Bayes Theorem Quotes

My favourite fellow of the Royal Society is the Reverend Thomas Bayes, an obscure 18th-century Kent clergyman and a brilliant mathematician who devised a complex equation known as the Bayes theorem, which can be used to work out probability distributions. It had no practical application in his lifetime, but today, thanks to computers, is routinely used in the modelling of climate change, astrophysics and stock-market analysis. ~ Bill Bryson

Imagine you're sitting having dinner in a restaurant. At some point during the meal, your companion leans over and whispers that they've spotted Lady Gaga eating at the table opposite. Before having a look for yourself, you'll no doubt have some sense of how much you believe your friends theory. You'll take into account all of your prior knowledge: perhaps the quality of the establishment, the distance you are from Gaga's home in Malibu, your friend's eyesight. That sort of thing. If pushed, it's a belief that you could put a number on. A probability of sorts. As you turn to look at the woman, you'll automatically use each piece of evidence in front of you to update your belief in your friend's hypothesis Perhaps the platinum-blonde hair is consistent with what you would expect from Gaga, so your belief goes up. But the fact that she's sitting on her own with no bodyguards isn't, so your belief goes down. The point is, each new observations adds to your overall assessment. This is all Bayes' theorem does: offers a systematic way to update your belief in a hypothesis on the basis of the evidence. It accepts that you can't ever be completely certain about the theory you are considering, but allows you to make a best guess from the information available. So, once you realize the woman at the table opposite is wearing a dress made of meat -- a fashion choice that you're unlikely to chance up on in the non-Gaga population -- that might be enough to tip your belief over the thresho ~ Hannah Fry

The analysis of variance is not a mathematical theorem, but rather a convenient method of arranging the arithmetic. ~ Ronald Fisher

Those who have more power are liable to sin more; no theorem in geometry is more certain than this. ~ Lord Acton

Bells theorem dealt a shattering blow to Einsteins position by showing that the conception of reality as consisting of separate parts, joined by local connections, is incompatible with quantum theory ... Bells theorem demonstrates that the universe is fundamentally interconnected, interdependent, and inseparable. ~ Fritjof Capra

Therefore Godel's theorem had an electrifying effect upon logicians, mathematicians, and philosophers interested in the foundations of mathematics, for it showed that no fixed system, no matter how complicated could represent the complexity of the whole numbers: 0,1,2,3,..... ~ Douglas R. Hofstadter

I realized that anything to do with Fermat's Last Theorem generates too much interest. ~ Andrew Wiles

I took care to replace the Compendium in its correct pamphlet, and in doing so dislodged a slim pamphlet by Grastrom, one of the most eccentric authors in Solarist literature. I had read the pamphlet, which was dictated by the urge to understand what lies beyond the individual, man, and the human species. It was the abstract, acidulous work of an autodidact who had previously made a series of unusual contributions to various marginal and rarefied branches of quantum physics. In this fifteen-page booklet (his magnum opus!), Grastrom set out to demonstrate that the most abstract achievements of science, the most advanced theories and victories of mathematics represented nothing more than a stumbling, one or two-step progression from our rude, prehistoric, anthropomorphic understanding of the universe around us. He pointed out correspondences with the human body-the projections of our sense, the structure of our physical organization, and the physiological limitations of man-in the equations of the theory of relativity, the theorem of magnetic fields and the various unified field theories. Grastrom's conclusion was that there neither was, nor could be any question of 'contact' between mankind and any nonhuman civilization. This broadside against humanity made no specific mention of the living ocean, but its constant presence and scornful, victorious silence could be felt between every line, at any rate such had been my own impression. It was Gibarian who drew it to my attention, ~ Stanisław Lem

In mathematics, in place of characters, you have variables or unknowns. If I'm trying to plot a theorem, I try to imagine these variables interacting with each other. The boundary of their interaction is the theorem. ~ Manil Suri

In a world in which the price of calculation continues to decrease rapidly, but the price of theorem proving continues to hold steady or increase, elementary economics indicates that we ought to spend a larger and larger fraction of our time on calculation. ~ John Tukey

The most famous Diophantine equation in history is the one known as Fermat's last Theorem, the celebrated statement by Pierre de Fermat (1601-55) that there are no whole number solutions to the equation x^n + y^n = z^n, where n is any number greater than 2. When n = 2, there are many solutions (in fact an infinite number). For instance, 3^2 + 4^2 = 5^2 (9 + 16 = 25); or 12^2 +5^2 = 13^2 (144 + 25 = 169). Miraculously, when we go from n = 2 to n = 3, there are no whole numbers x,y,z that satisfy x^3 + y^3 = z^3, and the same is true for any other value of n that is greater than 2. Appropriately, it was in the margin of the second book of Diophantus's Arithmetica, which Fermat was eagerly reading, that he wrote his extraordinary claim-one that took no fewer than 356 years to prove. ~ Mario Livio

It is a common misconception, in the spirit of the sentiments expressed in Q16, that Godel's theorem shows that there are many different kinds of arithmetic, each of which is equally valid. The particular arithmetic that we may happen to choose to work with would, accordingly, be defined merely by some arbitrarily chosen formal system. Godel's theorem shows that none of these formal systems, if consistent, can be complete; so-it is argued-we can keep adjoining new axioms, according to our whim, and obtain all kinds of alternative consistent systems within which we may choose to work. The comparison is sometimes made with the situation that occurred with Euclidean geometry. For some 21 centuries it was believed that Euclidean geometry was the only geometry possible. But when, in the eighteenth century, mathematicians such as Gauss, Lobachevsky, and Bolyai showed that indeed there are alternatives that are equally possible, the matter of geometry was seemingly removed from the absolute to the arbitrary. Likewise, it is often argued, Godel showed that arithmetic, also, is a matter of arbitrary choice, any one set of consistent axioms being as good as any other. ~ Roger Penrose

The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas. Thus a serious mathematical theorem, a theorem which connects significant ideas, is likely to lead to important advances in mathematics itself and even in other sciences. ~ G.H. Hardy

Context enables us to determine which of several meanings is in play in a particular text. The verb in "I see" means something quite different if uttered by a formerly blind man healed with spittle and dust, by a student who has just received an extended explanation of a difficult mathematical theorem, or by a skeptical wife whose husband offers a lame explanation for the lipstick on his collar. In the first context, see refers to physical sight, while in the latter two it refers to understanding, and in the last it could hardly be said without a heap of sarcasm. ~ Peter J. Leithart

You're right: if there's sentient life behind the border, it probably won't share my goals. Unlike the people in this room, who all want exactly the same things in life as I do, and have precisely the same tastes in food, art, music, and sex. Unlike the people of Schur, and Cartan, and Zapata - who I came here in the hope of protecting, after losing my own home - who doubtless celebrate all the same festivals, delight in the same songs and stories, and gather every fortieth night to watch actors perform the same plays, in the same language, from the same undisputed canon, as the people I left behind.

"If there's sentient life behind the border, of course we couldn't empathize with it. These creatures are unlikely to possess cute mammalian neonate faces, or anything else we might mistake for human features. None of us could have the imagination to get over such insurmountable barriers, or the wit to apply such difficult abstractions as the General Intelligence theorem - though since every twelve-year-old on my home world was required to master that result, it must be universally known on this side of the border.

"You're right: we should give up responsibility for making any difficult moral judgments, and surrender to the dictates of natural selection. Evolution cares so much about our happiness that no one who's obeyed an inherited urge has ever suffered a moment's regret for it. History is full of joyful case studies of people who followed their na ~ Greg Egan

Gradually, at various points in our childhoods, we discover different forms of conviction. There's the rock-hard certainty of personal experience ("I put my finger in the fire and it hurt,"), which is probably the earliest kind we learn. Then there's the logically convincing, which we probably come to first through maths, in the context of Pythagoras's theorem or something similar, and which, if we first encounter it at exactly the right moment, bursts on our minds like sunrise with the whole universe playing a great chord of C Major. ~ Philip Pullman

I swear the older I get, the more I value bad examples over good ones. It's a good thing too, because most people are egotistical, neurotic, self-absorbed peons, insistent on wearing near-sighted glasses in a far-sighted world. And it's this exact sort of myopic ignorance that has led to my groundbreaking new theory. I call it Mim's Theorem of Monkey See Monkey Don't, and what it boils down to is this: it is my belief that there are some people whose sole purpose of existence is to show the rest of how not to act. ~ David Arnold

To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well. ~ Albert Camus

A theorem is a proposition which is a strict logical consequence of certain definitions and other propositions ~ Anatol Rapoport

The overall structure of the calculus is simple. The subject is defined by a fantastic leading idea, one basic axiom, a calm and profound intellectual invention, a deep property, two crucial definitions, one ancillary definition, one major theorem, and the fundamental theorem of the calculus.
The fantastic leading idea: the real world may be understood in terms of the real numbers.
The basic axiom: brings the real numbers into existence.
The calm and profound invention: the mathematical function.
The deep property: continuity.
The crucial definitions: instantaneous speed and the area underneath a curve.
The ancillary definition: a limit
The major theorem: the mean value theorem.
The fundamental theorem of the calculus is the fundamental theorem of the calculus.
These are the massive load-bearing walls and buttresses of the subject. ~ David Berlinski

In that moment, the future - uncontainable by any Theorem mathematical or otherwise - stretched out before Colin: infinite and unknowable and beautiful. ~ John Green

In 1931, Kurt Godel proved in his famous second incompleteness theorem that there could be no finitary proof of the consistency of arithmetic. He had killed Hilbert's program with a single stroke.

So should you be worried that all of mathematics might collapse tomorrow afternoon? For what it's worth, I'm not. I do believe in infinite sets, and I find the proofs of consistency that use infinite sets to be convincing enough to let me sleep at night. ~ Jordan Ellenberg

I know logically that I can live without him, but loving him has become such an integral, necessary part of my life; I am not sure I could stop, even if we parted. ~ Anna Bayes

It is an illusion that we were ever alive,
Lived in the houses of mothers, arranged ourselves
By our own motions in a freedom of air.

Regard the freedom of seventy years ago.
It is no longer air. The houses still stand,
Though they are rigid in rigid emptiness.

Even our shadows, their shadows, no longer remain.
The lives these lived in the mind are at an end.
They never were . . . The sounds of the guitar

Were not and are not. Absurd. The words spoken
Were not and are not. It is not to be believed.
The meeting at noon at the edge of the field seems like

An invention, an embrace between one desperate clod
And another in a fantastic consciousness,
In a queer assertion of humanity:

A theorem proposed between the two -
Two figures in a nature of the sun,
In the sun's design of its own happiness,

As if nothingness contained a métier,
A vital assumption, an impermanence
In its permanent cold, an illusion so desired

That the green leaves came and covered the high rock,
That the lilacs came and bloomed, like a blindness cleaned,
Exclaiming bright sight, as it was satisfied,

In a birth of sight. The blooming and the musk
Were being alive, an incessant being alive,
A particular being, that gross universe. ~ Wallace Stevens

Can the difficulty of an exam be measured by how many bits of information a student would need to pass it? This may not be so absurd in the encyclopedic subjects but in mathematics it doesn't make any sense since things follow from each other and, in principle, whoever knows the bases knows everything. All of the results of a mathematical theorem are in the axioms of mathematics in embryonic form, aren't they? ~ Alfred Renyi

It may be appropriate to quote a statement of Poincare, who said (partly in jest no doubt) that there must be something mysterious about the normal law since mathematicians think it is a law of nature whereas physicists are convinced that it is a mathematical theorem. ~ Mark Kac

Mathematical knowledge is unlike any other knowledge. While our perception of the physical world can always be distorted, our perception of mathematical truths can't be. They are objective, persistent, necessary truths. A mathematical formula or theorem means the same thing to anyone anywhere – no matter what gender, religion, or skin color; it will mean the same thing to anyone a thousand years from now. And what's also amazing is that we own all of them. No one can patent a mathematical formula, it's ours to share. There is nothing in this world that is so deep and exquisite and yet so readily available to all. That such a reservoir of knowledge really exists is nearly unbelievable. It's too precious to be given away to the "initiated few." It belongs to all of us. ~ Edward Frenkel

In a companion essay to "Continuity and I.N." that's usually translated as "The Nature and Meaning of Numbers," Dedekind evinces a remarkable proof for his "Theorem 66. There exist infinite systems," which runs thus: "My own realm of thoughts, i.e., the totality S of all things which can be objects of my thought, is infinite. For if s signifies an element of S, then the thought s', that s can be an object of my thought, is itself an element of S,..." and so on, meaning that the infinite series ([s] + [s is an object of thought]+['s is an object of thought' is an object of thought] + ...) exists in the Gedankenwelt, which entails that the Gedankenwelt is itself infinite. With respect to this proof, notice (a) how closely it resembles the various Zeno-like VIR back in paragraph 2a, and (b) how easily we could object that the proof establishes only that Dedekind's Gedankenwelt is 'potentially infinite' (and in precisely Aristotle's sense of the term), since nobody can ever actually sit down and think a whole infinite series of (s+s'+s")-type thoughts-i.e., the series is a total abstraction. ~ David Foster Wallace

By now, he was also a 'Protestant Atheist', which he remained all his life. ~ John Ellis

Any good theorem should have several proofs, the more the better ~ Michael Atiyah

Colin,
I hate to fulfill the Theorem, but I don't think we should be involved romantically.
The problem is that I secretly in love with Hassan.
I can't help myself. I hold your bony shoulder blades in my hands and think of his fleshy back.
I kiss your stomach and I think of his awe-inspiring gut. I like you,
Colin, I really do. But-I'm sorry. It's just not going to work.
I hope we can still be friends.
Sincerely,
Lindsey Lee Wells
P.S. Just kidding. ~ John Green

The combination of Bayes and Markov Chain Monte Carlo has been called "arguably the most powerful mechanism ever created for processing data and knowledge."
Almost instantaneously MCMC and Gibbs sampling changed statisticians' entire method of attacking problems. In the words of Thomas Kuhn, it was a paradigm shift. MCMC solved real problems, used computer algorithms instead of theorems, and led statisticians and scientists into a worked where "exact" meant "simulated" and repetitive computer operations replaced mathematical equations. It was a quantum leap in statistics. ~ Sharon Bertsch McGrayne

This skipping is another important point. It should be done whenever a proof seems too hard or whenever a theorem or a whole paragraph does not appeal to the reader. In most cases he will be able to go on and later he may return to the parts which he skipped. ~ Emil Artin

It is an unfortunate fact that proofs can be very misleading. Proofs exist to establish once and for all, according to very high standards, that certain mathematical statements are irrefutable facts. What is unfortunate about this is that a proof, in spite of the fact that it is perfectly correct, does not in any way have to be enlightening. Thus, mathematicians, and mathematics students, are faced with two problems: the generation of proofs, and the generation of internal enlightenment. To understand a theorem requires enlightenment. If one has enlightenment, one knows in one's soul why a particular theorem must be true. ~ Herbert S. Gaskill

If a customer asks you to build a system that handles netsplits while staying consistent and available, you know that you need to either calmly explain the CAP theorem or run away (possibly by jumping through a window, for a maximal effect). ~ Fred Hebert

What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men. ~ G.H. Hardy

The only way to enjoy architecture is not just to look at it but to move around it and through it. ~ Kenneth Bayes

Being a writer brings out a sassy woman inside me, and I love her so. ~ Anna Bayes

Note for a Textbook

The question is never answered, never resolved,
The circle of love and anger never squared,
The stubborn instinct not translated yet
In decimals, accurate and predictable
In union dues or payroll cuts or blood...

Always a symbol lost in the lovely theorem,
A fraction that will not fit in the sum of the system,
A jutting thrust in the graph of the commissar's forecast,
A troublesome blank in gauleiter's careful accounting,
An awkward hitch in the plans of the second vice-president.

The answers worked on the slate are never the same,
And the answers proved in the back of the book are wrong. ~ Charles Bruce

I tend to regard the Coase theorem as a stepping stone on the way to an analysis of an economy with positive transaction costs. ~ Ronald Coase

I think mathematics is a vast territory. The outskirts of mathematics are the outskirts of mathematical civilization. There are certain subjects that people learn about and gather together. Then there is a sort of inevitable development in those fields. You get to the point where a certain theorem is bound to be proved, independent of any particular individual, because it is just in the path of development. ~ William Thurston

About Thomas Hobbes: He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentleman's library, Euclid's Elements lay open, and "twas the 47 El. libri I" [Pythagoras' Theorem]. He read the proposition "By God", sayd he, "this is impossible:" So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that truth. This made him in love with geometry. ~ John Aubrey

Dullards would have you believe that once you have eliminated the impossible, whatever remains, however improbable, must be the truth ... but to a mathematical mind, the impossible is simply a theorem yet to be solved. We must not eliminate the impossible, we must conquer it, suborn it to our purpose. ~ Kim Newman

Fain would I turn back the clock and devote to French or some other language the hours I spent upon algebra, geometry, and trigonometry, of which not one principle remains with me. Stay! There is one theorem painfully drummed into my head which seems to have inhabited some corner of my brain since that early time: "The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides!" There it sticks, but what of it, ye gods, what of it? ~ Jessie Belle Rittenhouse

To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples ... ~ John B. Conway

Poetry is a theorem of a yellow-silk handkerchief knotted with riddles, sealed in a balloon tied to the tail of a kite flying in a white wind against a blue sky in spring. ~ Carl Sandburg

The future is the result of actions, and actions are the result of behavior, and behavior is the result of prediction. ~ David McRaney

No part of mathematics is ever, in the long run, "useless." Most of number theory has very few "practical" applications. That does not reduce its importance, and if anything it enhances its fascination. No one can predict when what seems to be a most obscure theorem may suddenly be called upon to play some vital and hitherto unsuspected role. ~ C. Stanley Ogilvy

No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it also gives the impression of being beautiful. ~ George Boole

Schelling's God is the totality of Nature struggling towards consciousness, and Man is as far as the struggle has got, with the animals not too far behind, vegetables somewhat lagging, and rocks nowhere as yet. Do we believe this? Does it matter? Think of it as a poem or a painting. Art doesn't have to be true like a theorem. It can be true in other ways. This truth says there is a meaning to it all, and Man is where the meaning begins to show. ~ Tom Stoppard

Any effect, constant, theorem or equation named after Professor X was first discovered by Professor Y , for some value of Y not equal to X. ~ John C. Baez