Game Show Problem

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Re: Game Show Problem

Postby Gofer » Mon Mar 20, 2017 3:16 pm

Not nearly as interesting as hearing Jeff explain the phrase "The possible outcomes for one coin toss can be described by the sample space {Heads,Tails}.".

Note that it is the coin toss that is described by the sample space, and NOT, as Jeff wants it, that the sample space is described by {Heads,Tails}.
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Re: Game Show Problem

Postby JeffJo » Mon Mar 20, 2017 4:05 pm

And yet still not recognizing the words "described by" or "can be."

Note that it is the outcomes that are described by what, if we follow Gofer's preferred sources, the state space {Heads,Tails}. But since it is almost never necessary to be as pedantic as Gofer wants us to be, many people equate the two.

AGAIN: The sample space is the set of all possible outcomes. The model of the sample space that we use, properly called the state space if you want to be that pedantic, is the set of all possible combinations for the set of random variables we choose to use to describe it.

Isn't it interesting that Gofer never tries to rebut this? That he only asks me to explain ambiguous references that he cherry-picks to maybe, possibly, if you ignore enough definitions, justifies his position? Which he never clearly states?
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Re: Game Show Problem

Postby Gofer » Tue Mar 21, 2017 7:46 am

JeffJo wrote:
> And yet still not recognizing the words "described by" or "can be."

Right, our coin toss in the real world is "described" by our model. And there "can be" other sample spaces in our model describing a coin toss in the real world. Surely this cannot be that hard to comprehend.

JeffJo wrote:
> if we follow Gofer's preferred sources, the state space {Heads,Tails}.

That's funny because the sentence I just quoted doesn't say "state space" but "SAMPLE SPACE".

JeffJo wrote:
> AGAIN: The sample space is the set of all possible outcomes.

.. in our model.

JeffJo wrote:
> Isn't it interesting that Gofer never tries to rebut this?

There's nothing to rebut because no source supports your position.

JeffJo wrote:
> That he only asks me to explain ambiguous references that he cherry-picks to maybe,
> possibly, if you ignore enough definitions, justifies his position? Which he never
> clearly states?

I have stated it "clearly" many times. Here it is again: If we wish to model an experiment in the real world using measure-theoretic mathematics, there are two objects that need to be properly defined: the sample space and the probability space. Random variables are not really needed though.
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Re: Game Show Problem

Postby JeffJo » Tue Mar 21, 2017 8:39 am

> Right, our coin toss in the real world is "described" by our model. And there "can be"
> other sample spaces in our model describing a coin toss in the real world. Surely
> this cannot be that hard to comprehend.

Not at all. I've been saying that for months. It is you who objected when I said that these "others" can comprise a "plethora."

> That's funny because the sentence I just quoted doesn't say "state space" but "SAMPLE SPACE".

But your source does. Random variables map the sample space to the state space. The difference is that a true sample space distinguishes every possible outcome from every other. And like I said before, and you ignored, the description that we conveniently call the sample space is really a state space, but it is too pedantic to use two names so we call it the sample space.

>JeffJo wrote:
>> AGAIN: The sample space is the set of all possible outcomes.
>... in our model.

Our model doesn't include all possibilities, just those we are interested in.

> If we wish to model an experiment in the real world using measure-theoretic mathematics,

One thing you never make clear is why we should wish to use measure-theoretic mathematics.

>... there are two objects that need to be properly defined: the sample space and the
> probability space.

Another thing you never make clear is what you think "well-defined" means here. The sample space, as you use the term, or state-space, as you ignore in your source, is "well-defined" by identifying mappings from the set of truly-all possibilities, to a set that includes only the aspects we are interested in. These are called random variables.

+++++

Just to try to make Gofer understand, if he will let himself:

All of the possible outcomes of an experiment form distinct elements of a sample space. Theoretically, each instance of the experiment produces an outcome that is distinct from every other one. (Wikipedia: “Each possible outcome of a particular experiment is unique, and different outcomes are mutually exclusive.”)

We can’t deal with such a set, so we instead describe each outcome by a collection of what, in simple probability theory, are called random variables. These can be, but don’t have to be, numbers. So, for example, a coin flip almost always is described by the values “Heads” and “Tails.” But if you need to use measure-theoretic tools, they do need to be numbers. There is no such need in the Game Show Problem.

Usually, there is no need to make the extreme terminology distinctions that Gofer seems to insist on. So different explanations may simplify some of these details. Some do it differently. I’ll just quote one article from Wikipedia, that supports my terminology:

> In probability theory and statistics, a probability distribution is a mathematical function
> that, stated in simple terms, can be thought of as providing the probability of occurrence
> of different possible outcomes in an experiment. For instance, if the random variable X is
> used to denote the outcome of a coin toss ('the experiment'), then the probability
> distribution of X would take the value 0.5 for X=Heads, and 0.5 for X=Tails.

See? Heads and Tails are values of the random variable X.

> A probability distribution is defined in terms of an underlying sample space, which is
> the set of all possible outcomes of the random phenomenon being observed. The
> sample space may be the set of real numbers or a higher-dimensional vector space,
> or it may be a list of non-numerical values; for example, the sample space of a coin
> flip would be {Heads ,Tails}.

See? The possible values of the random variable is what defines the sample space.

> A univariate distribution gives the probabilities of a single random variable taking on
> various alternative values; a multivariate distribution (a joint probability distribution)
> gives the probabilities of a random vector—a list of two or more random variables—
> taking on various combinations of values.

See? The values of the random variables are the arguments of the distribution function.
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Re: Game Show Problem

Postby Gofer » Tue Mar 21, 2017 9:58 am

JeffJo wrote:
> But your source does. Random variables map the sample space to the state space. The
> difference is that a true sample space distinguishes every possible outcome from
> every other. And like I said before, and you ignored, the description that we conveniently
> call the sample space is really a state space, but it is too pedantic to use two
> names so we call it the sample space.

That's even more funny because nowhere in the W. article on "random variables" occurs the phrase "state space".

JeffJo wrote:
> Our model doesn't include all possibilities, just those we are interested in.

which is why there "could be" other sample space in our model describing the coin toss.

JeffJo wrote:
> One thing you never make clear is why we should wish to use measure-theoretic mathematics.

Irrelevant! But you would do well excogitating the reasons for the development of measure-theoretic probabiltiy theory.

JeffJo wrote:
> Another thing you never make clear is what you think "well-defined" means here.

The same way the set {1,2,3} is well-defined!

JeffJo wrote:
> The sample space, as you use the term, or state-space, as you ignore in your source,
> is "well-defined" by identifying mappings from the set of truly-all possibilities,
> to a set that includes only the aspects we are interested in.

No, a random variable is a mapping, requiring both source and target to be well-defined.

> But if you need to use measure-theoretic tools, they do need to be numbers.

False! Look up the definition of "probability space" on W.

JeffJo wrote:
> For instance, if the random variable X is
> > used to denote the outcome of a coin toss ('the experiment'), then the probability
> > distribution of X would take the value 0.5 for X=Heads, and 0.5 for X=Tails.
>
> See? Heads and Tails are values of the random variable X.

See the words "used to denote"? It simply means that a random variable COULD be used to describe the sample space.


JeffJo wrote:
> See? The values of the random variables are the arguments of the distribution function.

Yeah, so? I've never denied that one could describe a probability space "backwards" by defining random variables and their distributions on it.
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Re: Game Show Problem

Postby JeffJo » Fri Mar 24, 2017 4:00 pm

> That's even more funny because nowhere in the W. article on "random variables" occurs
> the phrase "state space".

Wikipedia, under the "Random variables" section of the article on "Probability space," says:

] A random variable X is a measurable function X: Ω → S from the sample space Ω
] to another measurable space S called the state space.

Many of these terms are interchangeable, or have narrow definitions when narrow fields-of-view are required (which is not the case in the GSP). The point is that you don't get to cherry-pick the ones that don't mention my usage, BUT DON'T CONTRADICT IT EITHER, to claim my usage is wrong. I have given ample evidence that my terminology is used, and you have not found anything that says it isn't. Just examples where it isn't mentioned.

>> One thing you never make clear is why we should wish to use measure-theoretic mathematics.
> Irrelevant!

If you insist we must use it, you do indeed need to justify why. But you don't even know what narrow field of view it uses, do you?

>> Another thing you never make clear is what you think "well-defined" means here.
>
> The same way the set {1,2,3} is well-defined!

And since you don't say what those number represents, you have not "defined" anything, let alone "well-defined" a sample space.

Hint: do provide such a definition, you need to define those values as the range of a random variable.

> A random variable is a mapping, requiring both source and target to be well-defined.

Which you don't do. And, according to Wikipedia, it is a mapping from the sample space to the state space.

>> But if you need to use measure-theoretic tools, they do need to be numbers.
>
> False! Look up the definition of "probability space" on W.

True! look up the definition of "measure" on W. And your favorite quote, from the "random variable" article, under the heading "Measure-theoretic definition":

] The most formal, axiomatic definition of a random variable involves measure theory.
] Continuous random variables are defined in terms of SETS OF NUMBERS...

> See the words "used to denote"? It simply means that a random variable COULD be used
> to describe the sample space.

And it also means that when the sample space is denoted {Heads, Tails}, that those are values of a random variable.
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Re: Game Show Problem

Postby Gofer » Fri Mar 24, 2017 5:51 pm

> Many of these terms are interchangeable, or have narrow definitions when narrow fields-of-view
> are required (which is not the case in the GSP).

I disagree! If we wish to analyze the problem measure-theoretically, then all those things need proper definitions.

> The point is that you don't get
> to cherry-pick the ones that don't mention my usage, BUT DON'T CONTRADICT IT EITHER,

I pick the coin-toss example supporting of my position, not yours. You need to provide proof where your theory is substantiated.

> to claim my usage is wrong. I have given ample evidence that my terminology is used,

only in the area of statistics, according to some definition, and not in strict math (measure-theoretic).

> and you have not found anything that says it isn't. Just examples where it isn't
> mentioned.
>
> >> One thing you never make clear is why we should wish to use measure-theoretic
> mathematics.
> > Irrelevant!
>
> If you insist we must use it, you do indeed need to justify why. But you don't even
> know what narrow field of view it uses, do you?

I don't insist we use it; but if we do, things need to be properly defined.

> >> Another thing you never make clear is what you think "well-defined" means here.
>
> >
> > The same way the set {1,2,3} is well-defined!
>
> And since you don't say what those number represents, you have not "defined" anything,
> let alone "well-defined" a sample space.

I don't need to. They just need to be distinct, and could be sets.

> Hint: do provide such a definition, you need to define those values as the range
> of a random variable.
>
> > A random variable is a mapping, requiring both source and target to be well-defined.
>
> Which you don't do.

Sure I do. I have demonstrated it many times.

> >> But if you need to use measure-theoretic tools, they do need to be numbers.
> >
> > False! Look up the definition of "probability space" on W.
>
> True! look up the definition of "measure" on W. And your favorite quote, from the
> "random variable" article, under the heading "Measure-theoretic definition":
>
> ] The most formal, axiomatic definition of a random variable involves measure theory.
> ] Continuous random variables are defined in terms of SETS OF NUMBERS...

That's because it says "CONTINUOUS random variables", my emphasis!
Besides, that section onward should really be part of the article on probability spaces.

In my opinion, W. gives a rather poor description of a measure-theoretic random variable, so here I provide a better one:

A function f:A→B is said to be a random variable if for every b∈BB there exists an a∈AA such that f(a)=b where AA and BB are sigma-algebras defined on A and B. Note that, since a is a set, f(a) needs to be defined, which however is trivial.
So a random variables is really nothing more than a measurable function.

> > See the words "used to denote"? It simply means that a random variable COULD be
> used
> > to describe the sample space.
>
> And it also means that when the sample space is denoted {Heads, Tails}, that those
> are values of a random variable.

No, they could be the values of a random variable.
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Re: Game Show Problem

Postby Gofer » Fri Apr 07, 2017 6:29 am

So to summarize, for practical purposes and clarity, Jeff probably has the best solution for this problem, notwithstanding a slightly ambiguous term, namely Pr(H=3), where H, describing the host action, really needs an underlying probability space describing the experiment, to make sense. So I'm going ahead and tweaking his solution a bit, adding some clarity:

The solution to the Game Show Problem is:

Pr(Hc) vs Pr(Hf), in favor of switching wins the car, where Hc (host has a choice) and Hf (host is forced) are events (H=h & C=p) and (H=h & C=c) respectively, p and h are doors just chosen by the player and the host respectively, c is the third door, H and C are random variables describing the host and the car action respectively, and Pr is the probability function operating on a probability space describing an experiment starting AFTER the player made his choice and ending at the revelation of the car C.

Under reasonable assumption, we'd expect to see Hc occur half the times compared to Hf, because the host has a choice between two doors in the former, but not in the latter, making the relative odds 1:2 in favor of switching.
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Re: Game Show Problem

Postby JeffJo » Fri Apr 07, 2017 8:29 am

"Gofer"> I disagree! If we wish to analyze the problem measure-theoretically, then all
> those things need proper definitions[1].

"Doctor, Doctor, it hurts if I do this!"

"Don't do that!"

Why should we wish to "analyze the problem measure-theoretically?" You don't even understand what it means, why you should want to do so, or how to do so.

(Hint #1: after many months of insisting on it without justification, you still haven't.)

(Hint #2: From Wikipedia: "The raison d'être of the measure-theoretic treatment of probability is that it unifies the discrete and the continuous cases, and makes the difference a question of which measure is used.")

(Hint #3: It requires you to unify all random variables as continuous variables.)

> You need to provide proof where your theory is substantiated.

Theory? What Theory? I have used accepted terminology, meant for non-mathematicians, and that is easily understood. But you refuse to understand it because you want to argue about the semantics of that terminology, AND YOU ARE WRONG ABOUT IT.

> I don't need to ("well-define a sample space").

See [1].
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Re: Game Show Problem

Postby Gofer » Fri Apr 07, 2017 1:34 pm

No, the reason for a measure-theoretic framework is to establish a solid mathematical foundation of probability theory, WHICH INCLUDES HAVING TO DEFINE THE SAMPLE SPACE - get that?
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Re: Game Show Problem

Postby JeffJo » Sat Apr 08, 2017 12:47 pm

Gofer> No, the reason for a measure-theoretic framework is to establish a solid mathematical foundation of probability theory, WHICH INCLUDES HAVING TO DEFINE THE SAMPLE SPACE - get that?

Isn't interesting how Gofer selects which sources he thinks provide absolute truth, and which he chooses to dismiss, based on whether they agree with him? Even to the point of selecting which passages in a source to subject to his filters?

Andrey Kolmogorov's set of axioms establish a solid mathematical foundation for probability theory. Nothing more is needed, but it does handle discrete and continuous cases separately. The same source that Gofer uses to defend his insistence on a well-defined sample space in a probability space - well, ignoring that it says it is part of the model of the space, and not the space itself - also says that the reason to use a measure-theoretic framework is to unify the treatment of discrete and continuous cases. And Gofer disagrees with a direct quote from that source.

The probability space of an experiment is an abstract that cannot be defined. I'm sorry that no source explains this to Gofer, to Gofer's satisfaction, but (A) none contradict it and (B) all are consistent with it. It can;t be defined. The "well-defined" part is the model of the probability space, comprising the same basic elements in a definable manner, so that we can use it in mathematics. And the sample space is defined in terms of random variables.
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Re: Game Show Problem

Postby Gofer » Sat Apr 08, 2017 1:24 pm

>Isn't interesting how Gofer selects which sources he thinks provide absolute truth, and which >he chooses to dismiss, based on whether they agree with him?

Isn't it interesting how Jeff never has provided evidence supporting his position, whereas I have provided a direct quote supporting mine?

>Andrey Kolmogorov's set of axioms establish a solid mathematical foundation for probability >theory. Nothing more is needed, but it does handle discrete and continuous cases separately.

Hence "establish a solid mathematical foundation of probability theory"

>The same source that Gofer uses to defend his insistence on a well-defined sample space in a >probability space - well, ignoring that it says it is part of the model of the space, and not the >space itself

Isn't it interesting how Jeff lies about what W. says about "sample space". Here's what it really says: "A well-defined sample space is one of three basic elements in a probabilistic model (a probability space)"

Thus, it isn't a "model of the space" but a "probabilistic model".

Contrary to Jeff's delusional statements, the truth is so simple it almost shames me to state it: In order to have a WELL-DEFINED probabilistic model within the framework of measure-theoretic mathematics, we are REQUIRED to define its three components, those being a sample space, an event space, and a probability measure.

That wasn't so hard, now was it, Jeff?
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Re: Game Show Problem

Postby JeffJo » Mon Apr 10, 2017 7:22 am

Gofer> Isn't it interesting how Jeff never has provided evidence supporting his position, whereas I have provided a direct quote supporting mine?

Isn't it interesting how Gofer ignores the evidence I provide, as well as the explanations of how he misinterprets the evidence he thinks he has provided?

Example:

>>>The same source that Gofer uses to defend his insistence on a well-defined sample space in a >>>probability space - well, ignoring that it says it is part of the model of the space, and not the >>>space itself
>>
>> Isn't it interesting how Jeff lies about what W. says about "sample space". Here's what it
>> really says: "A well-defined sample space is one of three basic elements in a probabilistic
>> model (a probability space)"
>
> Thus, it isn't a "model of the space" but a "probabilistic model".

What Gofer thinks his point is - as always - is unclear. The probability space underlying any probabilistic model is an abstract concept that can be modeled in many different ways based on what random variables you choose. The model is the only thing you can "well-define," and the only thing this source says needs to be "well-defined." This contradicts everything that Gofer is trying to say here, yet he won't admit it.

> Contrary to Jeff's delusional statements, the truth is so simple it almost shames me to state
> it: In order to have a WELL-DEFINED probabilistic model within the framework of
> measure-theoretic mathematics, we are REQUIRED to define its three components,
> those being a sample space, an event space, and a probability measure.

Contrary to what Gofer wants to believe, all his sources say is that you need to define the three components of the model.

And also contrary to what Gofer wants to believe, you don't need to use measure-theoretic mathematics to approach the Game Show Problem.

That wasn't so hard, now was it, Gofer?
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Re: Game Show Problem

Postby Gofer » Fri Apr 14, 2017 2:25 pm

> Isn't it interesting how Gofer ignores the evidence I provide, as well as the explanations of how he misinterprets the evidence he thinks he has provided?

You have never supported your theories on measure-theoretic mathematics with ANY evidence!

>The model is the only thing you can "well-define," and the only thing this source says needs to be "well-defined." This contradicts everything that Gofer is trying to say here, yet he won't admit it.

On the contrary! Gofer has always said that it is the MODEL, hence its three components including "the sample space", that needs to be well-defined.

>Contrary to what Gofer wants to believe, all his sources say is that you need to define the three components of the model.

Duh!

> And also contrary to what Gofer wants to believe, you don't need to use measure-theoretic mathematics to approach the Game Show Problem.

Gofer never said you do, but actually recommended not to for practical purposes. Just a reminder for the audience: it was actually Jeff who first introduced the measure-theoretic concept in this thread, presumably done to show off his math skills.

> That wasn't so hard, now was it, Gofer?

Correct! It wasn't so hard!
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Re: Game Show Problem

Postby JeffJo » Sat Apr 15, 2017 8:43 am

Gofer>You have never supported your theories on measure-theoretic mathematics with ANY evidence!

I have always rejected your claim that there is any reason to use measure-theoretic mathematics! I have even shown you the quote, from your own source, that explicitly states what it is useful for! Which doesn't apply to the GSP! And you called that wrong, with no evidence!

You have not supported your claim that we need to use it! I'll also point out that you have not supported your claim, that what I say is not right in measure-theoretic mathematics! All you have found is one parenthetical comment; that the model of a probability space can, itself, be called a probability space! That doesn't mean other things can't! It isn't the definition! Which is provided earlier in the same article!

What I have provided evidence for, and you have ignored, is that my terminology is used for problems like this one!

> Just a reminder for the audience: it was actually Jeff who first introduced the
> measure-theoretic concept in this thread, presumably done to show off his math skills.

Just a reminder for the audience. Eight months ago, I said "A random variable is any measurable quantity of your system." I did not then, and never have, claimed that this had to fit the definition of formal measure theory. I did not then, and never have, meant this to " show off math skills." I don't know that I had ever, before responding to Gofer then, even used the term "formal measure theory." It's not my field. But since Gofer took the word "measure" to mean this formal definition, it seems he was trying to show off that way.

As was clearly implied then, I only meant you had to be able to assign, in each instance of the experiment, a label to the concept represented by the random variable. My word "measure" was intended only to say the door chosen by the host was one random variable, regardless of what door the contestant choose. The point was that Gofer was claiming, then, that there were different random variables depending on which door the contestant choose[1], that seemed to exist only for that case. That is wrong, another error Gofer has never admitted to. Although with all his research, he seems to have corrected it.

It was Gofer who took this to mean the measure-theoretic concept of "measure," and introduced that concept. And has not let go of that misunderstanding, or claimed that I started it, ever since. And now lies by saying I started it. That concept is that you have to map each "label" value in a non-measure-theoretic sample space, to a real-valued number, so that you can derive formal mathematical properties of the formally-defined mathematical fields in order to demonstrate formally-defined closure.[2]

That wasn't so hard, now was it, Gofer?

+++++

[1] "You really have 3 random variables H, one for each door chosen by the contestant."

[2] None of which serves any purpose in understanding the GSP.
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