5:50 PM
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Q: The Principle of Indifference in Overlapping Sets

Adam GoldingI have revised this question to make it more obviously answerable: MY QUESTION: Does anyone know of any scholarly sources which argue for claims that would bear directly upon answering the 'under-defined' problem below? THE QUESTION MY QUESTION IS ABOUT: Suppose that $A$ and $B$ are sets such...

 
Well, I have an application in mind, but I am also interested in the abstract philosophical question as to what probability model is appropriate here if you really don't know anything other than the propositions stated in the question. (Perhaps invoking the principle of indifference in some way.)

The application I have in mind involves performing two google queries, looking at the total results for each query, and estimating the number of results in the intersection, and that sort of thing.
 
Adam, if you are unwilling to say anything about your application nor to provide any more detail, then this question will have to be closed as overly broad and unanswerable. The commenters (cardinal and Henry) have ably and clearly indicated what is missing and what needs to be done to improve the question.
 
whuber please do not confuse an ignorant question with a question about ignorance--this is a question about how to apply the principle of indifference--they have not given any reason to believe that there is no way to apply some form of the principle of indifference in this case. (Also, I did describe the application, in my second comment.)
cardinal your use of A and B as though they are sets of events means you are answering a slightly different question, but your intutition is similar to mine, except that I proceeded more arithmetically:

if |A|=|B|, then we have identical information about A-B, B-A, and AnB, so we should set them all to be equal. If |A|<|B| or vice versa, making all three equal is not possible. I am inclined to say |AnB| = min(|A|,|B|)/2 but I find that intuition hard to justify and therefore suspect :-)
 
I can see, Adam, where you want to go with this, and I find it interesting, but the difficulty we're trying to cope with is that the question is ill posed. Until you can supply sufficient information (or assumptions) for there to be an answerable question, we cannot discuss indifference or ignorance.
 
I would like to remind everyone that I did give a concrete example in the second comment to the question: you search for two terms on google, and get two different numbers of search results--estimate the number of results in the intersection. Individual responses follow.
@cardinal sorry, I was actually answering Henry, I got your responses confused--what your response essentially points out is that all possible values from 0 to 1 are compatible with Kolmogorov's axioms. I am aware of this, but there is no good argument to the effect that only Kolmogorov's axioms should be consulted in determining probabilities. For instance, the various formulations that have been made of the Principle of Indifference in the literature are all logically independent of the Kolmogorov axioms. Also I had a typo, "as though they are sets of events" should have read "outcomes".
@Neil G: not true, you can measure the probability of something that is not an event by first choosing an appropriate sample space, and by modelling it as an appropriate event--as an appropriate subset of that sample space--there is a reason this question is tagged with 'modelling' and 'model-selection'--the probability axioms do not tell you what model to use but that does not mean all choices are correct.

Here the model/modelled distinction is particularly important since what we are modelling is itself already formally described, but the best model could be distinct from it.
@whuber that gets it backwards--if I supplied extra (biasing) information, I would be violating indifference--it is precisely when information is 'missing' that we require the principle of indifference. For what it's worth, I once heard second hand of a paper that attempts to solve something like this problem by trying to 'maximize' a few distinct measures of indifference, but I cannot find it. Suffice it to say that the question should not be voted down if it is actively discussed in the academic literature. Also, as mentioned above, I did provide a concrete example.
@NeilG because I would like to do this for larger numbers of terms as a heuristic for directing further search--if I have n search terms then you (in the worst case) need at least 2ⁿ queries to determine the values of all logical combinations of them. (i.e. all 'pieces' of the 'venn diagram'). Thus, I want to make the 'best' estimates possible based on the least information possible, in order to limit the total number of queries, and increase the value of n which is feasible. I.e. the purpose of making estimates based on nearly no information is to decide when to gather more information.
I am also interested in making inferences based on more complex combinations of queries, such as those mentioned in the more complex question linked to, but this is the simplest such question I could dream up.
 
5:50 PM
Regrettably, this thread has become a discussion: it may be interesting, but it has deviated far from the format and accepted practices of this site. Please carry out further discussion in chat. If that enables you to formulate an answerable question, then of course you can post it as a new question.
 
"We expect answers to generally involve facts, references, or specific expertise;" I have made the question into a request for references to relevant literature--please review the changes--is this sufficient?

Btw, with the stated policies, I fail to see how anyone can post a question about model-choice. Please also note that I do not yet have the reputation necessary to participate in chat, as I just joined here :-)
 
Adam, I really appreciate your efforts to help us understand the question and to keep editing it. But the comments have gotten to be such a mess (mass?) that they preclude any constructive engagement with the edits except through replies, which is not a good idea. Would you mind copying out the portions of your edited question that you wish to keep and starting a new thread so we can proceed with a clean slate? If that one starts filling up with comments, flag it and a moderator can create a chat room for you.
 
Ok, before I do that, is there a different combination of tags you would suggest for this question?
p.s. the existence of the 'philsophical' tag almost seems like reputation downvote bait at this point, lol
 
@Adam, As a bearer of a philosophy degree, I don't mind :-). But it might be wise to avoid that tag. You could settle for [probability](/questions/tagged/probability) now and add more tags later. In the meantime, I created a room for further discussion: it should be obvious which one (and ought to be visible through the 'chat' link). You should have full access; if not, please reply and let me know.
 
Ok, I think I'm in--this is a spiffy interface :p
 
5:53 PM
@AdamGolding Hi, Adam! Before we start, would you mind telling me how you managed to migrate everything to chat? Occasionally a prompt appears, but if it doesn't, I haven't found any other way to do it!
 
I saw "Please avoid extended discussions in comments. Would you like to automatically move this discussion to chat?" and clicked "automatically move this discussion to chat"
 
@AdamGolding OK, that explains it: somehow you managed to raise that prompt. Thanks for responding to it!
I don't have much time--my lunch break is over--but I do want to be as helpful as I can and I see you are working hard to formulate a good question.
 
oh, ok, well I was just about to make a new question with basically everything before the line "[Further Remarks (from the older less 'meta' version of this question):"
 
One thing most commenters are looking for is more guidance. You have resisted that, challenging us to apply a Principle of Indifference or something like it. Fair enough, but I think that there's not even enough information to get started.
For instance, if an indifference principle is to be applied, we need a clearly specified set of alternatives among which we can be indifferent.
If the alternatives are not specified, there's not much one can do.
Another thought I wanted to share is that the reformulation of your question is starting to sound a bit like capture-recapture sampling.
 
Well, ET Jaynes's article handles precisely that problem, although its not clear how easily it can generalize to this situation
(not capture-recapture, but your previous remarks)
 
5:58 PM
A good example is to estimate how many fish are in a pond. You catch a bunch, mark them, put them back, let them swim around, then catch some more, among which some are marked. You can estimate the entire population from that information.
I'm familiar with Jaynes' book, but I'm not sure how his approach applies when the sample space is completely unspecified.
 
Well, me neither, but the entire reason that I wasn't specifying a sample space was that I wanted advice on the appropriate choice of a sample space.
 
It might also be worth adding--because you're fairly new here--that successful questions tend not to be abstract (although they might include some mathematical formulation). You're starting to go in a good direction that way with your example of Internet searches. Concrete examples help a lot. (Sometimes in the effort to make our problem simple, we leave out the essentials.)
@AdamGolding OK, choice of sample space is a good thing to bring up then. Answers will likely involve extra-statistical considerations, so you ought to think about providing useful background information to guide people there.
Something else that may come up concerns the possibility of statistical dependence between the two Internet searches: we would expect that. So, it might not be valid to think of A and B as being independent samples of the same set.
 
Yeah, I'm worried that people will answer some very emprirical question, rather than the indiferent quesiton I"m more intersted in--for isntance, in that Jaynes article, he shows that three different answers can be appropriate for different experimental setups, but only one of those three is appropriate if you don't know what the experimental setup is. I wouldn't want ONE of those three answers on the basis of the example I provide, I woudl want the indifferent more abstract answer :p
I suppose I should say what I just said there, lol
 
@AdamGolding I appreciate that difficulty. You could emphasize the general nature of the question at the outset and offer the example with some diffidence. You could also offer two or three examples to show the generality of the question.
Just be a little careful not to go overboard. I know that when a question fills more than one page, I become much less inclined to read it through.
Others might have similar reactions.
 
yeah, that's why I tried to make it like three lines initially :p
I hadn't realized how much people would assume any mention of a set was a mention of a set of outcomes in a sample space :p
 
6:05 PM
@AdamGolding Yes, it's not an easy situation. I'm sure that some of my advice may seem contradictory, because you need to balance multiple aims.
 
kind of like my attempt to satisfy different a priori aims when creating an appropriate 'indifferent' estimate :p
 
@AdamGolding Well, a set sure sounds like an event, doesn't it? Especially when you ask for its probability. Axiomatically, the only things that can have probabilities are (measurable) sets of outcomes.
 
yeah, but not all sets are sets of outcomes--you can't ask, "What's the probability of the set of Real Numbers", for instance :p
 
@AdamGolding Actually, the probability of the set of real numbers makes perfect sense in many applications! When the sample space is the set of reals, the probability (axiomatically once more) is 1.
 
well, yeah, but only once you add a sample space, etc :p
 
6:08 PM
I guess I don't understand the sense of your "you can't ask."
 
similarly: I have a set of books on my table--that set does not have a probability.
 
Also, if there is no sample space in evidence, then it's impossible to talk about probability.
 
i know, but a choice of sample space is a choice of model that can be good or bad, and I was hoping for advice on choosing a good model by choosing a good sample space :p
 
@AdamGolding OK, let's discuss the books. If we nominate them as a sample space, the first step is to indicate which subsets are measurable. Typically, for finite sets, we consider all subsets to be measurable.
 
ok (and I am much more familiar with finite sample spaces, btw)
 
6:10 PM
The next step is to assign probabilities to the measurable sets consistent with the axioms of probability. I'll grant that often that is not given. This is where people start looking to maxent and Bayes methods.
The choice of sample space is then obviously basic. I guess you could consider it as part of model building, but in some ways it is a non-statistical part of model building.
 
maybe. I'm sure statisticians have thought about it more than most other professions I could consult :p
 
That is, usually by "model building" we mean we already have (or contemplate) data that are obtained in some way, so the possible sample spaces are heavily constrained.
 
I could conceivably also ask on an AI/machine learning forum of some kind
 
We then worry about constructing appropriate probabilities on the sample space.
@AdamGolding Well, you're in the right place! This is also a machine learning forum (see the FAQ).
 
6:13 PM
I doubt, though, that ML offers any of the insight you seek. I believe it focuses more on methods: damn the probabilities and full speed ahead! That places ML at a greater philosophical remove from your more basic questions.
 
well, incidentally I recently heard of some recent AI work in 'open world probability models' that don't assume a given size on the universe--I hope to dig around for some of that at some point
apparently for ages AI when doing explicit probability on predicate logic sentences, would have to pick a size for the universe to get started, and this has only recently become less necessary
 
@AdamGolding I'm afraid I have no idea what "open world probability models" are.
 
I think that they are precisely that--models in which you don't know the size of the universe of objects, but in which you can still specify a sample space of configurations of that universe
 
Have we discussed enough to help you with a reformulation of your question?
 
absolutely, you should get to work, I don't know if it's as flexible as mine ;-)
 
6:16 PM
(Note that if necessary you can reference this chat thread in the question.)
 
ok cool
 
I'm flexible, but I'm also behind on submitting a report to a client. It's beginning to distract me, so time to get it done...
Best of luck!
 
ditto!