[podcast]https://www.wmbriggs.com/audio/wmbriggs_com_lecture_0004.mp3[/podcast]
Today’s lecture:
Format change The regular and statistics podcasts are undergoing some changes. Stay tuned to this page for details. Today’s show is a little rushed because of this.
Randomness Why don’t we need to add “a fair die” or “an unbiased die” to our list of premises, “We toss a six-sided die, just one side will show, and just one side is labeled 6”? The probability of the conclusion “A 6 will show” is 1/6 without the addition of “unbiased”. If we add it, then our argument becomes circular, for “unbiased” means each side is equally likely.
We’ll talk more about the mysterious hold the word “random” has on experiments once we get to how to map our logical arguments to reality. That is, how do we determine whether a given die is biased or not?
Examples, finally! Give that three out of four dentists prefer Veldensteing Weissbier, what is the probability that your dentist prefers it? Obviously, 3/4. If you further know that your dentist is a teetotaler, then you must modify the entire argument to: “Given that three out of four dentists prefer Veldensteing Weissbier and my dentist is a teetotaler”, the probability that your dentist prefers that beer is 0—you have deduced it.
But you must always explicitly state your premises. You cannot criticize the conclusions of an argument for not including your favorite premises. Arguments stand on their own. Failure to heed this simple rule has led to more grief than anything else in understanding logic and probability. Research the sad story of Laplace’s Rule of Succession for an example.
Incidentally, in most of the arguments that matter to us (politics, religion, etc.), it is difficult to fully state our premises, a situation which makes it appear that probability is subjective. See the notes for more detail on this complicated subject.
Many multiculturalists are fond of saying, “There is no truth.” What is the probability the statement “There is no truth” is true? If the probability is 1, then the statement is false, for we have just found a truth. If the probability is 0, because there are no truths, then there are truths because the statement is false.
In other words, this is just another in a long list of asinine propositions put forward by half-wits who are anxious to get away with something.
But notice in our proof of the idiocy of that statement, we still used certain logical connectives, or steps, the validity of which we assumed true. We can never escape the fact that all truth eventually rests on our intuitions.
Last, my insurance company asked me to “prove that I do not have an additional policy with another company.” What is the probability I can prove this? Clearly state your premises.
Last, my insurance company asked me to “prove that I do not have an additional policy with another company.†What is the probability I can prove this? Clearly state your premises.
Zero. You can’t prove a negative. It’s impossible to prove nonexistence.
The probability of you objectively proving such a point, is currently zero. Beyond a few purely mathematical contrivances, non-existence is inherently impossible to prove.
Your chances could be greatly improved by requesting further information from your insurance company.
First, and most important, would be a comprehensive definition of the phrase, “additional policy.”
A second requirement would be a complete list of every entity that your company would count as, “another company.”
No objective proof can be possible without agreement, and finite limitations, on both of the above. In the absence of such constraints, your insurance company could claim that an intergalactic planetowner’s policy listing you as property, may exist. However far fetched, that assertion does have a nonzero probability, invalidating any “proof” that fails to address it.
A third requirement, though possibly not as fundamental as the first two, would be some method of compelling all potential insurers to provide complete and accurate information regarding their possible dealings with you, or with any agency working on your behalf, with or without your knowledge or approval.
Once those conditions are met, the probability that you can prove this point would be a function of the number of the number of potential insurers, the number of agreements that would be considered relevant policies, your own tenaciousness, your personal resources, and the willingness of everyone involved to help or hinder. As all of those are currently unknowns, at least to myself, I can only conclude that your chances of an objective proof would be raised to some level above zero.
My personal feeling, however, is that the probability would be only trivially above zero. Roughly the same as the probability of an intergalactic planetowner’s policy listing you as property.
The probability that the sun will rise tomorrow is 1/2, and I can prove it using Bayes’ theorem.
http://saltzafrazz.blogspot.com/2009/11/bayes-laplace-and-sun.html
I remain unconvinced. The claim that you have deduced that the probability that a 6 wll show given a six-sided die just one side if which shows a six is 1/6 is missing a premise. What is to prevent me claiming that it’s 1/5? Since you posit a “die” your argument also permits the same conclusion about the other five numbers. You may claim that these are conclusions but where is the information in your premisses that constrains us to these probabilities? “There are six faces”? How have you excluded the case where the probabiliy of a 6 is 6/21?
And if there is none then your 1/6’s are not logically deduced and if there is then your argument is circular anyway.
By the way, proving a negative is easy:
No man is a hero to his valet.
I am a man
I am not a hero to my valet
It’s universal negatives that can’t be proved deductively.
Write to your insurance company saying, “Find enclosed photocopies of all my policies with other insurance companies”. The envelope should contain only the letter.
I think the notion “you can’t prove a negative” depends on the context. There is generally a provable negative statement of every provable statement. Let A be any provable statement. Let B=~A. Then the negative statement ~B is provable, because ~B=A. It is also possible to prove “universal negatives” deductively, depending on the universe. For example, it is possible to prove that there does not exist a solution to exp(x)=0 in the universe of the real (or complex) number system. The statement “there does not exist a two ton elephant in my dishwasher” seems pretty provable to me too, where “pretty provable” might mean there is a high likelihood. On the other hand, the current question about insurance seems “not so provable” in the sense that it would be difficult to compile enough evidence to convince someone that it is highly likely to be true.
But convincing someone of the likelihood of something is a far cry from a proof. To steal your analogy, I’m personally 100% certain that the sun will rise tomorrow. But that’s an inherently unprovable position. There are non-zero likelihoods of both the sun and the earth ceasing to exist in the next 20 hours, along with some even crazier possibilities like a spontaneous reversal of G. No aspect of the future can ever be objectively proven, as the entire universe could theoretically disappear at any moment.
As with the insurance example, the elephant in the dishwasher requires rigid definitions. I’m not a lawyer, but it helps to think like one. Maybe there’s a very small, very dense, two ton elephant in your dishwasher. Without a fully comprehensive definition of a “two ton elephant,” there can be no absolute proof regarding anything but it’s weight. And even then, since weight is a function of G, certain other parameters may need to be specified. A toy elephant may well weigh two tons, on a massive enough planet. The dishwasher aspect is much simpler. By referring to a single, specific object, you’ve fully defined that parameter. An objective proof can only exist when every parameter is finite. (fully predictable patterns can sometimes be substituted for finite, as in the exp(x)=0 example)
I agree for the most part. In the elephant and in the sun example, there is no real proof. I also agree the estimation of likelihood may be non-objective. Suppose we accept this non-objectivity and define “X is pretty provable†to mean that it is possible to compile enough evidence to convince nearly everyone that X is highly likely to be true. This seems to be close to the working definition that is used in science. Unlike in mathematics, I’m afraid that’s about the best we can do.
But if we can adopt such a definition, then it seems to me we can say that statement “there does not exist a two ton elephant in my dishwasher†is pretty provable.
Agreed.
Objective proof in real world circumstances is as rare as it is unnecessary.
But until I see your dishwasher with my own eyes, I’ll always be suspicious of you.
From a very long time ago:
Q: “How do you know when an elephant’s been in your fridge?”
A: “Footprints in the butter.”
All,
You cannot be “100% certain that the sun will rise tomorrow”. Nobody can, because that event is contingent and all contingent events have probabilities between 0 and 1. Future or unknown contingent events never are false or true, never 100% certain. To prove something means to show that is it so with 100% probability.
There is no such things as practically true, or true with 100% – epsilon probability.
Some aspects of the future can be predicted logically. Suppose you had the premises “All objects dropped fall to the ground. This coin is an object that will be dropped”. The conclusion “This coin when dropped will fall to the ground” is certain.
But isn’t a coin drop a contingent event? Ordinarily, yes. There is nothing in the real universe which guarantees a coin will always fall to the ground. But our argument was a logical one with a premise that was assumed true, even those it may be false in fact.
This example illustrates how careful we must be in exactly specifying our premises. Most arguments in most situations can be avoided if people more carefully listed their assumptions.
Re: the insurance example. I might, as Rich suggested, mail my insurance company an empty envelope. But they does not prove I do not have another policy, because I could be lying to my insurance company. I can logically prove only to myself that I do not have another policy. I do this by conditioning on the premise (or similar premises) “I do not recall having another policy, and my memory on this point is infallible.” If I don’t have that second proviso, it is of course possible that I do have another policy, but I forgot it.
Whether I do have another policy is a contingent event to the insurance company. They have different evidence than I do. They will have some probability in mind below which they judge it unlikely enough that I do not have another policy so that they do not to take further action.
And for me, knowing I do not have another policy does not allow me to prove (which means show with certainty) that I do not have one. There is no experiential event, no observation, that can prove it. In other words, we cannot learn from experience that I do not have one. This will be important to remember.
Rich,
Re: randomness. Your argument forms the basis of subjective probability. It remains convincing to many, but not to me. You might say, “why not 6/21?” or any other probability not equal to 1/6, but I claim that nobody believes that kind of statement. We can get more technical about this, but for now here is a paper I wrote on the subject.
We’ll meet with “randomness” many times in the course of these lectures.
Unless such propositions are not absolutely true ;). Then, it makes sense. Basically, what relativists are saying is that the concept “Truth” is a platonic device, or a religious one. It infers that words have a special relationship with “Reality”. What they are not saying is that “Truth” doesn’t exist, but that it is a darned word with many problems, problems which could be solved if we just forget about that word, and started being a little more humble about “Truth”.
Science theories are about what’s useful and predictive, and people just like to mumble “Trooth” as if they owned the meaning of the universe. It’s merely pounding the table, “but it’s the truth!! Yeah, go convince people how to behave better with that!
“but for now here is a paper I wrote on the subject.” Where? I don’t see a link. Sounds interesting.
It’s link on the words “here is a paper.” Maybe I should change the color of the links under comments. The blue is difficult to see unless multiple words are linked.
William
Re : paper you linked .
Symetries .
This is as far as I am concerned the most powerful and the best link between statistics and the physical reality .
Symetries are mathematical statements about invariance of a physical system following some transformation .
As they apply to physical systems , they can be validated by observation .
For instance I have always been fascinated by the power of isotropy which is a statement that space is invariant by rotations or that some stuff stays the same regardless how you look at it (f.ex a sphere) .
But it is at the same time a statistical statement saying that something happening in 1 spatial direction can happen in any other direction as well . E.g different directions which are well defined by different vectors have the same PROBABILITY .
The isotropic Mother Nature doesn’t care about directions , hence you did well to talk about indifference 🙂 . Of course sometimes she does care and the wisdom is in being able to make the difference .
.
One can but be impressed how this simple intuitively easily understandable notion leads to the cathedral of statistical thermodynamics and to the Kolmogorov turbulence theory (even if the latter is subject to caution) .
Symetries are at the core of all modern physics (Lorenz symetry in relativity , group symetries in quantum chromodynamics and supersymetries in string theory) .
I think that the part of statistics applied to physics is objectivated by symetries while the other words like “uncertainty” and such are much weaker and “fuzzier” .
Briggs, I suspect that when “we get more technical later” we discover the detail where the devil is hiding.
This probability assignment, made explicit in the form of the statistical syllogism, is
derived from assuming uniform probability across the individual events that make up
the “sample space”
Just for now, could you explain how the statement above about an assumption mentioned only afterward is different to the assumption, tradtionally included, “and it is a fair coin”? After all, “it is a fair coin” actually distributes the probability uniformly over the sample space. Doesn’t it?
Rich,
Fair enough.
First, let us agree that by a “fair die” or “unbiased die” (coin, n-sided object, whatever) we implicitly mean “this die has equi-probability of coming up any side”; or, equivalently, “this die has probability 1/6 of showing ‘6’.” I cannot think of any other meaning of the words “fair” or “unbiased.”
Now let’s write our argument:
P1 – A six-sided die will be thrown and only one side will show
P2 – Just one side is labeled “6”
————————————————————————————————————————
C – A “6” shows.
I claim Pr( C | P1 & P2) = 1/6.
Now let’s add the premise about “fairness.”
P1 – A six-sided die will be thrown and only one side will show
P2 – Just one side is labeled “6”
P3 – The probability of seeing a “6” is 1/6
————————————————————————————————————————
C – A “6” shows.
Now, it is true that Pr( C | P1 & P2 & P3) = 1/6, but we can also eliminate P1 and P2 and write Pr( C | P3) = 1/6, but that’s not very interesting, is it? Especially you write it all out:
“The probability of a ‘6’ showing is 1/6 given that the probability of a ‘6’ showing is 1/6.”
That’s a circular argument. It’s still valid, as circular arguments are, but it doesn’t do much for you because you have assumed what you hoped to show.
In Pr( C | P1 & P2) = 1/6 we have invoked an unproven principle of logical probability, no question about it. So perhaps (and in the class notes I make reference to this), it would be better to write it as
Pr( C | P1 & P2 & I) = 1/6
Where I represents our intuitions about logic and so forth. Even when we don’t write it, it’s always there.
Tom,
Symmetries are of course extremely important.
I understand what you mean by the word “uncertain”, but perhaps I am more comfortable with it because of long familiarity.
William
.
“First, let us agree that by a “fair die†or “unbiased die†(coin, n-sided object, whatever) we implicitly mean “this die has equi-probability of coming up any sideâ€; or, equivalently, “this die has probability 1/6 of showing ‘6′.†I cannot think of any other meaning of the words “fair†or “unbiased.—
.
Yes and here is why .
The die is left invariant by rotations by 90° around the axis passing through the centers of opposing faces .
4 rotations x 3 axis = 12 states
3 of them are identities (rotations by 360°) .
12 – 3 = 9 different states
All rotations around 1 axis can be expressed as combination of rotations around the other 2 axis .
9 – 3 = 6 independent states .
Therefore due to the symmetries there are 6 dynamical states that can’t be distinguished because they are deduced from each other by a combination of 6 different rotations that let the die invariant and that is true regardless if the die is rolling or already stopped .
So the probability of any final state is 1/6 .
Note that if there was not this particular rotationnal symmetry , the probabilities of the final states would not be 1/6 …
.
So I reformulate .
P1 : the die is invariant by 6 independent transformations
P2 : the die can have 6 states
Conclusion : the probability of any state is 1/6 🙂
When discussing probability, a “six sided die” to me means a real-valued function p defined on the integers 1,2,…6 such that 0<=p(k)<=1 for all k and p(1)+p(2)+…+p(6)=1. When you ask for the probability of the conclusion "A 6 will show" given that "We toss a six-sided die, just one side will show, and just one side is labeled 6," to me this means you are asking for p(6), given only the above knowledge about p. You can't conclude p(6)=1/6 from just that knowledge alone, can you? Adding "a fair die" to the premise means to me that p(k)=p(j) for all k,j. This additional knowledge lets you conclude p(k)=1/6 for all k.
Suppose, starting from scratch, I told you that p is a function on the set of integers 1,2,…,6 with 0<=p(k)<=1 and p(1)+p(2)+…+p(6)=1, and asked you "what is p(6)?" Would you reply p(6)=1/6? Wouldn't the "statistical syllogism" principle in this case tell us that p itself has a probability distribution that is uniform on the 5 dimensional simplex associated with the properties of p? (yipe!) If so, then "what is p(6)?" doesn't seem answerable to me.
So what stops me saying, “… assuming uniform probability across the individual events that make up the sample space” or equivalently, “this die has equi-probability of coming up any side†or, equivalently, “this die has probability 1/6 of showing 6″?
Note I’m not saying any of these formulations are right or wrong. I’m saying that your probability syllogism, with its necessary assumption, is equivalent to the subjectivist definition you reject.
Or has the horse stopped moving?
Mind you, I quite like the symmetry idea. It seems to me to be saying, “There is no logical way to distinguish between the faces of a die therefore they have the same probability (else we could distinguish between them)”.
The symmetries of the die have nothing to do with subjectivity or with logics .
It is BECAUSE of the symmetries that the probability is 1/6 .
The nature and the number of symmetries determine the value of the probabilities .
Here is an example how it works and this time I will take a special die .
2 opposing parallel sides of the die are a big square and a small square . The 4 sides connecting these 2 opposite sides are identical trapezoids . The die looks like a Maya pyramid .
This die admits only 4 rotations that keep it invariant – 90° rotations around an axis passing through the centers of the square sides .
One is identity (360° rotation) so that leaves 3 different rotations .
Why is the invariance by transformations so important ?
.
Because when you throw this die , you may apply on it at any moment the invariant rotation and it doesn’t change at all the trajectory of the die .
The dynamics of the die don’t care about which trapezoid is up or down because you can’t distinguish the states that you obtain by applying the rotations which keep the die invariant .
Obviously what is true during the evolution is also true for the final states when the die stops .
.
So this die has also 6 states . 4 of them can be obtained by rotations which keep the die invariant .
Therefore this die will have 3 (6-3) different probabilities (as opposed to only one in the previous case with 6 symetries) .
P1 for the big square up , P2 for the small square up and P3 for a trapezoid up .
Of course we will have P1 + P2 + 4.P3 = 1 .
After the examination of the specific geometrical features of the die , one finds the solution f.ex : 1/3 + 1/6 + 4.1/8 = 1 .
There is not necessary to know anything else than the symmetries to draw these conclusions .
.
This is equivalent to degenerated states in quantum mechanics .
The states with different quantum numbers (labels on the die) but the same energy (trajectory of the die) are called degenerate .
All degenerate states for a given energy are equiprobable .
The degeneracy of an energy level is due to particular symmetries .
If you apply a perturbation , you destroy some symmetries and the energy states split reducing the degeneracy .
That’s what I did here . I took a 6 times degenerated die (probability of each state = 1/6) and applied a perturbation (making a Maya pyramid out of it) .
Doing that I destroyed 3 symmetries and the full degeneracy reduced to 2 non degenerate states and one 4 times degenerate state .
Of course as I only slightly perturbed the fully degenerate case , the total number of states is still 6 in both systems .
TomVonk: http://www.webamused.com/blogosophy/archives/002064.html
(in reference to “After the examination of the specific geometrical features of the die , one finds the solution “)
🙂
Tom,
I argue that the die being symmetric—rather, our knowledge of whether it is or not—is irrelevant to the argument (which I repeat for ease):
P1 – A six-sided die will be thrown and only one side will show
P2 – Just one side is labeled “6”
————————————————————————————————————————
C – A “6” shows.
Then Pr( C | P1 & P2) = 1/6.
There is no information about symmetry present in those premises. If you like, we are inferring symmetry in stating the probability of the conclusion. If we add the premise that “symmetry exists”, it is no different than adding the premise “the die is fair.”
Expand the original to P1 = “A n-sided object…” Then the conclusion has logical probability Pr( C | P1 & P2) = 1/n. I show this example because maybe it’s easier to disconnect ourselves from the physics: it’s too easy in the former example to imagine a hypothetically perfectly symmetric die; with an n-sided object, it’s less natural to think about symmetries.
Or how about this example:
P1 – In a room there are three beings
P2 – One of these beings will be selected
P3 – Being 1 is a centaur
P4 – Being 2 is a leprechaun
P5 – Being 3 is a fairie
————————————————————————————————————————
C – The leprechaun is selected
Then Pr( C | P1 & P2 & P3 & P4 & P5) = 1/3.
Not only is there no symmetry, there can’t in reality be an actual selection based on the fact that none of these beings exists. Yet we can still compute the logical probability.
Rich,
Quite right: everything eventually rests on our intuition, which you can call “subjective”. I only do not use that word because of its negative connotations, though there is, as you suggest, nothing wrong with it.
However, there is only one principle of logical probability we have to (subjectively, intuitionally) accept. It is sometimes call the “Principle of Indifference”, another name I don’t love (another name is “Ignorance”). It would say, in the examples so far, “There is no reason to suppose any side is more likely than another, etc. etc.” I instead say there is a positive reason to assign the probability we did based on the definite evidence we have. We are not completely ignorant as the name of the principle suggests.
For example, try this argument:
P1 – T
————————————————————————————————————————
C – M
where T is any tautology (and therefore true no matter what: incidentally, how do we know this? it’s proof still rests on our intuition) and M is some statement. That argument shows true ignorance. We know nothing about M. So
Pr( C | P1) is undefined.
Now change T to this:
P1′ – T’ = “M will happen or it will not”
————————————————————————————————————————
C – M
Then
0 < Pr( C | P1') < 1 .
This is because T contains implicit information that M is contingent—and we recall from Lecture #2 that all contingent events have probabilities different than 0 and 1. Equivalently, if we accept this implicit knowledge P1′ = “M is contingent”. If we do not accept it, we’re back to the first argument.
Now, a true subjectivist is allowed to say
Pr( C | P1′) = 0.48761
or any other number that suits their fancy. This is the subjectivism I reject.
There’s still more to it. The so-called Keynesisn Principle of Indifference has difficulties once we allow infinity to creep in arbitrarily. That is, if you start from scratch by saying you have some proposition X which can take any continuous value, then you have problems in how to assign a probability to the values X can take. There is an enormous literature here.
We can, and we do, bypass all these problems by staying in a discrete, finite world (this is accepted by all), and that we should follow Jaynes’s advice and not jump to infinity until the end. We’ll talk more about this idea when we come to “priors.”
Rich I said ‘f.ex” what means that I made the numbers up 🙂
But as the probabilities depend on the ratios of surfaces , it would be really fast to describe the particular Maya dice that has these particular probabilities I mentionned .
.
William
.
That is precisely why I constructed the Maya dice which is also fair .
By taking the same P1 and P2 as you did and observing C , my answer would be :
Then Pr( C | P1 & P2) = 1/8 .
And if you ask why , I answer because of the symmetries .
Then you would probably tell me “Ah but your die is slightly different from mine !”
And I would say “Ah but it is said neither in P1 nor in P2 that it shouldn’t ” .
My point being that if you have no information about the symmetries you can tell nothing about probabilities of a dynamical state be it a dice , an electron or a falling feather .
But yes it can be inverted – having the probabilities (infinite number of throws) , I could deduce from there what if any symmetries exist . It would be impossible with 1 try .
.
An n sided object changes nothing for me . It only increases the number and the complexity of potential symmetries and there is only 1 case among infinity where the probability would be 1/n .
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The Leprechaun case is indeed amusing 🙂
I had to think a bit to find out what was the necessary (unsaid) symmetry for your conclusion to be true .
It is actually because you didn’t specify the selection process – in the case of the die it is by throwing it .
If your selection process is “pick the male with beard” , then the probability to pick the Leprechaun is 1 .
If it is “close the eyes and point a finger” then all depends on the spatial configuration and the probability is anywhere between 0 and 1 .
There is only a small class of selection processes where the probability is 1/3 and all of them would present 3 symmetries .
Briggs,
I’m not happy with a subjectivist interpretation either, any more than I am with moral relativism so I’d really like to see the answer that evades it.
I have a feeling that your example, so beloved of authors of books on probability, is actually contaminating the discussion with considerations of real-world dice that you’d announced would come later. So how about this:
P1 – S is the set {1,2,3,4,5,6}
P2 – M is a member of S
—
C – M is 6
Then P(C|P1 & P2) feels like it has to be 1/6. It’s very difficult to see how it could be anything else.
So, are we using our intuition – unavoidably subjective – as the start point but then constraining it with logic so that the result is not abitrary?
“Then P(C|P1 & P2) feels like it has to be 1/6. It’s very difficult to see how it could be anything else.”
Tell me how you pick M and I will tell you how it could be anything else .
“Tell me how you pick M and I will tell you how it could be anything else .”
Like this: M is a member of S. There really isn’t anything else.
“Like this: M is a member of S. There really isn’t anything else.”
OK . Like 1 or 3 . Nothing has been picked , C doesn’t exist and nothing happened . There are no probabilities you only keep looking at a box with 6 things in it with M staying undefined . As soon as you want to have something particular for M , you tell me how you do .
Maybe it could be more explicit:
P1 – S is the set {1,2,3,4,5,6}
P2 – M is an arbitrarily chosen member of S
—
C – M is 6
P(C|P1 & P2) = 1/6
I don’t believe this adds more information; I do believe the information is sufficient.
“C doesn’t exist and nothing happened “. This isn’t true. I drew the conclusion “M is 6” which now exists. It is neither definitely true nor definitely false (it could be true but it might be false) so it’s uncertain and the measure of its uncertainty is the probability 1/6.
It adds much because of the tautology of “arbitrarily” in P2 .
If I insist on a precise description of the process of “picking” or “choosing” it is because this process doesn’t happen in the void .
It is and must be a precisely decribed physical process .
And it is important because all and any of these processes will modify the probability of the result .
See what I said about William’s Leprechaun’s example .
So how do you choose “arbitrarily” ?
“It is and must be a precisely decribed physical process.” Well, no. A set is not a physical object so no physical procedure is involved. You could always represent the set with tickets in a box or whatever but then you introduce extra considerations about how they’re going to be selected. (See what I did there :-))
I don’t believe “arbitrarily” introduces anything quantifiable so I can’t see that it’s tautological. It’s very like “ignorantly” but with an added dash of “but I don’t care!” To be honest, I think Briggs’s paper linked above essentially dances round this point without finding a completely convincing resting place but, rather, invites us to adopt the point of view the he thinks works best. And it works best because, even if subjectivity isn’t driven out the door it at least consigned to a corner out of the way.
(Had we been at dinner our wives would have left us the port would all be gone).
Rich you didn’t read carefully .
I talk about the PROCESS of choosing (what you correctly quote) and you talk about sets . It is not the same thing .
So you still didn’t answer how you choose arbitrarily .
It is not important whether you choose Leprechauns or Calabi Yau manifolds , it is important HOW you do it or even whether you can do it at all .
Because if you can’t answer this question , then you can’t choose either and your P2 that uses the words “arbitrary choice” (it is not me who has introduced the process of selection it is the P2) becomes meaningless .
Once you begin to really think about what it means choosing , you will see that it is a precise perfectly described physical process .
And you will also see that the probabilities will change when the process of choosing changes like the probabilities changed when I used the Maya pyramid die instead of the cube die some posts above .
William,
your implicit ‘I’ axiom seems to be “unless otherwise stated, a flat distribution is implied”. That might be intuitively reasonable, but you need to clearly state it either as one of your assumptions or as part of your proof procedure.
By the way, where are these class notes? I’ve hunted all over this site and had no luck locating them.
Cheers!
— Rafe
PS If you haven’t already, you should try “Old Speckled Hen”. Cracking beer.
Tom:
“Never surrender. Never, never, never.” as Churchill said. I shouldn’t have given in and diluted the original. I retract version 2 and return to my original syllogism. P2 is “M is a member of set S”.
You’re determined to have all this made physical somehow and, I think, are demanding to be told what the exact physical process is that selects M from S. My entire point (since I added nothing else to what the good Briggs said) is that the logic works in the realm of abstract sets and so dispenses with the problems that arise when you try to find physical instances that resemble it. Not being a teacher I don’t have a better explanation.
(I was going to mention Zermelo-Frankel Axiomatic Set Theeory till I realised I couldn’t remember any of it).
Rich:
Yes you are right about where I aim .
Namely that probabilities are not a property of a set but of the action of choosing .
That’s why it is not important that you don’t mention ZF , it has not much to do with probabilities .
Well actually there is a weak link – if you want a subset , you must exactly specify how you choose/define the elements of the subset .
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That’s also why I pointed out several times that unless you specify HOW you choose , you must not talk about probabilities .
Btw I do not necessarily ask for a physical process . I only think that experience shows us that it is hard to choose unphysically (e.g by not using mass , energy , momentum or space-time) .
But of course if you know how to choose unphysically and it makes logical sense , I accept it too .
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So in the initial version
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P1 – S is the set {1,2,3,4,5,6}
P2 – M is a member of S
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C – M is 6
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C – M is 6 is still undefined as I said many posts above .
You have then only 2 choices :
1) You let it undefined and we stop there . The trivial consequence is that the probability is also undefined . It can be 1 or 0.5 for all we now . Only 0 is excluded .
2) You tell me how you choose and I tell you EXACTLY what the probability is and why it is so . I can also do it the other way round , take any number in ] 0 , 1] and I tell you how I choose so that the probability is this number . If you want , it can also be equal to 1/6 in few very special cases .
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If I intervened in this discussion it was because clearly the only variant which is interesting and makes progress is 2) . 1) permits only to say that the probaility is not 0 what would be a rather poor result for a so long thread 🙂
Tom,
I don’t entirely follow you. In 1), how is 0 excluded, but not 1?
If I’m reading you correctly, you’re saying that the probability of getting a 6 is wholly dependent on the method of choosing. But if that method is left undefined, I can envision several selection methods that would guarantee 6, or not 6. Most obvious, of course, is simply to consciously choose 3. Or 6.
It seems to me that any method allowing a probability of 1 for a given M would correspond to a probability of 0 for all other Ms.
Tom, Here’s a thought. A classical syllogism:
P1 – All M’s are a member of S
P2 – Some member of S is 6
C – Some M is 6
Before you object that this is manifestly a tautology I will draw your attention to my earlier post where I said just that. Better philosophers than me have argued that (and got paid for it!)
Assuming you accept the validity of this classical syllogism I will add two things: a definition of S – {1,2,3,4,5,6} – and our intuition. (Notice that a basic intuitive logic is also necessary for us to make a start on ordinary syllogisms). Then having extra information, we can draw a more definite conclusion. We go from “Some M is 6” to “an M is 6 with probability 1/6”.Not completely definite but more than a mere “some”.
Hmm. I don’t see anything here to change your mind and I’ve begun to repeat myself. I think I’ll wait and see what’s in part 5.
Ted right .
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A process of choice that picks 6 with a probability of 1 is indeed a process that can’t pick anything else . It’s easy to construct one and there is actually an infinity of them .
And indeed that proves that the probability is totally dependent on the choice process .
If you read what I said about the probability of choosing a Leprechaun in William’s example you clearly see how it works . This probability is undefined untill I decide how I choose .
And once I decide that , I can get any number I wish .
It is also a good analogy how quantum systems “choose” when they proceed from a spectrum of possible values of a parameter to the actual value that you observe .
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Rich I have no problem with your syllogism . It is indeed a tautology but that is not a problem . Tautologies exist .
The follow up is precisely what was my point . “Extra information”
“Extra information” doesn’t mean intuition . It just means more information because you notice yourself that there is no path from the tautology mentioned to a probability .
I only go one step farther and precise what kind of “extra information” is needed .
The precise definition how one goes from some M to an M what is nothing else than the process of choice (because you see/measure or otherwise acknowledge this particular M)
And then I’d go still one step farther and show that one of the most important characteristics in the choice process are its symmetries upon which I commented earlier .
Unless I missed something , I didn’t need intuition at no stage of the chain .
And it actually teaches us a bit more than only a tautology .