Talk:Principle of indifference
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From an email:
I had to handle the Principle of Indifference a bit... I like your article but wonder how a novice would like it? May be a simpler introduction would help. At he end of the article, I suggest to introduce Dempster-Shafer Theory.
- Hrm, yeah, not too novice-friendly. A nice example would go a long way. I'll do that eventually, if someone doesn't beat me to it.
- Cyan 18:25, 11 Aug 2003 (UTC)
- I've got a book by Martin Gardner on paradoxes (the name eludes me at this time) that presents a user-friendly discussion on the Principle. If I remember, I'll paraphrase an example or two from it.
- CHz 20:12, 2 Jan 2004 (UTC)
- The book is "aha! Gotcha" by Martin Gardner. Here's the full citation:
- Gardner, Martin (1982). aha! Gotcha. New York: W. H. Freeman and Company. ISBN 0-71-671361-6.
- The following example occurs on pages 107-108:
- "Let's see how contradictions arise if the principle is carelessly applied to our questions about Titan and atomic war. What is the probability there is some form of life on Titan? We apply the principle of indifference and answer 1/2. What is the probability of no simple plant life on Titan? Again, we answer 1/2. Of no one-celled animal life? Again, 1/2. What is the probability there is neither simple plant life nor simple animal life on Titan? By the laws of probability we must multiply 1/2 by 1/2 and answer 1/4. This means that the probability of some form of life on Titan has now risen to 1 - 1/4 = 3/4, contradicting our former estimate of 1/2.
- "What is the probability of an atomic war before the year 2000? By the principle of indifference we reply 1/2. What is the probability of no atom bomb dropped on the United States? Answer: 1/2. Of no atom bomb on Russia? Answer: 1/2. Of no atom bomb on France? Answer: 1/2. If we apply this reasoning to ten different countries, the probability of no atom bomb falling on any of them is the tenth power of 1/2, or 1/1024. Subtracting this from 1 gives us the probability that an atom bomb will fall on one of the ten countries--a probability of 1023/1024.
- "In both of the above examples the principle of indifference is aided by an additional assumption in yielding such absurd results. We have tacitly assumed the independence of events that clearly are not independent. In light of the theory of evolution, the probability of intelligent life on Titan is dependent on the existence there of lower forms of life. Given the world situation as it is, the probability of an atom bomb falling on, say, the United States is not independent of the probability of such a bomb falling on Russia."
- (forgive any typographical errors: I'm tired)
- This is a great example that explains both how the Principle is applied and how it can be incorrectly used. I don't think I can rephrase in any better way, so does anyone have an objection to me slapping this into the article?
- I was also thinking about adding a simplified definition, maybe something like "In layman's terms, the Principle of indifference states that, if we have a list of several independent events and have no reason to believe that any are more or less likely to occur than others, then we should assume that each has an equal chance of occurring." Any thoughts? CHz 05:31, 5 Jan 2004 (UTC)
Quoting such a large chunk of text may be a copyright violation, so check up on fair use rules before you paste it into the article. Also, feel free to go ahead and make any changes you think are appropriate - typically, people use the talk page for proposing really big changes, but relatively small improvements can be discussed after the fact, if anyone feels it's necessary. Cheers, Cyan 15:45, 5 Jan 2004 (UTC)
- You do have a point there; I forgot all about fair use. You can tell I'm still gettin' used to all this. "Just because you quote something and give the source doesn't mean your reference is legal." I'm working on some legal examples for the page. I should hopefully have them up within the next couple of days, providing I don't go on unexpected vacation again like I did on Monday-Thursday. CHz 03:43, 10 Jan 2004 (UTC)
- Ta-da! I'm finally done. I aded a somewhat simpler definition, extended the sections on coins and dice, and added a new section on misuse. Feel free to dissect my work as you see fit. I do have a tendency to write things that make perfect sense to me but no sense to anyone else. CHz 03:48, 18 Jan 2004 (UTC)
Not bad, not bad. However, I believe that one of the explanations is flawed. I am refering to the section under the heading "Ranges". You wrote, "These conflicting results arise because the middle of the ranges is not the "best" guess." I believe this is incorrect. From the point of view of decision theory, there is no best guess until a loss function is specified; once it is, the best guess is the one that minimizes the expected loss. If the loss function is symmetric, then the median (also the mean) of the bounded uniform distribution is the optimal guess.
The unknown box example does demonstrate why the principle of indifference cannot be applied to a continuous variable. The reason why it fails is that the example implicitly assumes a uniform epistemic probability distribution both for the length of a side and for the volume; these two assumptions are contradictory. Choosing the middle value as the best guess for both variables and arriving at a contradiction is simply a consequence of these two implicit, contradictory assumptions.
In general, for continuous variables, the principle of indifference does not indicate the which variable (e.g. in this case, length or volume, or surface area, even) is to have a uniform epistemic probability distribution. This is the reason why the principle of indifference can't be applied to continuous variables.
What do you think? -- Cyan 05:43, 18 Jan 2004 (UTC)
- Yeah, I struggled with that section a bit while writing. I read about the unknown cube paradox in a book, but it didn't have an explanation about how to resolve it, so I had to come up with my own. (It also didn't help that I couldn't find any information on applying the principle of indifference on continuous variables.) I tried to get around the loss theory issue (in my mind, anyway) by assuming that "better" means "more likely." The statement "The principle of indifference is normally misused by being applied to dependent events. It is also used as part of a faulty argument involving ranges of values." (which has a grammar error in the first sentence... whoops) could be changed to "The principle of indifference is normally misused in one of two ways: it is misapplied to dependent events and continuous variables."
- And, of course, your explanation is good. It stands up to examination more than mine does. =P CHz 04:40, 19 Jan 2004 (UTC)
Shall I work my explanation into the text of the article, or do you want to do it? -- Cyan 04:47, 19 Jan 2004 (UTC)
- Well, you seem to have somewhat more experience with the application of the principle of indifference to continuous variables, so it's probably best that you do it. I don't mind you tampering with my baby... CHz 03:59, 20 Jan 2004 (UTC)
Okay, all done. -- Cyan 04:48, 20 Jan 2004 (UTC)
- "Not bad, not bad." I just rephrased the introduction to the "Ranges" section; no new content. (I like the addition of surface area, by the way.) CHz 04:12, 22 Jan 2004 (UTC)
Hello. This is a nice article. I think the part about continuous variables can be improved. The principle of indifference doesn't imply that all functions of a quantity must be uniformly distributed if one of them is. If I assume surface area is uniformly distributed, then the p.i. does not apply to volume, since I am not assuming that ranges of the volume of equal size have equal probability. Likewise if I assume volume is uniformly distributed, the p.i. doesn't apply to surface area. The point is that if I make an assumption about one variable, I'm no longer ignorant about the other; the p.i. has nothing to say about the other, so I can't construct a contradiction. I agree that there is some subtlety to working with continuous variables, but it doesn't follow that the p.i. is not applicable. I invite your comments. FWIW, Wile E. Heresiarch 17:04, 16 Mar 2004 (UTC)
- You are correct, of course. I think that historically, the essential complaint is that nothing in the P.I. tells one which parameterization of the problem is the one in which P.I. applies. If one uses it in a real data analysis problem, one immediately invites the criticism, "Why did you choose to make that parameter's prior uniform? Why not its square, or its square root, or some other arbitrary function?" A principle that applies to continuous parameters really ought to provide the answer to this criticism, but P.I. doesn't. (The principle of transformation groups claims to, but I haven't tried to write that article because (i) I have misgivings over some of the applications, and (ii) my math skills aren't quite there yet. Good username by the way. "Heresiarch." Heh.) -- Cyan 17:37, 16 Mar 2004 (UTC)
Hello. I have a comment on the "life on Europa" example. The inconsistent result is due to the erroneous assumption of independence, as pointed out by Gardner himself -- he's quoted above as saying "We have tacitly assumed the independence of events that clearly are not independent." So although this example is a cautionary tale -- you have to be careful about dependence -- it's not clear to me that this example tells something about the principle of indifference. Is there some way to rephrase it so that the problem is clearly with the p.i. and not with an unjustified assumption about independence? Happy editing, Wile E. Heresiarch 14:16, 29 Mar 2004 (UTC)
- Hmm. Well, to me the example seems perfectly clear, but that's likely a side effect of the fact that I wrote that portion. How about this revision to the last paragraph?
- "This probability of 3/4 contradicts the previous probability of 1/2. These contradictory results occur because (assuming evolution is valid) the principle of indifference has been applied to dependent events. The probability of multi-celled life is not independent of the probability of single-celled life; multi-celled life would develop from single-celled life."
- How's that? CHz 04:37, 30 Mar 2004 (UTC)
- I agree the example is perfectly clear, but what is clear is that a mistaken assumption of independence leads to a mistaken conclusion. I don't see that the example, either in the original version or the rephrased version above, says anything about the principle of indifference. Whether the probabilities were assigned by the p.i. is immaterial; the answer will be wrong whatever kind of assignment is made. Can we find an example that speaks directly about the p.i. ? Wile E. Heresiarch 04:49, 15 Apr 2004 (UTC)
- On a related note, the p.i. can certainly be applied to dependent events, so that can't be the source of the problem. For example: you roll two dice and then you get to spin a roulette wheel with a number of segments equal to the sum of the dice. The second outcome depends on the first, yet a conventional analysis (as correct as can be without making a detailed physical analysis) would be to apply the p.i. twice. Wile E. Heresiarch 04:49, 15 Apr 2004 (UTC)
Sorry about not replying for a month; I somehow managed to remove this article from my watchlist. And then I went on vacation. =P
I read your comment yesterday and did some thinking about it: working it out, realizing I need a probability textbook, and so forth. Suddenly, while I was working on something else, I got a flash of inspiration, so I ran over to the computer and typed up the next two paragraphs in a minute or two. I then went back to what I was doing. It made sense when I wrote it, and hopefully it'll make sense to you.
- I think I figured out why the principle of indifference works on the dice-and-wheel example but not the Europa one. The problem lies not in the application on dependent events in general but rather on dependent events that are unaccounted for. In the dice-and-wheel example, the p.i. can be used with no problems on the roll (although a single die might be better because the probability distribution of the total of two rolled dice isn't uniform) and can therefore also be applied to a wheel whose number of segments is equal to the number rolled by the dice. In this case, the certainty of the probabilities of one event allows the p.i. to be applied to another event dependent on it.
- In the Europa example, however, the principle of indifference can't simply be applied to the existence of single-celled or multi-celled life because each is dependent on a large list of other factors that need to be accounted for, including temperature, moisture, the existence of certain elements, etc. The probability of life cannot be boiled down to a simple "either there is or there isn't" dichotomy, which is why the p.i. cannot be applied without further information.
Does that sound right? CHz 17:57, 7 May 2004 (UTC)
Following up on this discussion from last spring -- I've struck out the section titled "Dependent Events". The essential difficulty is that Gardner's puzzle hinges on an inappropriate assumption of independence, and doesn't say anything about the principle of indifference. It may be possible to reformulate the puzzle to directly address the p.i., but, absent any evidence that such a mistake or misapplication has been made more than once, we're veering off into original research. The implication in the previous revision was that application of p.i. to dependent events is a common mistake (the article said "normally misused") but I've yet to see that in a decade of reading, and there's no evidence presented that this misuse occurs "in the wild". If anyone has such evidence, we can discuss how to restore the section, otherwise, I just don't see the point of it. For what it's worth, Wile E. Heresiarch 17:56, 15 Aug 2004 (UTC)