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# Subjective Probability

## Question

A discrete, uniform random number generator, which produces any number between 1 and 10 inclusive, all outcomes of which are equally likely, gives Bob a random number, which we shall call x.

Bob knows what x is (he knows that it is 7).

Chris does not know what x is, but knows that the random number generator produces whole numbers between 1 and 10, inclusive, equally likely.

Alice poses the following question to both Bob and Chris.

"What is the probability that x is 5?"

Chris says: 1/10

Bob says: 0

Who is right?

What answer should the person who was wrong have given?

Does your definition of probability depend on how much knowledge you have (i.e. subjective), or is it independent of knowledge (i.e. objective)?

Does the concept of relative frequency make sense for one-time, non-repeatable experiments, for which an outcome is not known?

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Both Bob and Chris are correct.

Probability is contingent upon given information.

If you create a hypothetical scenario in which you stipulate that some relative frequency distribution exists and that the experiment can only be performed once, then it is valid. Beyond mere stipulation, though, how are we to determine a relative frequency distribution if we have never performed the experiment before?

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Bob is right. Chris should have said 0, as there is no stipulation that x must be an integer; thus the probability is infinitely close to 0.

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Bob is right. Chris should have said 0, as there is no stipulation that x must be an integer; thus the probability is infinitely close to 0.

"A discrete, uniform random number generator..."

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Neither was wrong. Both were right. Google "conditional probability."

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Both Bob and Chris are correct.

Probability is contingent upon given information.

If you create a hypothetical scenario in which you stipulate that some relative frequency distribution exists and that the experiment can only be performed once, then it is valid. Beyond mere stipulation, though, how are we to determine a relative frequency distribution if we have never performed the experiment before?

I was trying to see how many people would argue for a frequentist interpretation (probability is objective ratio of occurrences to infinite number of experiments) vs. a bayesian one (probability is a subjective degree of certainty).

I had a debate about this with my sister, and we couldn't seem to meet in the middle.

This all originally came out of something a statistics professor said to my class about two years ago.

We were talking about confidence intervals regarding the population mean of a set of measurables.

He asked the class, for a 90% confidence interval, what is the probability that the population mean is within our calculated interval?

Some people said 0.9, but he said wrong - it is either 1 or 0, since the population mean can either be inside the interval or it cannot be.

The book agreed with him, and I disagreed with both him and the book.

I thought to myself, what if knowledge wasn't a concern - what if something knew everything, i.e. could determine whether any statement was true or false.

Wouldn't the answer to any probability question then be one or zero? Can't all probabilistic events be restated as a question of whether some statement is true or false? I concluded probability as a tool only makes sense in the context of missing knowledge, and that this is in essence the same thing as subjectivity.

I have some philosophical qualms with the frequentist perspective and was hoping to get some representation from someone who better understands it.

It seems like the responders all believe that probability is subjective though.

I don't believe the frequentist interpretation can be used to answer any question about any specific event occurring and still be consistent with the claim of objectivity. By specific event, I mean like: what is the probability that Bob got a 5 from the machine at this point in space-time?

There is no such thing as repetition for Bob drawing from the machine at this point in time (I don't think it matters if the time under discussion is past, present, or future) , so is the frequentist definition of probability even valid in this case?

I agree it may be valid for characterizing the output of the random number generator over many requests, but how can it ever be justifiably used to answer any question about any specific event?

Yet I don't think a frequentist would hesitate to answer "what is the probability that my next coin flip is heads?", which is a specific event.

To be objective, the answer must be independent from any specific subjective mind. If you took two "objective" frequentists and asked them the same question, but showed one of them that the coin had double-heads and did not show the other, then with certainty they would have different answers. What if the outcome was not so obvious as double-heads, but in general, I would expect someone with extraordinary predictive power or relevant hidden knowledge to not always have the same probability response those with duller minds and lack of knowledge.

I am convinced that probability cannot be objective, but feel there must be a good reason that so many people believe it to be.

I am interested in the insight of others.

Thanks,

Edited by mmiguel

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I don't really see that there is much of a difference between the subjectivist and objectivist views. We can state the problem you've provided in frequentialist terms and stil conclude that Chris was correct to say 1/10.

Given an infinite number of trials in which the only information which is guaranteed to stay the same is that which Chris is given, the relative frequency of x=5 would be 1/10.

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I don't really see that there is much of a difference between the subjectivist and objectivist views. We can state the problem you've provided in frequentialist terms and stil conclude that Chris was correct to say 1/10.

Given an infinite number of trials in which the only information which is guaranteed to stay the same is that which Chris is given, the relative frequency of x=5 would be 1/10.

Alice's question is specific to only the "trial/run of the experiment" mentioned above.

Let's call that trial 1 (the one mentioned above where Bob gets a 7).

Imagine that Alice noticed that something else important happened at the same time as Trial 1, which makes Trial 1 of dire importance to Alice, but any other trials throughout time are not important at all to Alice.

In Bob and Chris's perspective we could have more trials: 2, 3, ...N

and it is true that roughly 1/10 of the time the random variable could be 5. But this is a characterization of trials 1 through N and Alice only cares about trial 1.

Is it possible we are actually considering 2 random experiments (which overlap in trial 1)?

There is a random experiment related to the number that the machine spits out - and relative frequency is easily defined here.

There is another random experiment for which repetition is not defined - the one that is important to Alice.

Can relative frequency even be defined here? There is no possible repetition here, only the lack of knowledge on Alice and Chris's part.

Going by relative frequency, and assuming complete objectivity, the probability by definition must be the number of occurrences of the event divided by number of experiments. The experiment only occurs once (by my definition above), and the event x=5 never occurs for any (of the one) times the experiment is run, therefore the probability must be 0 (as Bob stated).

Generalizing this concept, although we as people, treat things as repetitions, and use that in our definition of relative frequency, there is no such thing as repetition from a truly objective standpoint (only from a subjective one). (On a side note, there is no such thing as true sameness from a truly objective standpoint).

This implies to me, that any rigorous probability theory that can claim objectiveness must have only two possible values for any event (1 or 0, depending on whether the statement of the event is true or false).

There is no probability theory in practice that restricts the values to 1 or 0 (that can be called probabilistic at least), and hence all probability theories must be subjective. All Objectivists would say the probability of heads in a coin flip is 1/2, yet this does not make sense because if they had more knowledge (wind currents, specific distribution of weight on the coin, tendency of flippers thumb to flip in a certain way, ...etc) then they would surely adjust their probability evaluation to reflect what they believe is more likely.

My ultimate argument here is that all probability theories are subjective regardless of any claims to the contrary.

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There seems to be a bit of a problem with the question. You have stated the properties of the random number generator, for example, that it is unbiased. However, if we were to make an objective interpretation, using an objectivist interpretation, I don't think we can say that the generator is unbiased if the experiment (i.e. seeing what number it produces) has never been done before, thus the only sensible interpretation would be a subjectivist one.

Note that once the generator has output the 7, the experiment has not yet completed for Chris, so he cannot yet make an objective assessment.

To say "there is no such thing as repetition from a truly objective standpoint" is not true if we allow for abstraction, and we communicate for the most part in abstractions. You could, for example, say that there is objectively a repetition of things which are blue. The only time you cannot have repetition is if you define something enumeratively (i.e. an exhaustive extensional definition) to have only one instance. Confusion might arise if we are inconsistent with the scope in which we ascribe objectivity. Ultimately, whenever somebody states that something is objective, this is necessarily a subjective assessment with many contingencies. However, this does not mean that we cannot talk about something being objective, because we can narrow the scope by assuming various axioms, such as that we have commonality in our language.

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There seems to be a bit of a problem with the question. You have stated the properties of the random number generator, for example, that it is unbiased. However, if we were to make an objective interpretation, using an objectivist interpretation, I don't think we can say that the generator is unbiased if the experiment (i.e. seeing what number it produces) has never been done before, thus the only sensible interpretation would be a subjectivist one.

Note that once the generator has output the 7, the experiment has not yet completed for Chris, so he cannot yet make an objective assessment.

To say "there is no such thing as repetition from a truly objective standpoint" is not true if we allow for abstraction, and we communicate for the most part in abstractions. You could, for example, say that there is objectively a repetition of things which are blue. The only time you cannot have repetition is if you define something enumeratively (i.e. an exhaustive extensional definition) to have only one instance. Confusion might arise if we are inconsistent with the scope in which we ascribe objectivity. Ultimately, whenever somebody states that something is objective, this is necessarily a subjective assessment with many contingencies. However, this does not mean that we cannot talk about something being objective, because we can narrow the scope by assuming various axioms, such as that we have commonality in our language.

Thank you for your input in this discussion.

Quote: "There seems to be a bit of a problem with the question. You have stated the properties of the random number generator, for example, that it is unbiased. However, if we were to make an objective interpretation, using an objectivist interpretation, I don't think we can say that the generator is unbiased if the experiment (i.e. seeing what number it produces) has never been done before, thus the only sensible interpretation would be a subjectivist one."

There are two "experiments" under discussion here though, one of which has been done before (to determine distribution), and one of which can only ever be done once. The first is the one for which I laid out the uniform distribution, which I agree is repeatable (though only due to abstraction ultimately coming from a subjective perspective), and for which there should be no confusion.

The second experiment is a unique measurement in the real world at a defined point in time.

There is a distinction between the abstract model (first experiment) and a real, physical measurement that coincides with some trial of this abstract model.

I believe the purpose of probability is prediction of such real, physical measurements.

My problem is that the claim of objectivity in probability fails when applied to a statement of any single real, physical measurement, unless the absolute truth of statement is known.

Relative frequency works fine if you are talking about a large number of physical measurements, but fails to say anything useful if applied only to any specific one (you can only assign it a value of 1 or 0, but only if you know which one it is). The Bayesian interpretation however easily applies to any specific measurement.

I agree with your statement that the only sensible interpretation here is subjective, but I think that in general, the only sensible use of probability in general is inherently subjective.

Quote: "Note that once the generator has output the 7, the experiment has not yet completed for Chris, so he cannot yet make an objective assessment."

What part is not complete? Which of the two experiments?

Quote: "To say "there is no such thing as repetition from a truly objective standpoint" is not true if we allow for abstraction, and we communicate for the most part in abstractions. You could, for example, say that there is objectively a repetition of things which are blue. "

I think that abstraction itself is subjective. Let us ponder the concept of abstraction. Abstraction is the opposite of specification. It is essentially the removal of details. As an example, there could be a classroom full of children, each child is different in at least some of their characteristics (e.g. height, weight, name, gender, spatial position, personality, ...etc). Instead of thinking about each of them as completely dissimilar entities, which would be impractical, we can, from observing their common characteristics, abstract out the concept of student by removing from consideration details which we designate as not important. We make spatial position not important in the definition of the concept of a student i.e. a student may have any spatial position and still be classified as a student. We leave important characteristics, i.e. such as being enrolled in the school. By removing information (i.e. unimportant characteristics) we have established an abstract group of specific entities.

Another example is numbers. Numbers are abstract concepts as well. You can communicate the concept of the number three by showing me three apples, then telling me to remove all characteristics associated with apples.

What is the point of this rambling? ---> Abstraction is the removal of unimportant information, however the judgment of what characteristics are unimportant is subjective. Hence the concept of abstraction can only exist in a subjective perspective. All hypotheticals that we might imagine or define, come from real, physical observations which we have watered down (i.e. removed details i.e. abstracted) to form abstract objects for the sake of practicality (and this is very much a good thing).

Your example: Repetition of things which are blue ---> remove all characterstics from the concept except those relating to whether the emitted/reflected light is within a certain frequency range (the blue range).

So I agree, repetition is possible with abstraction, but I argue that abstraction itself is subjective, so I disagree about repetition existing from a truly objective standpoint.

I know this is all largely philosophical, however I think it is important for understanding probability, since how one believes in this determines whether or not they would answer questions like the probability of the population mean falling within a confidence interval. Different beliefs about these philosophical details yield completely different answers to questions like that.

It seems like you agree that probability is inherently subjective, although I'm not certain that you do.

Edited by mmiguel

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