**Example 1**

Not all probability is objectively strictly quantifiable. Premise: “Some X are F and x is X”. Conclusion: “x is F”. The objectivist can only say “The probability (given the premises) the conclusion is true is greater than 0 but less than or equal to 1.” This is because the logical “some” implies “at least some and perhaps all.”

The subjectivist is free to say, for example, “The probability (given the premises and my beliefs) the conclusion is true is 42.8%.” But he does so only by some mysterious introspection which, in effect, adds to or subtracts from the fixed premises. Of course, most subjectivists in practice would agree with the objectivist.

The frequentist, as with all his probabilities, must embed this fixed premise in an metaphysical infinite sequence of “identical but randomly different” premises, which is not exactly a coherent description. But again, like subjectivists, the frequentist would often in practice agree with the objectivist. If he disagrees, he is acting like the subjectivist because he is adding to or subtracting from the fixed premises in order to supply details of the infinite sequence. (There isn’t anything necessarily wrong with metaphysical infinite sequences: they are used all the time in analysis, for example. But in analysis, these sequences are precisely explained.)

In other words, the subjectivist and frequentist in disagreeing with the objectivist *depart* from the *fixed* premises, which is to say they change the evidence. But they speak as if they use the same evidence. To emphasize: they are not using the same premises. This is the primary reason for confusion and bad blood.

The problem is further highlighted by changing the question (keeping the same fixed premises and conclusion). “How much would you bet that x is F?” In this case, the problem unavoidably becomes subjective. Additional premises not supplied in the “fixed” list must be invented (though there is an infinitesimal bit of wiggle room because we *know* the probability is strictly greater than 0). These additional premises pertain to each individual’s idea of money or “utility” and the situation of the bet itself (who made it, why, the personal relationship between bettor and bookie, and on an on).

Subjectivists argue that because each individual can eventually come to a monetary amount (or utility), there exists a “true” probability for “x is F”. This is so, but only because the subjectivist supplies new premises to the official list. In other words, he changes the problem. It’s also the case that the subjectivist won’t be able to describe what these premises are; i.e., they will involve “gut feelings.” Frequentists in these cases also act like subjectivists.

Further examples are generated by changing “Some” in the premise with “Many”, “A few”, “Not that many”, “I’ve heard that a lot of”, “Most”, and others. This makes for a good exercise.

**Example 2**

Donald Williams proposed the label Statistical Syllogism, an example of which is this. Premises: “There is a n-sided object, just one side of which is labeled ‘Q’; the object will be tossed and only one side can show.” Conclusion: “A ‘Q’ will show.” The objective deduction (given the premises) is the probability the conclusion is true is 1/n.

The subjectivist is free to change this number, but only if he changes the premises. The frequentist is on more familiar ground here because it appears “tosses” are ready-made for infinite sequences.

Notice that there is no need to add words about “random” tosses. “Random” only means unknown, and it is already unknown (given the premises) which side will show. That is, there is nothing in the premises which tells us how to deduce the outcome.

Suppose n = 2. The deduced probability is 1/2, given the fixed premises. The premises say nothing about the object being a coin, particularly being *this* coin in *this* real situation. If we had a real coin in a real situation, and if we were able to list additional premises which were probative of the conclusion, we might be able to deduce whether the coin would land one side or the other with certainty. People have done this (see the work of Persi Diaconis, for example.)

But if all we know is that we have a two-sided real coin and we are not physicists enough to add any additional premises, then the probability of the real toss is still 1/2. If you want to know about this real coin in this real situation and you won’t learn physics, you can experiment with the coin and add the observations as new premises. That’s what the science of statistics does.

Incidentally, we do not need to add the premise “unbiased” to our tosses. “Unbiased” makes the argument circular, because it has in it the notion that the probability of the conclusion just is 1/n. That would make the argument into this: “Given the probability the outcome is 1/n, i.e. ‘unbiased’, the probability of the outcome is 1/n.”

Another statistical syllogism. Premise: “Two-thirds of the marbles in the bag are white and just one marble will be pulled from the bag.” Conclusion: “The marble pulled is white.” Notice we don’t need words on how marble was pulled, etc.

**Example 3**

David Stove (a follower of Williams) liked examples like this one. Premise: “Exactly two-thirds of Martians wear hats and George is a Martian.” Conclusion: “George wears a hat.” The objectivist deduces 2/3 for the probability; the subjectiist will usually agree, but etc. The frequentist must remain mute forevermore because there is no infinite sequence of events. This is because there is no event to be embedded. And that is because there are no Martians, hat-wearing or not.

**Example 4**

Premise: “If the bank would have made a loan to Jones and this list of Jones’s financial pertinents.” Conclusion: “Jones would have defaulted.”

This is a counterfactual, and a common one. The bank didn’t make the loan and is hoping that it was the right decision. Depending on the exact list of Jones’s financial pertinents, and probably given additional premises about how these pertinents are quantified, the objectivist could deduce a probability for the conclusion. So can the subjectivist. Here, both the objectivist and subjectivist are acting subjectively, unless that list of pertinents allows one to deduce—and not *guess*—what the relevant quantifications are.

The frequentist must again remain mute, for there is no embeddable sequence; there is no observation of any kind.

**Example 5**

Premise: “Between 1/2 and 2/3 of X are F and x is X.” Conclusion: “x is F.” Objectivist deduction: the probability is the interval [1/2, 2/3], or possibly (1/2, 2/3) depending on what meaning is supplied for “between”. Subjectivists would usually agree, but etc. Frequentists weep, for the idea of probability as in interval is unknown to them. However, we already showed in Example 1 that probability can be an interval.

**Example 6**

Another Stovian example. Premise: “Bob is a horse.” Conclusion: “Bob is a winged horse.” The objectivist probability of the conclusion is at least greater than the probability of the conclusion of this next argument. Premise: “Bob is a horse.” Conclusion: “It is not the case that Bob is a horse.” Now (quoting, *Rationality of Induction*, p. 166):

The latter has logical probability=0. So the former has logical probability > 0. But the schema for the former argument,

xis a horse

—————————————

xis a winged horse,has relative-truth-frequency=0. For the number of winged horses divided by the number of horses = 0/n, for some positive n.

Frequentism fails again.

**More examples**

There are many more examples which show frequentism fails but in which objectivism (i.e. logical probability) works (subjectivists in these cases usually agree with objectivists, but etc.). If there are any frequentists left, maybe some day we can go over these.

Statistics finally appeared (in Example 3), but we still haven’t got to the ideas of “priors” and all that. That will be next time. Told you it wasn’t easy! (If it was easy, we wouldn’t have some many arguments for so many years.)

**Crucial Update: Example 7**

Many people are understandably stuck on Example 2. This is for three reasons. The first is easy: because classical training emphasizes physical and not logical examples, it is natural to fall back to this training and to insist that all examples are physical in a sort of subjectivist frequentist manner.

Second. Change the example to this. Premises: “There is an n-state *device*, just one state of which is set at ‘Q’; the device will be used and only one state can attain.” Conclusion: “A ‘Q’ will attain.” The objective deduction (given the premises) is the probability the conclusion is true is 1/n. It is now very difficult (or impossible) to insist on adding premises about “equal sides” or “unbiasedness” or any other thing which makes the argument circular (see the comments).

Third. We are making *logical* not *physical* arguments. This distinction was easy to make in the counterfactual and Martian examples, but is often blurred in the cases of the statistical syllogism. Objective probability is a strict matter of logic. We must take the argument exactly as it is given, and add or subtract nothing from it. Just think: what if (as I am doing now) you were given just this premise and just this conclusion and asked to give a probability for the conclusion. What would you say? Why? On what grounds would you insist the probability cannot be known?

Lastly, if you’re still stuck, change “device” to this premise: “George the Martian will select from the numbers (integers) 1 through n and pick only one and Q is a number between 1 and n.” Same conclusion. It should now be starkly obvious that this is a logical and not physical argument. Note that we need no premise on *how* George will pick Q, just that he will pick it.

Incidentally, this example also works if we change “George the Martian” to “Briggs the statistician.”

The Siren Song of Reason. If we merely prove them wrong, they shall see reason and relent.

NO.

The motive is POWER and this is merely one of many tools.

Among the other tools is – ignoring, suppressing, refusing to give you Salt [work].

They’re certainly not going to abandon their Salt for any argument.

So …The Great Question..Now What?

Is it correct then that in example 6, given the premise, the conclusion â€œBob is a winged teapot” has nonzero probability as well? (because there might be teapots which are also horse, in some sense, e.g. horse-shaped teapots, belonging to the class of ceramic horses?).

Is it not that at some point, the ambiguity of everyday language will limit the utility of such arguments?

Cees de Valk,

Ambiguity in language can never be removed completely. This is just tough luck. Later we’ll see that attempts to remove it, by introducing symbolic math, does not remove and can make things worse.

The major point today is that

anychange in the premises is a change in the argument. It is an evasion (not malicious usually, nor always conscious) of the original argument.So let me ask something probably very stupid. Is the example six formulating a “black swan” event and how frequentism utterly fails by not even acknowledging it?

Luis,

I would read example six as just a basic non sequitur.

â€œThere is a n-sided object, just one side of which is labeled â€˜Qâ€™; the object will be tossed and only one side can show.â€ Conclusion: â€œA â€˜Qâ€™ will show.â€ The objective deduction (given the premises) is the probability the conclusion is true is 1/n.You are fond of saying this but it can only be true for some n-sided objects. Take a small cutting board, for example, or maybe a cell phone. Two of the sides should have a higher probability of showing after a toss (on the top, I guess) than the other four sides.

It seems that assuming 1/n is also adding a premise. The best one can say with only the knowledge that the object is n-sided is: don’t know.

DAV,

You are a subjectivist: “Take a small cutting board, for example, or maybe a cell phone. Two of the sides should have a higher probability of showing after a toss (on the top, I guess) than the other four sides.”

Your example very important because we need to see how you are

addingpremises to the original argument. The original premise was (in brief): “N-sided object, just one side with Q.” One of yournewpremises could be: “N-sided object, just one with Q, and one side longer than other.”These are radically different.

You are adding details about real objects, which we can only know about via experimentation, or perhaps through other physical deductions.

Luis,

Not quite. Problem is in defining what you mean by “Black Swan.” Maybe one way is this. P: “x can be either X, Y, or Z.” C: “x is W.” Given P, probability of Z is 0. But then suppose we see that x

isW. Well, what’s more black swan than that? Something that had 0 probability but which appeared.Seems that what most people mean is a stunted set of premises.

More on this later.

Briggs,

I was just giving two examples of an n-sided object where one shouldn’t assume equal probability. I maintain equal assignment of probability based on only the knowledge that the object is n-sided involves a hidden assumption. Lacking that assumption the deduction can only be: don’t know. Granted, it’s often a safe assumption.

DAV,

If you add a premise which says “equal sides” or “unbiased” or the like, you are making your argument circular. That works, of course, but defeats the purpose.

Maybe this example will illustrate better:

P: “A Martian toy has n sides, just one of which etc.” C: “A Q shows.” Now you can see that we don’t need to think about cell phones or cutting boards, or indeed about any

physicalobject because, of course, there are no such things as Martian toys. We take the premises just as they are.Brigs,

Article: “There is a n-sided object, just one side of which is labeled â€˜Qâ€™; the object will be tossed and only one side can show.â€ Conclusion: â€œA â€˜Qâ€™ will show.â€ The objective deduction (given the premises) is the probability the conclusion is true is 1/n”

In reply to DAV: “You are a subjectivist: â€œTake a small cutting board, for example, or maybe a cell phone. Two of the sides should have a higher probability of showing after a toss (on the top, I guess) than the other four sides.â€

Your example very important because we need to see how you are adding premises to the original argument. The original premise was (in brief): â€œN-sided object, just one side with Q.â€ One of your new premises could be: â€œN-sided object, just one with Q, and one side longer than other.â€”

You also need to add unstated premises. Given the original stated premise of: “There is a n-sided object, just one side of which is labeled â€˜Qâ€™; the object will be tossed and only one side can show.”

The only way that they probability of ‘Q’ showing can be exactly 1/n is if you assume a regular solid polygon of n sides(all edges of equal length). Any other shape will result in skewed probabilities.

Briggs,

Just because the assumption is forced doesn’t mean it isn’t being made.

MattS, DAV,

You are making your argument circular by adding a premise which states, in different language, the probability of the conclusion. You are thinking of real objects.

In the original argument, there just is nothing about the shape of the object save the n sides. It could be skewed, it could be even. One side may be 10,000 times as wide as all the others. There are an infinite number of possibilities, not just the two or three you have imagined. There is no way to know which is correct because there is no premise which says. If you add a premise, you change the argument.

As it is, imagine a real situation where that is all the information you were given (the original argument). How would you bet? Why?

So, how can you arrive at 1/n without assuming yourself a “mostly” regularly shaped object?

“You are making your argument circular by adding a premise which states, in different language, the probability of the conclusion. You are thinking of real objects.”

No I am not. I am making an argument that you can only reach the conclusion that you give (1/n probability) based on an unstated premise that the object is a regular solid polygon. No other shape will allow even 1/n probabilities for all sides.

“As it is, imagine a real situation where that is all the information you were given (the original argument). How would you bet? Why?”

I wouldn’t, because the original argument does not contain enough information to make any conclusions about the probability. Your original argument relies on the fact that nearly everyone (especially anyone familiar with dice) will subconsciously assume that the object is a regular solid polygon.

Various,

I have to support Briggs here. His argument is based on the degree of uncertainty given the premises. The probabilities may be wrong for any particular real object. A similar argument is often made in statistical mechanics with respect to accessible energy states. In the end probabilities are adjusted as new knowledge, i.e. a physical measurement, is added but it is amazing what you can get away with. Briggs just talks a little funny, but that’s because he is a statistician.

William Sears,

“The probabilities may be wrong for any particular real object”

The problem is that the probabilities ARE wrong for all but a very particular sub-set of real objects.

MattS,

If you view the probability as a level of certainty in the outcome, without any other information other than the number of sides, you are pretty much forced to assume an equal likelihood for each outcome. It’s still an assumption though.

I’ve never bought into Briggs’ position on the 1/n debate. If a_1, a_2, … , a_n are n non-negative real numbers that sum to 1, what is a_1? Briggs seems to say a_1=1/n given no other information. I’d say only that a_1 is in the range [0,1].

Dav,

“If you view the probability as a level of certainty in the outcome, without any other information other than the number of sides, you are pretty much forced to assume an equal likelihood for each outcome.”

But we do have more information.

From the original argument, we know the following:

It is a physical object.

It can be thrown.

It has sides and there fore is not a sphere.

From here is is rational to conclude that the precise shape of the object will significantly affect the probabilities of any one side showing. There fore, without any information on the shape it is inappropriate to conclude more than that the probability is greater than zero and less than one.

MattS,

Indeed. I’ll leave the esoteric reasoning for someone else but a practical reason for the assumption is that is the easiest to overcome with future observations than one that has a built-in bias if the initial assumption is erroneous.

I’m used to working with discrete nodes in a Bayes Network. The starting counts must be set to something non-negative and zero causes a lot of problems. The usual practice is to start with counts of one for each possible outcome. So right from the start, I must use 1/n.

I’m with Briggs too. The only thing you know about the object is that it can be in in one of n states. The best you can do is assign equal probabilities to these states, if you must give them.

But I would not bet on outcomes with so little information on the exact nature of the object.

All,

See the most crucial update in the main post. This is too important to bury in the comments. Plus, thanks to everybody for helping me clarify this. It is a major point.

The answer is obvious, you don’t know which facets are bigger or more inclined to be on top, so if someone comes to you and says the following premises:

“I rolled an object at home yesterday, and it has N faces.”

“I painted each face with a different letter.”

“What is the odd that the face on top is *A*?”

Knowing nothing more, you are forced to pick 1/N. There’s just no way out of this.

You are overthinking this. Taking your reasoning to the extreme, there are no “odds” even in a “regular polygon”, since physical laws determine that a certain face will be on top with 100% probability.

The odds are just the measure of our

lack of knowledge. If you don’t know the shape of the object, then you have no idea if there’s more chance that face =A= will be on top than face *B*. Because of this, you are forced to bet equally in all of them.Joining in the fun:

You have an eight-horse race. You have no other information about the race, no form, no starting prices. What is your estimate of the probability that “Sunny Jim” will win?

Ok, I didn’t get that at all, except if I correct your paragraph into this:

Now if that is what you meant to write, then yes that was what I was thinking. The fact that black swans exist are a huge argument against frequentism, and for bayesianism.

Luis,

The answer is obvious, you donâ€™t know which facets are bigger or more inclined to be on top, so if someone comes to you and says the following premises: …I was (clumsily) trying to make the point that the logical deduction requires an assumption of equal likelihood for all outcomes effectively adding a premise. Otherwise, the only possible answer is “don’t know”. It’s a forced assumption but an assumption nonetheless.

Yes. I do think however such a logical case could be geometrically proven. If you “average out” every possible geometric solution for a N face solid, then it becomes clear the odds of face X being on top becomes 1/N.

That is, if in case hypothetical 15124, face “A” “really” has (due to its geometrical conditions) only 1/2N chance of being on top, you will always find an “opposite” case where face “A” “really” has 2/N chance of being on top. Not knowing which one we are dealing with here forces us to pick 1/N.

What is your estimate of the probability that â€œSunny Jimâ€ will win?Zero. The Thoroughbred Sunny Jim died in the early twentieth century.

http://www.allbreedpedigree.com/sunny+jim9

But maybe you’re thinking of Sunny Jim the Arabian (also deceased).

http://www.allbreedpedigree.com/sunny+jim

The other Sunny Jim’s:

http://www.allbreedpedigree.com/index.php?query_type=check&search_bar=horse&h=SUNNY+JIM&g=5&inbred=All+Duplicates

Luis,

Interesting but I don’t think it works with the Martian number picker in the update. We don’t know if George the Martian has any bias for particular numbers (I guess because he’s a Martian — people tend to pick either 3 or 7 when asked for a number between 1 and 10). The answer should still be “don’t know” but to move on requires assuming he has none.

DAV,

“Don’t know” is incorrect.

I’ll add this above, too, but consider the premise: “Q will either be picked or it won’t”. Same conclusion.

Nowyou can say you don’t know, or that the probability is unknown.What you are saying with “don’t know” is that the premise “George the Martian, etc.” conveys

noinformation whatsoever, as in none. And that is just not so. There is definite information in that premise which differs substantially and markedly from the tautological premise “Q will either etc.”What you are

necessarilyclaiming by saying you don’t know is that this tautology and the original premise are logically equivalent, and that is false.Briggs,

I guess but I gave two real world examples where the assumption is erroneous. “Don’t know” is my equivalent to “insufficient information”. You only need the 1/n when you must supply an answer. The problem as stated has no definite answer at all. You can’t get to the 1/n answer without yourself assuming no bias. If you think you can explain how (saying you just do won’t cut it).

DAV,

No, sir. The burden of proof is on you to demonstrate how any tautology is logically equivalent to the seemingly information-rich “George the Martian etc.” premise. You must prove this logical equivalence to conclude no probability is deducible.

Attempts have been made: see esp. Section 6 of this poorly written paper (mine).

Matts,

Luis gave my answer, which is that the sum over all states gives 1/n. It is probability as uncertainty. This reminds me of the information theory approach to statistical mechanics (Claude E. Shannon) and its use as a definition of entropy. It is difficult to separate out probability from a physical object or event which is probably why Briggs wants to use the term frequentism. Thus I think that uncertainty is a better general term and that we might consider using probability only for specific cases.

Briggs,

Not sure I can. You keep claiming I’m stating a tautology when I say there’s insufficient information. I don’t see it. Please explain. I think I’m saying you convert the problem into one to arrive at an answer.

It is possible it is just too abstract a problem, and I can see that if one is not inclined to accept logic to work exactly that way, one can always say we are assuming too much to accept 1/N.

This reminds me of a very recent rant by Nassim Taleb warning us that real life isn’t anything like these abstract exercises, and these will only create delusions which turn out to be false and expensive. (His main example was a particular case where odds were something like 70/30, and how statisticians keep saying we are so dumb to go to the “30%” some 30% of the time, we should rather go for the “70%” 100% of the time. What he says is that we are hard wired to work in the real world, not in the abstract platonic one, and that in the real world, “odds” do not really really exist at all, they are “unknown” and so a strategy that ignores the 70/30 split is probably the correct one to take, in the real world).

Briggs,

Put another way: X+Y=1. What is X? You say 1/2 but that assumes X=Y. It IS an answer but hardly the only one. You may have a host of valid reasons for making the assumption but they are still assumptions about a hidden premise.

DAV,

This last example could use some more thought. This and the example in the text are not even close to being equivalent.

It’s a good example for the material to come, though, on “improper” priors, the dangers of infinities, and that sort of thing.

You’ve read that paper?

Briggs,

I haven’t but I’ll try to do so soon. I think my example is the same as the n-sided object one and Ol’ Geroge except with fewer states and boiled down to an equation.

DAV, you added another premiss to the argument, to wit, “Sunny Jim is dead” so your conclusion is the conclusion to a different argument.

Then you added further dead Sunny Jim’s providing conclusions to still other arguments.

But in the sunlit alpine meadows of Formal Logic Sunny Jim grazes undisturbed.

William Sears,

As long as we are dealing with an n sided physical object that is to be tossed/thrown as specified in the original argument, with no information about the shape of the object is is my contention that stating the probability as 1/n understates the uncertainty to a significant degree.

Rich,

I say it’s time to clean the premises.

Rich,

“You have an eight-horse race. You have no other information about the race, no form, no starting prices. What is your estimate of the probability that â€œSunny Jimâ€ will win?”

Since we don’t have any information on which horses are running in the race, the probability would be 1/n where n=the total number of registered race horses world wide.

As n is likely to be a very large number, zero is a reasonable approximation.

Matts,

Quote “is my contention that stating the probability as 1/n understates the uncertainty to a significant degree”. This implies that you have a value in mind. What is it? Earlier you said that an answer was impossible due to a lack of information. However, it is this very lack of information that forces the 1/n result.

I agree with DAV 14 May 2013 at 8:59 am. If X and Y are real and non-negative, and X+Y=1, we can’t conclude X=1/2. The idea that we have to conclude this because we are “forced to” seems wrong to me because we don’t have to make any conclusion at all.

I am reminded of a criticism of the 1/2 approach of the commenter named Rafe many moons ago. Suppose we flip a coin we know *isn’t* fair. Now what is Prob(Heads)? Doesn’t the symmetry or entropy argument still lead us to conclude Prob(Heads)=1/2? But we know that’s not true!

William Sears,

“This implies that you have a value in mind. What is it? Earlier you said that an answer was impossible due to a lack of information. However, it is this very lack of information that forces the 1/n result.”

No, the lack of information does not force a 1/n result, it does not force any result at all.

Given as taken from the information that we have a physical object of n sides:

For any n > 3, there is a nearly infinite number of possible shapes.

For any n > 3, one and only one of the possible shapes yields 1/n probability for all sides.

Given this, stating that concluding a probability of 1/n in the absence of any information about the shape of the object properly accounts for the uncertainty is nonsense.

To all the commenters who still doubt Briggs’ example of the n-sided object: the probability of 1/n expresses the UNCERTAINTY about the outcome “Q shows” (and not a property of an actual object, about which you don’t know anything else). I think that should be enough.

Cees de Valk’s words are true.

SteveBrooklineMA,

The idea that we have to conclude this because we are â€œforced toâ€ seems wrong to me because we donâ€™t have to make any conclusion at all. … Suppose we flip a coin we know *isnâ€™t* fair. Now what is Prob(Heads)?Your right. The best answer would be to wait and see. But if you must proceed …

If you don’t know how the coin is biased , you still have to assume it isn’t if only until you get more information. You don’t know which side it is biased toward or by how much. Any guess you make could be just as wrong and, if erroneous, harder to overcome later than assuming no bias at all.

But you really don’t have to assume no bias if you have some basis for assigning it. In his book, E. T. Jaynes mentions that his belief in ESP is such that it would take an extraordinary amount of evidence to overcome his belief in its nonexistence. He doesn’t say how he arrived at his belief. Maybe he was one of those people who take a lack of evidence as evidence.

Briggs will tell you that he is just expressing his lack of knowledge but hasn’t said why he must actually assign a value before collecting more information. He probably coming to that. I hate being led blindfolded by the nose. Academics do this all the time. In my business, trying to do that with your audience gets you tossed out on your ear.

DAV-

Define a “super-biased coin” to be a coin such that Prob(Heads)<1/100 or Prob(Tails)<1/100. If you flip a super-biased coin, what is Prob(Heads)? Is Prob(Heads)=1/2? Now I agree that if I were initializing some Bayesian learning program that will update my estimate

of Prob(Heads) after we collect data, I would set Prob(Heads)=1/2. But Briggs seems to say that this is required by the Rules of Logic. But Prob(Heads)=1/2 doesn't seem logical at all, since one of the very few

things we know about a super-biased coin is that Prob(Heads) is nowhere near 1/2!

MattS, the mention of “Sunny Jim” was intended to imply that he was one of the eight. On your interpretation – a valid one so far as I can see – the information that there are eight runners would be redundant.

DAV: Quite so. “One more such proof shall make me say ‘bah!'”

Briggs, while your update did make your point clearer, I don’t think it addressed the main concern expressed about the post.

Consider “Some Martians wear hats. George is a Martian. Conclusion: George wears a hat.” That seems to be an instance of your example 1, where we can only say the probability of the conclusion is between 0 and 1. The question is what makes examples 2 & 7 different from this? (Or, if you disagree that it is like example 1, what makes it different from that?)

SteveBrooklineMA, I expect you would agree that if we had two coins, on with P(H)=1/100 and one with P(T)=1/100, one was chosen (with equal probability if you prefer; Briggs is saying this is unnecessary) and flipped, then P(H)=1/2, completely from logic. The answer is not illogical – the debate can only be whether we can say that much in a more general situation.

Johnathan D- Sure, there is no ambiguity: P(head coin & heads)=1/2*99/100, P(head coin & tails)=1/2*1/100, P(tail coin & tails)=1/2*99/100, P(tail coin & heads)=1/2*1/100. Briggs gets 1/2 even when none of the probabilities are specified.

Steve, my point was that the fact that P(H) is nowhere near 1/2 when the direction of bias is one of the premises is irrelevant to whether Briggs should come to that conclusion when we don’t any premises beyond a choise between two sides.

Well, I don’t get it. There are 1000 marbles in a bag, each is either black or white. How many are white? Briggs seems to say we can logically conclude there are exactly 500. I say I don’t know how many are white, only that there are somewhere between 0 and 1000. Maybe we’re debating the number of angels on a pinhead.

SteveBrooklineMA,

(Haven’t read comments here in two days…too busy; but I just saw this, and only this). Briggs concludes nothing of the kind. I have no idea where you got the 500. Here is the original argument.

The premise accepts there are exactly 2/3—of whatever number of marbles there are—which are white. There is no indication

how manymarbles are in the bag. There could be 3, there could be 3,000, or even 3,000,003. Who knows.It is a different, entirely different, question: I just pulled out a white marble from this bag which has N marbles. What is the probability that the next I pull is white, too.

That’s a good question! It will show us where parameters come from.

Probability is the logical measure of

uncertainty.I like to use the word “non-arbitrary” better than “objective.” Absolute objectivity seems unobtainable when it comes to data analysis.

Hi Briggs. Nice of you to take the time to reply. I was posing a different problem in response to Jonathan D. The premise is that there are 1000 balls in a bag, each of which is either white or black, and a single ball is pulled from the bag. The conclusion is that a white ball is pulled. The deduced probability is p=1/2. Since there are 1000 balls in the bag, and p=#white/1000, we have 1/2=#white/1000, so the deduced number of white balls in the bag is 500.