Consider a situation where a person has partial knowledge, but we have a more complete picture. For example, suppose that we want to know the probability that a fish is red. Suppose that the person with partial information knows 1/3 of all fish are red, but we know that the particular species is actually red 2/3 of the time. Do these two separate probabilities have special names? If they don't have any standard names, what would you call them?
Terminology for handling probabilities with partial knowledge
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terminology
probability
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0So, one guy knows that at least 1/3 are red, the other guy knows that exactly 2/3 are red? These are just different probabilities. The first one isn't fully specified. All the guy really knows it that the chance of a fish being red is $\ge 1/3 $, which doesn't uniquely determine a probability measure... – 2010-09-08
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1They are called "personal probabilities" in the literature on Bayesian statistics, which is the field where the possibility of different probability assessments among different observers is considered. – 2010-09-08
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1When you say "the person might know that 1/3 of fish are red," is this for *all* fish, or for the same species as "we" know about? Maybe your question could be more clear if you name the people and the fish. – 2010-09-10
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1@Kaestur: Updated question. @whuber: I think that is the answer. Do you want to post it as an answer? – 2010-09-10
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0@Casebash: I apologize for circumventing the usual procedure; you're right, it does constitute an answer, so I'll post it as one. – 2010-09-10
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0@whuber: There is no need to apologise - I wanted it as an answer so that I could accept it – 2010-09-11
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0Is the request for terminology to express only the fact that persons A (p=1/3) and B (p=2/3) hold *different* beliefs, or the distinction between A's partially-informed belief and B's having the true probability and knowing it to be the true one (e.g., because he is a superbeing, or runs the fish farm and genetically engineers the fish colors, or the "fish" are objects in a computer game B has written). As a third possibility, are you looking for language that expresses degrees of belief/confidence/information/evidence, which A and B happen to possess in different amounts? – 2010-09-11
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0@T: I was asking for terminology to make it clear that a certain probability was based on the knowledge that a particular individual posessed, rather than the complete knowledge available from the question. I am using simply "probability" to refer to the probability from our perspective – 2010-09-14
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0@Casebash: it seems there are several different concepts expressed in the question: (1) states of knowledge, levels of information, degrees of evidence, etc together with a (partial) ordering where some states are "more complete" than others, (2) assignment of probability models based on states of knowledge (and possibly also based on other things that are assigner-dependent), and (3) a "true" probability model that the assignments might converge to as the state of knowledge becomes increasingly complete. Concepts and setting of statistics (esp. inference) are highly relevant here. – 2010-09-14
2 Answers
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They are called "personal probabilities" in the literature on Bayesian statistics, which is the field where the possibility of different probability assessments among different observers is considered. You could check out the Wikipedia article on Bayesian probability, for instance.
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0I don't think that answers the question. If persons A and B assign different values to a probability, calling both assignments "personal" gives both values equal status. In the question, B knows that 2/3 is the true parameter ("actually red 2/3 of the time") and one wants a term that distinguishes the true model parameters from assumed, estimated, or Bayesian-priored parameters that are not necessarily the true ones. The question as written seeks a term expressing the difference between the 1/3 might-be, partially informed probability and the certainly-is, full information, value of 2/3. – 2010-09-11
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0I interpreted the question slightly differently, T. It appears to ask for the terminology, if it exists, in which a probability "we" have can be distinguished from a probability "you" have. According to the Bayesian/personalist/subjective philosophy, there is no such thing as a "certainly-is, full information," probability. Regardless of our different interpretations, I don't think you can validly dispute that my response actually answers the question; the point of difference appears to be whether the response is full or correct. If you believe not, then what alternative are you proposing? – 2010-09-11
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1There are certainly interpretations (arguably, the standard ones) where the concept of absolutely certain complete information on model parameters makes sense, i.e., classical statistics and statistical inference. Whether or not Bayesianism uses that concept has no bearing on whether the OP is allowed to use it or ask for terms expressing it. If you want only to to express the idea that persons A and B have two different values of $p$, or two different priors, there is already a standard term: "different". Anyway, the poster should clarify his question. – 2010-09-11
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0@T: I won't disagree. In the back of my mind, though, is the thought that a search on "personal probability" would be much more fruitful than a search on "different" or even "different probability" ;-). – 2010-09-11
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I would call the second of these a conditional probability when trying to emphasise the difference:
The probability of a chosen fish being red is 1/3.
The conditional probability of a fish being red, given that it is species X, is 2/3.
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0That doesn't seem to capture the distinction the OP appears to be trying to draw. I got the impression that both guys knew it was a fish of species X, but differed in their state of knowledge vis-a-vis colours of that species... – 2010-09-08
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0That isn't quite it – 2010-09-08