conditional contrapositives (the not-F quantities) are governed by random population frequencies. So, E.G., P(A∣not-F) = .13 (since .13 is the random chance of the ABO type), and P(A∣H & not-F) = .08 (since among type O blood men, 8 of 100 have the matching HLA characteristic and that matter equals the random chance of type O given the populational frequency of the HLA characteristic). Likewise, P(H∣A&F) is .7 (since among men having the HLA characteristic, 7 of 10 have type O blood).

The Bayesian equation is:

P(F∣A&H) =

P(F)P(A∣F)P(H∣A&F) ÷

[P(F)P(A∣F)P(H∣A&F) + P(not-F)P(A∣not-F)P(H∣A & not-F)].

Or, alternatively, the equation is:

P(F∣H&A) =

P(F)P(H∣F)P(A∣H&F) ÷

[P(F)P(H∣F)P(A∣H&F) + P(not-F)P(H∣not-F)P(A∣H & not-F)]

— which reverses some aspects of the multiplication order of

P(F∣A&H) =

P(F)P(A∣F)P(H∣A&F) ÷

[P(F)P(A∣F)P(H∣A&F) + P(not-F)P(A∣not-F)P(H∣A & not-F)]

[The alternative puts the P(A) aspects second instead of first.]

If the statistician accounts for ABO type before HLA type, the Bayesian fraction is:

(.5 x .25 x 1) ÷

[(.5 x .25 x 1) + (.5 x .13 x .08)]

or

.125 ÷ (.125 + .0052),

which equals 96%.

But if the statistician accounts for HLA type first, the Bayesian fraction is:

(.5 x 1 x .25) ÷

[(.5 x 1 x .25) + (.5 x .001 x .7)]

or

.125 ÷ (.125 + .00035),

which equals 99.72%.

The not-F field condition-order's reversal altered the outcome — from insignificant to significant (or marginally significant to significant, if, as is uncommon, 95% is at least marginally significant).

You may argue that the order-change changes the quantities multiplied, rather than changes the product of multiplication of the same quantities. But the argument is a matter of perspective, NOT a matter of TRUTH.

One "reason" is that the Bayesian theorem treats P(F∣A&H) as interchangeable with P(F∣H&A), since the theorem holds that H&A = A&H — apparently because (X intersection Y) = (Y intersection X).

So, per the theorem,

P(F∣A&H) =

P(F)P(A∣F)P(H∣A&F) ÷

[P(F)P(A∣F)P(H∣A&F) + P(not-F)P(A∣not-F)P(H∣A & not-F)]

is identical to

P(F∣H&A) =

P(F)P(H∣F)P(A∣H&F) ÷

[P(F)P(H∣F)P(A∣H&F) + P(not-F)P(H∣not-F)P(A∣H & not-F)]

— despite P(A∣H & not-F) does not = P(H∣A & not-F).

The (supposed) "reason" is that P(F∣A&H) = P(F∣H&A) since H&A = A&H.

Another (actual) reason is emotional: love of "symmetry" and "consistency" AND the statistician's not being able to endure (hence the statistician's delusional denial of) the system's actual failures.

Here statisticians (or Bayesianists) appear to feel "consistency" requires that "intersection" means in Bayesian calculus exactly what it means in set theory [in which (X&Y) = (Y&X) and (X&Y&Z) = (Z&Y&X) = (Z&X&Y) = (Y&X&Z) = (Y&Z&X) = (X&Z&Y)], because Bayesian calculus derives from set-theory-like reasoned extension of conditional probability calculus. In set theory, "intersections" bear the "symmetry" of commutativity. Therefore, Bayesian calculus must follow. So, P(X&Y) must equal P(Y&X) [since in set theory, (X&Y) = (Y&X)] — despite in set theory (A&B)∣B = (B&A)∣B but may not = (A&B)∣A [or (B&A)∣A].

A third (actual) reason is fear of having to choose either overwhelmingly intractable complexity or risibly arbitrary avoidance of real inconsistency. If Bayesian buffs admitted the non-commutativity trouble my hypothetical illustrates, then Bayesian calculus would need to

(a) establish a method that tries to (but cannot) account the often infinite inconsistency of consequences of the non-commutativity my hypothetical illustrates

or

(b) establish an indefensible arbitrary (and invalid, immensely unreliable, and very practically dangerous) rule like "the statistician is stuck with whatever happens to be the statistician's first choice of variables-order (a rule just a little more contemnable than the principle of indifference & 0! = 1)

or

(c) trash the system (and lose oodles of money-compensation now obtained by estimating outcome odds for big businesses or financial institutions or giving expert testimony in litigations or......).

A much simpler reason dismisses the argument that the order-change changes the quantities multiplied. The things multiplied are not mere "quantities." They are probability-indicators.

One probability-indicator is the paternity-probability-indicating bearing of the ABO type (the A∣F, and the A∣not-F). The other probability-indicator is the paternity-probability-indicating bearing of the HLA-type (the H∣F and the H∣not-F). The order-change changes only the relative "timing" of integration of the two indicators — the order of the positive or contrapositive indicators' being multiplied with each other and with the relevant prior probability [the P(F) or the P(not-F)].

But we must not miss the elusively obvious reason: If the multiplication-order-change changed the quantities being multiplied, then, FOR THAT REASON, multiplication is non-commutative in the case hypothesized and any like case. The root-and-ultimate point is that multiplication-order alters the consequence of multiplication.

Cf. quaternion multiplication: e.g., (ij = k) BUT (ji = -k). In that pair of equations, the first equation's ij must be somehow different from the ji of the 2nd, since -k cannot = k.

Ain't that fun?

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