We made rather merry at the Chapter House last night, an annual tradition. Class begins in two hours. I can still hear the vuvus. Sheesh.
Statements about the unknown made with reference to the known are common enough. The example David Hume gave us is, since all the many flames I have observed before have been hot, I expect the next flame will be hot.
That kind of inference is inductive. The outcome itself—the future flame being hot—is not guaranteed by the universe to be true. By which I mean, there are no set of true premises known to us that allow us to deduce the statement “the next flame will be hot.” It could be, for example, that there exists some mystery physics which will come into play with the next flame, causing it not to be hot.
A deductive inference known to everybody (from yesterday) starts with the premises, “All men are mortal, and Socrates is a man.” We supply the conclusion of interest. If we supply, “Socrates is mortal”, then our intuitions and the given premises tell us the probability of the conclusion is one.
But if we supplied “Socrates the immortal man has the flu” then given just our stated premises we cannot say anything about the probability of this conclusion, because, based on our knowledge of common English words and names, the conclusion is not related to the premises.
In this situation, it is easy to see that the conclusion and premises are unrelated; but of course it isn’t always so easy to gauge the interrelatedness of premises and conclusions. Just think of politics to know what I mean.
Supplying reasonable premises—which are nothing but data, models, and judgments by another name—and relevant conclusions is the basis of all probability modeling. Two people with different premises (models, data) but the same conclusion can come to different probabilities of the truth of that conclusion.
Both of those probabilities are correct, given the premises. It is always the premises—the evidence—that are in dispute.
The important thing to take away is that the truth, falsity, or probability of any statement cannot be made without reference to something. We need some evidence, premises, information upon which to condition any statement.
The probability of a die showing a ‘6’ is not “1/6″l; but the probability of a die showing a ‘6’ given that the die is a six-sided object with just one side labeled ‘6’ and that only one side will show, is 1/6.
This probability, given these premises, has been deduced. If you want to bring up questions about this or that real, physical die, well, then you are adding premises, different evidence. If all—which means everything—you know is that you have a real, physical die with six sides, just one of which is painted with six dots, then the probability of the conclusion is 1 in 6.
If you’re interested whether this next toss will result in a six, well, that’s the same thing. If you’re interested in how many times a six will show in the next dozen tosses—its relative frequency—then that is easy to compute, too (we’ll learn how tomorrow).
If you add the premise that “Something might be wrong with this die; something which subtracts from its symmetry”, then this is equivalent to “The die is asymmetric or it is not.” That is a tautology and is always true, no matter what the state of the universe and no matter whether your real die is symmetric or not.
Tautologies add no information, they cannot change the probability of any conclusion. Pause and think hard about this, because this important fact confuses. If we want to learn about this die we have to change the argument, especially the conclusion. We also need to add premises in the form of observations of actual die tosses. We’ll learn how to do that later, too.
As we have long discussed, we do not need to add any premises about “randomness” or “fair dies” to our premises. Random only means “unknown” and nothing more. Of course the conclusion is not certain: we cannot deduce a ‘6’ will show (without adding premises). The outcome is uncertain, which means it has a probability greater than zero but less than one.
Adding words about “fair” die is equivalent to saying “the probability of a ‘6’ is 1/6” to the list of premises. This is not wrong, but it makes the argument circular. It puts the conclusion in the premises, from which we can deduce the conclusion.
Added a few paragraphs later is the important point that the die must be symmetric–i.e. all sides the same shape. Shaved dice are pretty amazing. It takes little shaving to make the numbers on the faces normal to the short sides to show much more frequently.
Mystery physics not required.
1. Housing prices have always gone up. Houses are assets. A loan on an appreciating asset is risk free. Therefore a loan on a house is risk free.
2. In every set of loans there has been a subset that was repaid according to the loan agreement. A CDO is a set of loans. Therefore every CDO contains a set of loans that will be repaid according to the loan agreement and are by definition risk free.
It’s obvious that “The die is asymmetric or it is not â€, just as “Socrates is a man or he is not†and “All men are mortal or they are notâ€. But adding the premise of “fair (six-sided) die†is circular and putting the conclusion in the premises?! Hmmm, I don’t get it. Without the additional premise of “fair die,†I fail to see how one can conclude that the chance is 1/6, to tell you the truth. Evidently, my newly acquired bifocals haven’t helped me see better in this case.
The die must be a special case of the six sided object. It must be a cube.
The die example seems to be a case where our ignorance of the physical details of the die leads to the estimate that the side marked with 6 is as likely to turn up as any of the others. This would not preclude a bias caused by an assymmetry, but in ignorance of whether such a bias would be in favor of or against 6 showing, we are still stuck with an estimate of 1/6 for the probability. Perhaps it could be said that it suffices for our ignorance to be symmetric, and not necessarily the die, for the estimate to be valid.
Make that “asymmetry”. I must have listened to too many presidential speeches.
Mr. DeWitt,
If I don’t know, then I don’t know. I am not Spock, but one logical conclusion I can come up with is that the probability falls between 0 and 1, which is basically “I don’t know.†No estimation on my part.
I’ll accept the conclusion that the probability is probably 1/6 since a regular six-sided die is probably fair (by symmetry), though.
The, or a, point of this item is that one must understand the facts of the case under study and be able to distinguish FACT from inference/supposition.
All too commonly people make assumptions regarding “facts” unconsciously & as a result fail to realize those weren’t really “facts” after all (i.e. their assumptions/inferences were wrong). Which is to say they changed, within their model, the physics of the real-world situation.
Typically, those assumptions/inferences are perceived as an either-or option, with the “obvious option being mindlessly selected & then incorporated into the analysis[model]. Example involving a six-sided dice:
– integer numbers one thru six with one number per side
– the dice is symmetric
– the dice is balanced evenly
– the numbers are legible (i.e. that a reading taken is accurate; perhaps a reflection of the dice itself and/or its interaction with other experimental equipment)
– the dice in your experiment is comparable (interchangable) in every relevant way to a dice in the other experimenter’s experiment
– etc.
If any of those is unknown AND relevant to one’s analysis then one MUST account for that in some fashion, either by collecting additional information, by conducting a sensitivity analysis to assess how significant such an influence might be, etc. (or just noting the facts known & assumed).
By the way, I will argue that Dr. Briggs made just such a mistake by stating:
“If you add the premise that “Something might be wrong with this die; something which subtracts from its symmetryâ€, then this is equivalent to “The die is asymmetric or it is not.—
…because ‘something wrong’ could be a lot more than a/symmetry, as noted above. Very often it is impossible to measure what/why an asymmetric result is observed (maybe due to physical dimensions, or inclusion of a material defect/weight, or maybe the dice IS symmetric by every valid criteria but appears asymmetric because the electro-optical device used to measure the number that arises cannot distinguish between a one & a five due to optical conditions that were unrecognized at the outset, etc. etc.).
Again, its human nature to take a complex situation and, where uncertainty is detected, perceive then presume Either-Or possibilities/options when the reality is much more diverse.
That sort of thing–incorrect unconsciouly made implicit assumptions–happens all the time in a variety of disciplines — which is partly why engineers (especially “old timers”) tend to be distrustful of analytical models & prefer to test a physical object as often as possible.
JH, I agree. This topic has come up a handful of times on this blog and I’ve never seen a convincing argument that P(we throw a six | conditions as stated) = 1/6. Perhaps the problem is with me, but I don’t think so. I suspect that Briggs is implicitly suggesting that statements of probability are statements of knowledge. That is, in the absence of information to the contrary, we will throw a six in 1/6 of all possible worlds compatible with our state of knowledge.
However, this kind of “reasoning from symmetry among possible worlds” doesn’t sit well with me since it suggests that a slight change of the problem statement to
P(we throw a six | conditions as stated AND no roll has prob. 1/6)
still equals 1/6! This may be true in the possible worlds interpretation of P(_), but that’s almost certainly not what most people think when they see statements of probability.
Note that in his example Dr. Briggs assumed only two of Ken’s assumed facts were true: that the die had six sides and that one (and only one) of the sides was marked with a 6. His probability estimate would have been equally valid if the sides had been marked with the numbers 6 through 11. Additional information regarding the shape, weighting and manner of throwing the die might provide a basis for anticipating that the probability of 6 being thrown would be other than 1/6, but in the absence of any such information our ignorance about any potential biases of the die would be symmetric, and an estimated probability of 1/6 would remain the rational choice.
I think that Rafe actually understands Briggs’ point, although talking about other possible worlds would seem to be a bit extravagant. Instead of multiple possible worlds, all that would be required to meet the conditions for the estimate to be valid would be to have a bucket full of dice of various kinds, all six sided with one side marked 6, but aside from that they might be asymmetric (I got it right this time) and otherwise biased in diverse ways, perhaps the products of an inept die manufacturer. Then, if a random die is drawn from the bucket and rolled, the probability of a 6 appearing would be 1/6. Of course, if we had an opportunity to examine the die prior to throwing it, we might discover a reason to revise the estimate, but let us suppose that the die is drawn and tossed by a trained chimpanzee to avoid those complications. The estimate applies to only the first throw of any particular die; experience with a die would provide evidence for revising the estimated probability of it coming up 6.
A friehauf has six thinns only one of which can show at a time and only one of which is marked plink. When a friehauf is munged the probability that a plink will show is 1/6.
Do I need to state that the friehauf is symmetrical?
Perhaps, Briggs can clear this up. Yes, they can be interpreted as statements of knowledge, understandably because it’s a measure of uncertainty due to not knowing. I simply fail to derive the same probability conclusion, in other words, I don’t understand the logic behind the rational estimation, as Raffe has put it, that the probability is 1/6.
I, a stubborn one, haven’t been convinced.
Hey, “Ignorance breeds prejudice†is applicable here. I don’t mean it in a bad way. *_^
JH.
For the purpose of my comment, let us suppose that the six sides of the die are labeled 1 through 6. Let us suppose that you wish to posit that the probability of a 6 appearing is less than 1/6. To which side or sides would your prejudice dictate giving the additional probability made available by the reduced probability for side 6? Or if you wish to posit that the probability for side 6 is greater than 1/6, for which side or sides would you reduce the probability in order to make up for the additional probability assigned to side 6? From the information provided, there is no way to answer these questions, any more than there is reason to anticipate a probability for tossing side 6 that is other than 1/6. Yes, it is a matter of information. If everything that we wanted to know were known, there would be no need for probability estimates.
My apology to Rafe for misspelling his name and for misquoting him. I didn’t wear my bifocals.
Mr. DeWitt,
Indeed, there is no way to answer the questions in your comment from the premises provided by Mr. Briggs. That I don’t know if the probability is greater or less than 1/6 doesn’t logically imply that it is 1/6. I just don’t know what it is based on the given premises.
We know that drawing conclusions from premises can be based on the basis of our understanding of premises. I cannot draw any valid conclusions (of full certainty), probability statements or not, when the premises are not sufficient, to me.
If you assume that no other properties of a six-sided subject can influence the outcome, then I’ll say that the statement P(C1|P1,P2) = 1/6 is valid. However, without specifying this assumption/premise or knowing it as a fact, one conclusion that I am fully certain is that P(C1|P1,P2) falls between 0 and 1.
JH,
Certainly one tries to avoid situations in which it is necessary to make a decision in the absence of adequate information, but in real life, such situations do occur, and it is then that we take recourse to probability. Saying that an event’s probability falls between 0 and 1 would provide little basis for any decision and to leave it at that amounts to dithering. As I understand Dr. Briggs’ point, he is presenting a much stripped-down example of how minimal information allows an estimate of probability better than that it is merely in the range 0 to 1. I don’t think he would disagree that a better estimate could be had by measuring the physical properties of the actual die prior to its being thrown, or by prior experience actually throwing the particular die. The estimate would be even better, indeed a perfect 0 or 1, if only we could defer making the estimate until after the event occurred. However, in reality we often must make do with what is available.
An interesting question that might be asked is what does a probability mean if it is assigned to a one-time event, i.e., to a single throw of the die. Clearly, if there are to be a succession of throws, it is an attempt to anticipate the frequency with which 6 will be successfully thrown. For the case of the one-time throw, I suspect one would have to appeal to betting odds to give meaning to the claimed probability.