# Terminology for handling probabilities with partial knowledge

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?

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So, 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... –  Seamus Sep 8 '10 at 12:19
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. –  whuber Sep 8 '10 at 22:19
When 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. –  Larry Wang Sep 10 '10 at 7:11
@Kaestur: Updated question. @whuber: I think that is the answer. Do you want to post it as an answer? –  Casebash Sep 10 '10 at 8:55
@Casebash: I apologize for circumventing the usual procedure; you're right, it does constitute an answer, so I'll post it as one. –  whuber Sep 10 '10 at 14:19

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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I 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. –  T.. Sep 11 '10 at 5:53
I 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? –  whuber Sep 11 '10 at 15:12
There 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. –  T.. Sep 11 '10 at 18:47
@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" ;-). –  whuber Sep 11 '10 at 21:48

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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That 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... –  Seamus Sep 8 '10 at 15:56
That isn't quite it –  Casebash Sep 8 '10 at 21:41